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Current, resistance, and electrical conduction

Follow signed charge through an oriented surface, distinguish a material law from a device resistance, and connect steady conduction to energy transfer. Derive an original tapered-wire model and keep carrier, charge-relaxation and signal times distinct.

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After reading, you can

  • Define current and current density as signed charge flux, including several carrier species and a reversed surface orientation.
  • Use local and integral charge continuity without assuming that every steady current region has zero charge or zero field.
  • Distinguish local Ohmic conductivity, two-terminal resistance and the assumptions behind a one-dimensional resistance integral.
  • Calculate the potential, average current density and power of a slowly tapered wire while retaining the limits of the reduced model.
  • Derive the Drude drift and complex conductivity with their charge signs, relaxation time and phasor convention.
  • Explain where Joule energy comes from and why bulk charge relaxation, carrier momentum relaxation and signal propagation require different physical assumptions.
01

What crosses the surface?

Electric current counts signed charge crossing an oriented surface per unit time. A positive charge moving one way and a negative charge moving the opposite way can make currents in the same conventional direction. A picture of particles alone is therefore incomplete until their signs, densities and the surface orientation are supplied.

For a chosen cross-section, I = dQ_crossed/dt, where Q_crossed is algebraic transferred charge rather than the total charge sitting in the wire. Current has units amperes, with 1 A = 1 C/s. The sign is relative to the chosen normal, not a separate physical species of positive or negative current.

The conducting material supplies carriers and a response to applied fields and other gradients. The current is a collective transport rate; it need not mean that each carrier travels quickly from one end of a device to the other. Establishing an electromagnetic signal is also a different question from following an individual carrier.

The original examples in this lesson are separate models: prescribed carrier drifts, a reduced steady tapered wire, a phenomenological carrier-relaxation response, and a homogeneous charge-relaxation limit. Their normalizations make the assumptions auditable without selecting empirical material constants.

02

Add signed carrier contributions

For species s with number density n_s, signed charge q_s and average drift velocity v_s, its convective current density is J_s = n_s q_s v_s. Add the vectors to obtain J = Σ_s n_s q_s v_s. The units are (1/m³)(C)(m/s) = A/m². Random equilibrium motion with zero average velocity does not by itself produce a net drift current.

Choose positive scales n₀, q₀ and v₀, with J₀ = n₀q₀v₀. In our bookkeeping example, positive carriers have density n₀ and velocity +2v₀x̂, giving +2J₀x̂. Negative carriers have density 3n₀, charge −q₀ and velocity −v₀x̂, giving +3J₀x̂. Their conventional currents add to +5J₀x̂.

A stationary positive background with density 2n₀ supplies +2n₀q₀ of charge density and no current. Together with +n₀q₀ from the positive mobile species and −3n₀q₀ from the negative one, the net charge density is zero. A neutral region can therefore carry a nonzero current.

The chosen drift magnitudes are prescribed to illustrate charge-flux bookkeeping. They are not a claim that one common bare Drude mobility creates both velocities. Actual species mobilities, diffusion and applied forces must determine those drifts in a material model. Changing a carrier sign changes q_s v_s; it does not change the definition of conventional current.

Separate charge and current ledgers for the prescribed carrier model
Speciesn/n₀q/q₀v_x/v₀ρ/(n₀q₀)J_x/J₀
Mobile positive11212
Mobile negative3−1−1−33
Stationary positive background21020
Total———05
03

Integrate the normal component

The current through an oriented surface S is I = ∫_S J·n̂ dA. A uniform J and a flat constant normal give I = J·n̂ A. The angle is measured with the normal, not with the surface itself. A tangential current density contributes zero crossing current even if its magnitude is nonzero.

For J = 5J₀x̂ and area A = 6L₀², a +x normal gives I = 30J₀L₀². Reversing that normal gives −30J₀L₀². For the same mathematical area with n̂ = (3/5,4/5,0), the normal component is 3J₀ and the current is 18J₀L₀². The vector is a unit normal because (3/5)² + (4/5)² = 1.

These are separately chosen equal-area patches in the prescribed uniform current. They are not a claim that a tilted cut across the same finite wire keeps the same physical area. If a tilted surface spans exactly the same uniform straight current tube, its larger area and smaller normal component compensate, provided it intercepts the whole tube and no side current escapes.

For a nonuniform current density or a curved surface, retain the integral and the actual domain. A schematic perspective drawing does not set the physical area or angle; the specified vectors and geometry do. Surface orientation changes the reported sign without changing the carrier motions.

Charge sign, drift and oriented current are different

Add the signed current contributions of positive and negative moving carriers, retaining the stationary charge background. Compare three separate equal-area patches with their specified normals; the tilted patch is not a cut of unchanged area through the same wire.

Opposite drift signs, the same conventional-current direction

  • Signed drift vₓ/v₀ (velocity ratio)
  • Signed conventional current density Jₓ/J₀ (current-density ratio)
Opposite drift signs, the same conventional-current directionAll three carrier populations, including the stationary positive background. Positive carriers drift at+2v₀ and supply+2J₀; negative carriers drift at−v₀ and supply+3J₀. Background velocity/current are zero.-10123Positive mobile22Negative mobile-13Stationary background00Net ρ/(n₀q₀) = 0; total Jₓ/J₀ = 5Separate prescribed carrier populationsSeparate normalized vₓ/v₀ and Jₓ/J₀ (not position)

Three distinct equal-area surfaces, three signed currents

  • Unit normal n: common 60px direction scale; exact components shown
  • Common J = 5J₀ x̂: direction-only arrow, schematic length
Three distinct equal-area surfaces, three signed currentsEdge-on schematic of three separate A=6L₀² patches. Normal arrows use exact retained (1,0,0),(−1,0,0),(3/5,4/5,0). J is common+5J₀ x̂. I is signed oriented flux; this is not a same-wire area-cut comparison.Separate patches; edge-on schematic, each A = 6L₀²Patch 1n = (1, 0, 0)A/L₀² = 6J·n/J₀ = 5I/(J₀L₀²) = 30Patch 2n = (-1, 0, 0)A/L₀² = 6J·n/J₀ = -5I/(J₀L₀²) = -30Patch 3n = (3/5, 4/5, 0)A/L₀² = 6J·n/J₀ = 3I/(J₀L₀²) = 18Tilted area is prescribed separately; it is not an unchanged-area cut through the same wire.Drawing lengths do not measure patch area or compare the units of n and J.
  • Prescribed carrier populations with positive scales n₀,q₀,v₀,L₀ and J₀=n₀q₀v₀. Signed conventional current includes q. The stationary background supplies charge but no current. Three separate equal-area patches have their stated oriented normals; they are not unchanged-area cuts of one finite wire.
  • In the first panel, horizontal arrows compare separately normalized velocity vₓ/v₀ and current density Jₓ/J₀. They are not spatial paths, microscopic speeds or vectors with the same physical units. Zero entries are dots with no arrow normalization.
  • The patch panel is an edge-on schematic. Each indicated area is A=6L₀² in three dimensions; the drawing itself does not measure area. Exact normals and J·n determine I=+30,−30,+18 in units J₀L₀². The selected drifts are prescribed bookkeeping, not the response to one common bare-Drude mobility.
  • Every plotted value is read directly from the admitted model. Complete exact fractions,12-significant-digit approximate displays, original function-specific arithmetic qualifications and retained binary64 values remain in the data disclosures. No empirical material constant or numerical solver is introduced.

Complete labelled data equivalents retain every admitted scalar, record, source condition and null. Original manuscript tables remain separate.

Separate charge and current ledgers for the prescribed carrier model

All three prescribed carrier species: signed charge and current (3 rows)

Prescribed carrier populations with positive scales n₀,q₀,v₀,L₀ and J₀=n₀q₀v₀. Signed conventional current includes q. The stationary background supplies charge but no current. Three separate equal-area patches have their stated oriented normals; they are not unchanged-area cuts of one finite wire.

All three prescribed carrier species: signed charge and current
RowSpeciesn/n₀Signed q/q₀Signed vₓ/v₀Signed ρ/(n₀q₀)Signed Jₓ/J₀
0mobilePositive1Exact fraction: 1Retained binary64: 1.01Exact fraction: 1Retained binary64: 1.02Exact fraction: 2Retained binary64: 2.01Exact fraction: 1Retained binary64: 1.02Exact fraction: 2Retained binary64: 2.0
1mobileNegative3Exact fraction: 3Retained binary64: 3.0-1Exact fraction: -1Retained binary64: -1.0-1Exact fraction: -1Retained binary64: -1.0-3Exact fraction: -3Retained binary64: -3.03Exact fraction: 3Retained binary64: 3.0
2fixedBackground2Exact fraction: 2Retained binary64: 2.01Exact fraction: 1Retained binary64: 1.00Exact fraction: 0Retained binary64: 0.02Exact fraction: 2Retained binary64: 2.00Exact fraction: 0Retained binary64: 0.0
All three oriented equal-area patches, including every normal component (3 rows)

Prescribed carrier populations with positive scales n₀,q₀,v₀,L₀ and J₀=n₀q₀v₀. Signed conventional current includes q. The stationary background supplies charge but no current. Three separate equal-area patches have their stated oriented normals; they are not unchanged-area cuts of one finite wire.

All three oriented equal-area patches, including every normal component
RowUnit normal nₓUnit normal nᵧUnit normal n_zSquared normal lengthA/L₀² (same for each separate patch)J·n/J₀Signed I/(J₀L₀²)
01Exact fraction: 1Retained binary64: 1.00Exact fraction: 0Retained binary64: 0.00Exact fraction: 0Retained binary64: 0.01Exact fraction: 1Retained binary64: 1.06Exact fraction: 6Retained binary64: 6.05Exact fraction: 5Retained binary64: 5.030Exact fraction: 30Retained binary64: 30.0
1-1Exact fraction: -1Retained binary64: -1.00Exact fraction: 0Retained binary64: 0.00Exact fraction: 0Retained binary64: 0.01Exact fraction: 1Retained binary64: 1.06Exact fraction: 6Retained binary64: 6.0-5Exact fraction: -5Retained binary64: -5.0-30Exact fraction: -30Retained binary64: -30.0
23/5Exact fraction: 3/5Retained binary64: 0.64/5Exact fraction: 4/5Retained binary64: 0.80Exact fraction: 0Retained binary64: 0.01Exact fraction: 1Retained binary64: 1.06Exact fraction: 6Retained binary64: 6.03Exact fraction: 3Retained binary64: 3.018Exact fraction: 18Retained binary64: 18.0
carrierAndArea: source scales, exact constants and scope flags (3 rows)

Prescribed carrier populations with positive scales n₀,q₀,v₀,L₀ and J₀=n₀q₀v₀. Signed conventional current includes q. The stationary background supplies charge but no current. Three separate equal-area patches have their stated oriented normals; they are not unchanged-area cuts of one finite wire.

carrierAndArea: source scales, exact constants and scope flags
Quantity/source conditionAdmitted value
total Charge Density Overn0q00Exact fraction: 0Retained binary64: 0.0
total Jx Over J05Exact fraction: 5Retained binary64: 5.0
scopeprescribeddrifts and stationaryneutralizingbackground; not onecommonbareDrudemobility or samefinitewirecut area claim

Focus a labelled graph or table region and use arrow keys, or scroll sideways. Fixed-size readable diagrams stay within their own regions. Open the complete data disclosures for exact values, retained binary64 numbers and precision qualifications.

04

Charge continuity is local and integral

Charge conservation for a fixed volume Ω states d/dt ∫Ωρ_q d³r = −∮∂ΩJ·n̂ dA, where ρ_q denotes charge density. In a smooth bulk region this becomes ∂ρ_q/∂t + ∇·J = 0. The outward normal convention makes net outward current reduce the charge remaining inside.

In a steady state, ∂ρ_q/∂t = 0, so ∇·J = 0 in the bulk. This does not imply that J is spatially constant, that E is zero, or that ρ_q itself is zero. Steady surface and bulk charges can help establish the field that guides the current through a geometry.

At a junction, incoming and outgoing currents sum algebraically to zero only when charge is not accumulating in the chosen node region. During charging transients, their imbalance is exactly the rate of charge accumulation. The familiar steady circuit rule is therefore a consequence of continuity under a stated approximation.

Surface charges require the corresponding surface accounting. With negligible lateral surface current, a difference between normal currents on the two sides changes the sheet charge. If tangential surface currents matter, their surface divergence must also be included. Deleting a thin charged boundary from the model does not delete its conservation law.

05

A driving field is not equilibrium screening

The ideal conductor in electrostatic equilibrium had zero total field in its material. A real wire carrying a dissipative steady current is not in that equilibrium state: a driving field can persist while carriers transfer momentum and energy to their surroundings. A time-independent macroscopic current can coexist with ongoing microscopic dissipation.

For a stationary electrostatic or appropriate electroquasistatic segment, the voltage drop along the current direction is the line integral of E. Surface charges on a wire and its surroundings contribute to that field, including its changes near bends, contacts and different cross-sections. The field is not confined to an imaginary collection of carriers moving independently of the electromagnetic boundary conditions.

A battery or another active source can maintain a potential difference by converting other forms of energy, such as chemical energy, while charge circulates. The charge is transported rather than consumed as fuel. The source’s internal non-electrostatic processes are distinct from passive Ohmic conduction in a selected wire segment.

Diffusion and electrochemical-potential gradients can also drive transport, especially near junctions or in inhomogeneous materials. J = σE is a restricted local constitutive relation, not a universal definition of current. A complete transport model must state which forces and gradients are present.

06

Local Ohm law and device resistance

For a homogeneous isotropic local Ohmic medium in a specified regime, J = σE, with conductivity σ in S/m and positive resistivity η = 1/σ in Ω m. We use η for resistivity to keep it distinct from charge density ρ_q. Conductivity is a material-response parameter under its stated temperature, bias and frequency conditions.

For a two-terminal linear passive element with a specified geometry and fixed state, ΔV = IR defines its resistance R in ohms. A voltage divided by current can be formed for a nonlinear device too, but that ratio alone does not establish that R is independent of voltage or current. Differential resistance d(ΔV)/dI is another quantity.

A uniform straight wire of length ℓ and cross-sectional area A, with negligible end effects and approximately uniform axial current, has J = I/A and E = ηJ. Integrating E along its length gives R = ηℓ/A. This is a derivation from a local law plus geometry, rather than a law applying unchanged to every component.

Anisotropy can require a conductivity tensor, and strong gradients or long mean free paths can invalidate a local relation altogether. Contacts, interfaces and self-heating can contribute additional resistance. A value of σ by itself is insufficient to predict the measured two-terminal resistance of an arbitrary apparatus.

07

A slow taper uses a resistance integral

Consider a wire extending from x = 0 to x = 60L₀ with radius r(x) = L₀[1 + x/(60L₀)], constant positive resistivity η₀, and a reduced quasi-one-dimensional steady description. Its radius grows from L₀ to 2L₀ with slope 1/60, while its length is much larger than either radius. The reduced model uses cross-section averages and neglects detailed end and radial corrections; no quantitative full three-dimensional error bound is claimed.

Write s = x/(60L₀) and f = 1 + s. The area is A = πL₀²f². Steady current conservation fixes the same I through each complete cross-section. The averaged axial relation gives dV_drop = η₀I dx/A(x), so R = ∫η₀ dx/A(x). A narrow section contributes more resistance per unit length than a wide section.

Define R₀ = η₀/(πL₀). Then the resistance from the left endpoint to position s is R(s)/R₀ = 60∫ from 0 to s du/(1 + u)² = 60[1 − 1/(1 + s)]. At s = 1 this gives total R = 30R₀. The antiderivative and every endpoint remain explicit.

The area-average axial J = I/A(x) is not, by itself, a complete purely axial vector field in three dimensions. A full solution generally needs small transverse components to follow the sloping boundary while satisfying local ∇·J = 0 and no side leakage. The reduced integral respects complete-cross-section current conservation; it does not claim those uncomputed transverse components vanish.

08

Recover potential, average current density and power

Set the left potential to 6V₀ and the right to zero. With I₀ = V₀/R₀, the total resistance gives I = I₀/5. Use E₀ = V₀/L₀ and J₀ = I₀/(πL₀²) = V₀/(η₀L₀) for this wire model. These scales are separate from the prescribed carrier example’s J₀.

The potential is V/V₀ = 6 − (I/I₀)R(s)/R₀ = 12/f − 6. It decreases from 6 to zero. Differentiation with respect to physical x gives the averaged axial E/E₀ = 1/(5f²), consistent with the local Ohmic relation and averaged J/J₀ = 1/(5f²). The factors from s = x/(60L₀) must be kept during differentiation.

The same current flows through all cross-sections, while average J falls by a factor of four between the narrow and wide ends. The average axial field falls by the same factor. Constant current and constant current density are therefore distinct claims.

Within the reduced model, the local electrical power density is η₀J² = p₀/(25f⁴), where p₀ = V₀²/(η₀L₀²). With P₀ = V₀I₀, integration of this density over the changing area gives total power 6P₀/5. It equals IΔV = (I₀/5)(6V₀), an independent global energy check.

A tapered wire through cross-section averages

Follow potential, accumulated resistance, average axial current density and reduced power density along the selected slowly tapered wire. The curves are exact within the stated one-dimensional model; transverse and end corrections have not been calculated.

Potential drop and accumulated resistance

  • V/(6V₀), normalized by the full terminal drop
  • R(s)/(30R₀), normalized by the reduced total resistance
Potential drop and accumulated resistanceAll129 source samples versus s=x/(60L₀). Separate ratios V/(6V₀) and R(s)/(30R₀), with three exact controls each. Their source divisors are the retained terminal drop and reduced total resistance.00.250.50.75100.250.50.751V/(6V₀) and accumulated R(s)/(30R₀)s=x/(60L₀)

Average axial response and reduced power density

  • Cross-section average axial J/J₀
  • Average axial E/E₀; same numerical curve, different units
  • Reduced power density p/p₀; not a computed 3D field
Average axial response and reduced power densityAll129 samples of average axial J/J₀, average axial E/E₀ and reduced p/p₀. These are cross-section-averaged/reduced quantities; no transverse or end correction is computed.00.250.50.75100.050.10.150.2Conserved I/I₀ = 1/5; transverse/end corrections uncomputedAverage J/J₀, axial E/E₀; reduced p/p₀s=x/(60L₀)
  • Exact results of the stated quasi-one-dimensional cross-section-averaged model only; transverse/end corrections and a full three-dimensional current field have not been computed. s=x/(60L₀), radius=L₀(1+s), positive constant resistivity.
  • Scales: A₀=πL₀²,R₀=η₀/(πL₀),I₀=V₀/R₀,J₀=I₀/A₀,E₀=V₀/L₀,P₀=V₀I₀,p₀=V₀²/(η₀L₀²). Left potential6V₀,right potential0; conserved I/I₀=1/5.
  • The first graph normalizes potential by the full terminal drop6V₀ and accumulated resistance by the total reduced resistance30R₀. Original V/V₀ and R/R₀ values remain in the complete tables. These distinct normalizations do not change the source data.
  • The second graph keeps average axial J/J₀ and E/E₀ separate in its legend even though their numbers coincide. Blue remains visible between the orange dashes, and nested controls retain both. The power density is a reduced-model quantity, not a computed 3D local field.
  • Replacing the area profile by its arithmetic mean gives180R₀/7 instead of30R₀, relative error−1/7 within the reduced model. This is not an error bound for the reduction against a full3D solution.
  • Every plotted value is read directly from the admitted model. Complete exact fractions,12-significant-digit approximate displays, original function-specific arithmetic qualifications and retained binary64 values remain in the data disclosures. No empirical material constant or numerical solver is introduced.

Complete labelled data equivalents retain every admitted scalar, record, source condition and null. Original manuscript tables remain separate.

Exact controls of the reduced tapered-wire model

All 129 exact reduced-wire spatial samples (129 rows)

Exact results of the stated quasi-one-dimensional cross-section-averaged model only; transverse/end corrections and a full three-dimensional current field have not been computed. s=x/(60L₀), radius=L₀(1+s), positive constant resistivity.

All 129 exact reduced-wire spatial samples
Rows = x/(60L₀)X = x/L₀Radius/L₀Cross-section area/A₀Accumulated R(s)/R₀V/V₀Cross-section average axial J/J₀Average axial E/E₀Reduced power density/p₀Accumulated reduced power/P₀Conserved I/I₀
00Exact fraction: 0Retained binary64: 0.00Exact fraction: 0Retained binary64: 0.01Exact fraction: 1Retained binary64: 1.01Exact fraction: 1Retained binary64: 1.00Exact fraction: 0Retained binary64: 0.06Exact fraction: 6Retained binary64: 6.01/5Exact fraction: 1/5Retained binary64: 0.21/5Exact fraction: 1/5Retained binary64: 0.21/25Exact fraction: 1/25Retained binary64: 0.040Exact fraction: 0Retained binary64: 0.01/5Exact fraction: 1/5Retained binary64: 0.2
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81/16Exact fraction: 1/16Retained binary64: 0.062515/4Exact fraction: 15/4Retained binary64: 3.7517/16Exact fraction: 17/16Retained binary64: 1.0625289/256Exact fraction: 289/256Retained binary64: 1.1289062560/17Exact fraction: 60/17Retained binary64: 3.529411764705882290/17Exact fraction: 90/17Retained binary64: 5.294117647058823256/1445Exact fraction: 256/1445Retained binary64: 0.17716262975778546256/1445Exact fraction: 256/1445Retained binary64: 0.1771626297577854665536/2088025Exact fraction: 65536/2088025Retained binary64: 0.0313865973826941712/85Exact fraction: 12/85Retained binary64: 0.14117647058823531/5Exact fraction: 1/5Retained binary64: 0.2
99/128Exact fraction: 9/128Retained binary64: 0.0703125135/32Exact fraction: 135/32Retained binary64: 4.21875137/128Exact fraction: 137/128Retained binary64: 1.070312518769/16384Exact fraction: 18769/16384Retained binary64: 1.14556884765625540/137Exact fraction: 540/137Retained binary64: 3.9416058394160585714/137Exact fraction: 714/137Retained binary64: 5.21167883211678816384/93845Exact fraction: 16384/93845Retained binary64: 0.1745857531035217616384/93845Exact fraction: 16384/93845Retained binary64: 0.17458575310352176268435456/8806884025Exact fraction: 268435456/8806884025Retained binary64: 0.03048018518672386108/685Exact fraction: 108/685Retained binary64: 0.157664233576642331/5Exact fraction: 1/5Retained binary64: 0.2
105/64Exact fraction: 5/64Retained binary64: 0.07812575/16Exact fraction: 75/16Retained binary64: 4.687569/64Exact fraction: 69/64Retained binary64: 1.0781254761/4096Exact fraction: 4761/4096Retained binary64: 1.162353515625100/23Exact fraction: 100/23Retained binary64: 4.3478260869565215118/23Exact fraction: 118/23Retained binary64: 5.1304347826086954096/23805Exact fraction: 4096/23805Retained binary64: 0.17206469229153544096/23805Exact fraction: 4096/23805Retained binary64: 0.172064692291535416777216/566678025Exact fraction: 16777216/566678025Retained binary64: 0.029606258333380764/23Exact fraction: 4/23Retained binary64: 0.173913043478260861/5Exact fraction: 1/5Retained binary64: 0.2
1111/128Exact fraction: 11/128Retained binary64: 0.0859375165/32Exact fraction: 165/32Retained binary64: 5.15625139/128Exact fraction: 139/128Retained binary64: 1.085937519321/16384Exact fraction: 19321/16384Retained binary64: 1.17926025390625660/139Exact fraction: 660/139Retained binary64: 4.748201438848921702/139Exact fraction: 702/139Retained binary64: 5.050359712230215516384/96605Exact fraction: 16384/96605Retained binary64: 0.1695978469023342616384/96605Exact fraction: 16384/96605Retained binary64: 0.16959784690233426268435456/9332526025Exact fraction: 268435456/9332526025Retained binary64: 0.028763429673907607132/695Exact fraction: 132/695Retained binary64: 0.189928057553956841/5Exact fraction: 1/5Retained binary64: 0.2
123/32Exact fraction: 3/32Retained binary64: 0.0937545/8Exact fraction: 45/8Retained binary64: 5.62535/32Exact fraction: 35/32Retained binary64: 1.093751225/1024Exact fraction: 1225/1024Retained binary64: 1.196289062536/7Exact fraction: 36/7Retained binary64: 5.142857142857143174/35Exact fraction: 174/35Retained binary64: 4.97142857142857151024/6125Exact fraction: 1024/6125Retained binary64: 0.167183673469387761024/6125Exact fraction: 1024/6125Retained binary64: 0.167183673469387761048576/37515625Exact fraction: 1048576/37515625Retained binary64: 0.02795038067471886836/175Exact fraction: 36/175Retained binary64: 0.20571428571428571/5Exact fraction: 1/5Retained binary64: 0.2
1313/128Exact fraction: 13/128Retained binary64: 0.1015625195/32Exact fraction: 195/32Retained binary64: 6.09375141/128Exact fraction: 141/128Retained binary64: 1.101562519881/16384Exact fraction: 19881/16384Retained binary64: 1.21343994140625260/47Exact fraction: 260/47Retained binary64: 5.531914893617022230/47Exact fraction: 230/47Retained binary64: 4.893617021276595516384/99405Exact fraction: 16384/99405Retained binary64: 0.1648206830642321716384/99405Exact fraction: 16384/99405Retained binary64: 0.16482068306423217268435456/9881354025Exact fraction: 268435456/9881354025Retained binary64: 0.02716585756576007252/235Exact fraction: 52/235Retained binary64: 0.221276595744680841/5Exact fraction: 1/5Retained binary64: 0.2
147/64Exact fraction: 7/64Retained binary64: 0.109375105/16Exact fraction: 105/16Retained binary64: 6.562571/64Exact fraction: 71/64Retained binary64: 1.1093755041/4096Exact fraction: 5041/4096Retained binary64: 1.230712890625420/71Exact fraction: 420/71Retained binary64: 5.915492957746479342/71Exact fraction: 342/71Retained binary64: 4.8169014084507044096/25205Exact fraction: 4096/25205Retained binary64: 0.162507439000198374096/25205Exact fraction: 4096/25205Retained binary64: 0.1625074390001983716777216/635292025Exact fraction: 16777216/635292025Retained binary64: 0.02640866773040319784/355Exact fraction: 84/355Retained binary64: 0.236619718309859151/5Exact fraction: 1/5Retained binary64: 0.2
1515/128Exact fraction: 15/128Retained binary64: 0.1171875225/32Exact fraction: 225/32Retained binary64: 7.03125143/128Exact fraction: 143/128Retained binary64: 1.117187520449/16384Exact fraction: 20449/16384Retained binary64: 1.24810791015625900/143Exact fraction: 900/143Retained binary64: 6.293706293706293678/143Exact fraction: 678/143Retained binary64: 4.74125874125874216384/102245Exact fraction: 16384/102245Retained binary64: 0.1602425546481490416384/102245Exact fraction: 16384/102245Retained binary64: 0.16024255464814904268435456/10454040025Exact fraction: 268435456/10454040025Retained binary64: 0.02567767632016503736/143Exact fraction: 36/143Retained binary64: 0.25174825174825181/5Exact fraction: 1/5Retained binary64: 0.2
161/8Exact fraction: 1/8Retained binary64: 0.12515/2Exact fraction: 15/2Retained binary64: 7.59/8Exact fraction: 9/8Retained binary64: 1.12581/64Exact fraction: 81/64Retained binary64: 1.26562520/3Exact fraction: 20/3Retained binary64: 6.66666666666666714/3Exact fraction: 14/3Retained binary64: 4.66666666666666764/405Exact fraction: 64/405Retained binary64: 0.158024691358024764/405Exact fraction: 64/405Retained binary64: 0.15802469135802474096/164025Exact fraction: 4096/164025Retained binary64: 0.0249718030787989654/15Exact fraction: 4/15Retained binary64: 0.266666666666666661/5Exact fraction: 1/5Retained binary64: 0.2
1717/128Exact fraction: 17/128Retained binary64: 0.1328125255/32Exact fraction: 255/32Retained binary64: 7.96875145/128Exact fraction: 145/128Retained binary64: 1.132812521025/16384Exact fraction: 21025/16384Retained binary64: 1.28326416015625204/29Exact fraction: 204/29Retained binary64: 7.0344827586206895666/145Exact fraction: 666/145Retained binary64: 4.59310344827586216384/105125Exact fraction: 16384/105125Retained binary64: 0.155852556480380516384/105125Exact fraction: 16384/105125Retained binary64: 0.1558525564803805268435456/11051265625Exact fraction: 268435456/11051265625Retained binary64: 0.024290019361470194204/725Exact fraction: 204/725Retained binary64: 0.28137931034482761/5Exact fraction: 1/5Retained binary64: 0.2
189/64Exact fraction: 9/64Retained binary64: 0.140625135/16Exact fraction: 135/16Retained binary64: 8.437573/64Exact fraction: 73/64Retained binary64: 1.1406255329/4096Exact fraction: 5329/4096Retained binary64: 1.301025390625540/73Exact fraction: 540/73Retained binary64: 7.397260273972603330/73Exact fraction: 330/73Retained binary64: 4.520547945205484096/26645Exact fraction: 4096/26645Retained binary64: 0.15372490148245454096/26645Exact fraction: 4096/26645Retained binary64: 0.153724901482454516777216/709956025Exact fraction: 16777216/709956025Retained binary64: 0.02363134533579034108/365Exact fraction: 108/365Retained binary64: 0.29589041095890411/5Exact fraction: 1/5Retained binary64: 0.2
1919/128Exact fraction: 19/128Retained binary64: 0.1484375285/32Exact fraction: 285/32Retained binary64: 8.90625147/128Exact fraction: 147/128Retained binary64: 1.148437521609/16384Exact fraction: 21609/16384Retained binary64: 1.31890869140625380/49Exact fraction: 380/49Retained binary64: 7.755102040816326218/49Exact fraction: 218/49Retained binary64: 4.44897959183673516384/108045Exact fraction: 16384/108045Retained binary64: 0.1516405201536396916384/108045Exact fraction: 16384/108045Retained binary64: 0.15164052015363969268435456/11673722025Exact fraction: 268435456/11673722025Retained binary64: 0.02299484735246640676/245Exact fraction: 76/245Retained binary64: 0.310204081632653061/5Exact fraction: 1/5Retained binary64: 0.2
205/32Exact fraction: 5/32Retained binary64: 0.1562575/8Exact fraction: 75/8Retained binary64: 9.37537/32Exact fraction: 37/32Retained binary64: 1.156251369/1024Exact fraction: 1369/1024Retained binary64: 1.3369140625300/37Exact fraction: 300/37Retained binary64: 8.108108108108109162/37Exact fraction: 162/37Retained binary64: 4.3783783783783781024/6845Exact fraction: 1024/6845Retained binary64: 0.149598246895544181024/6845Exact fraction: 1024/6845Retained binary64: 0.149598246895544181048576/46854025Exact fraction: 1048576/46854025Retained binary64: 0.02237963547422019612/37Exact fraction: 12/37Retained binary64: 0.324324324324324341/5Exact fraction: 1/5Retained binary64: 0.2
2121/128Exact fraction: 21/128Retained binary64: 0.1640625315/32Exact fraction: 315/32Retained binary64: 9.84375149/128Exact fraction: 149/128Retained binary64: 1.164062522201/16384Exact fraction: 22201/16384Retained binary64: 1.355041503906251260/149Exact fraction: 1260/149Retained binary64: 8.456375838926174642/149Exact fraction: 642/149Retained binary64: 4.30872483221476516384/111005Exact fraction: 16384/111005Retained binary64: 0.1475969550921129616384/111005Exact fraction: 16384/111005Retained binary64: 0.14759695509211296268435456/12322110025Exact fraction: 268435456/12322110025Retained binary64: 0.02178486115246321252/745Exact fraction: 252/745Retained binary64: 0.3382550335570471/5Exact fraction: 1/5Retained binary64: 0.2
2211/64Exact fraction: 11/64Retained binary64: 0.171875165/16Exact fraction: 165/16Retained binary64: 10.312575/64Exact fraction: 75/64Retained binary64: 1.1718755625/4096Exact fraction: 5625/4096Retained binary64: 1.37329101562544/5Exact fraction: 44/5Retained binary64: 8.8106/25Exact fraction: 106/25Retained binary64: 4.244096/28125Exact fraction: 4096/28125Retained binary64: 0.145635555555555554096/28125Exact fraction: 4096/28125Retained binary64: 0.1456355555555555516777216/791015625Exact fraction: 16777216/791015625Retained binary64: 0.0212097150419753144/125Exact fraction: 44/125Retained binary64: 0.3521/5Exact fraction: 1/5Retained binary64: 0.2
2323/128Exact fraction: 23/128Retained binary64: 0.1796875345/32Exact fraction: 345/32Retained binary64: 10.78125151/128Exact fraction: 151/128Retained binary64: 1.179687522801/16384Exact fraction: 22801/16384Retained binary64: 1.391662597656251380/151Exact fraction: 1380/151Retained binary64: 9.139072847682119630/151Exact fraction: 630/151Retained binary64: 4.17218543046357716384/114005Exact fraction: 16384/114005Retained binary64: 0.14371299504407716384/114005Exact fraction: 16384/114005Retained binary64: 0.143712995044077268435456/12997140025Exact fraction: 268435456/12997140025Retained binary64: 0.020653424944538906276/755Exact fraction: 276/755Retained binary64: 0.36556291390728481/5Exact fraction: 1/5Retained binary64: 0.2
243/16Exact fraction: 3/16Retained binary64: 0.187545/4Exact fraction: 45/4Retained binary64: 11.2519/16Exact fraction: 19/16Retained binary64: 1.1875361/256Exact fraction: 361/256Retained binary64: 1.41015625180/19Exact fraction: 180/19Retained binary64: 9.47368421052631578/19Exact fraction: 78/19Retained binary64: 4.105263157894737256/1805Exact fraction: 256/1805Retained binary64: 0.14182825484764544256/1805Exact fraction: 256/1805Retained binary64: 0.1418282548476454465536/3258025Exact fraction: 65536/3258025Retained binary64: 0.0201152538731286636/95Exact fraction: 36/95Retained binary64: 0.378947368421052641/5Exact fraction: 1/5Retained binary64: 0.2
2525/128Exact fraction: 25/128Retained binary64: 0.1953125375/32Exact fraction: 375/32Retained binary64: 11.71875153/128Exact fraction: 153/128Retained binary64: 1.195312523409/16384Exact fraction: 23409/16384Retained binary64: 1.42877197265625500/51Exact fraction: 500/51Retained binary64: 9.803921568627452206/51Exact fraction: 206/51Retained binary64: 4.0392156862745116384/117045Exact fraction: 16384/117045Retained binary64: 0.1399803494382502516384/117045Exact fraction: 16384/117045Retained binary64: 0.13998034943825025268435456/13699532025Exact fraction: 268435456/13699532025Retained binary64: 0.01959449822885464520/51Exact fraction: 20/51Retained binary64: 0.392156862745098031/5Exact fraction: 1/5Retained binary64: 0.2
2613/64Exact fraction: 13/64Retained binary64: 0.203125195/16Exact fraction: 195/16Retained binary64: 12.187577/64Exact fraction: 77/64Retained binary64: 1.2031255929/4096Exact fraction: 5929/4096Retained binary64: 1.447509765625780/77Exact fraction: 780/77Retained binary64: 10.12987012987013306/77Exact fraction: 306/77Retained binary64: 3.97402597402597424096/29645Exact fraction: 4096/29645Retained binary64: 0.13816832518131224096/29645Exact fraction: 4096/29645Retained binary64: 0.138168325181312216777216/878826025Exact fraction: 16777216/878826025Retained binary64: 0.01909048608340883156/385Exact fraction: 156/385Retained binary64: 0.40519480519480521/5Exact fraction: 1/5Retained binary64: 0.2
2727/128Exact fraction: 27/128Retained binary64: 0.2109375405/32Exact fraction: 405/32Retained binary64: 12.65625155/128Exact fraction: 155/128Retained binary64: 1.210937524025/16384Exact fraction: 24025/16384Retained binary64: 1.46636962890625324/31Exact fraction: 324/31Retained binary64: 10.451612903225806606/155Exact fraction: 606/155Retained binary64: 3.909677419354838716384/120125Exact fraction: 16384/120125Retained binary64: 0.1363912591050988616384/120125Exact fraction: 16384/120125Retained binary64: 0.13639125910509886268435456/14430015625Exact fraction: 268435456/14430015625Retained binary64: 0.018602575560274212324/775Exact fraction: 324/775Retained binary64: 0.41806451612903231/5Exact fraction: 1/5Retained binary64: 0.2
287/32Exact fraction: 7/32Retained binary64: 0.21875105/8Exact fraction: 105/8Retained binary64: 13.12539/32Exact fraction: 39/32Retained binary64: 1.218751521/1024Exact fraction: 1521/1024Retained binary64: 1.4853515625140/13Exact fraction: 140/13Retained binary64: 10.7692307692307750/13Exact fraction: 50/13Retained binary64: 3.84615384615384631024/7605Exact fraction: 1024/7605Retained binary64: 0.13464825772518081024/7605Exact fraction: 1024/7605Retained binary64: 0.13464825772518081048576/57836025Exact fraction: 1048576/57836025Retained binary64: 0.0181301533084267128/65Exact fraction: 28/65Retained binary64: 0.43076923076923081/5Exact fraction: 1/5Retained binary64: 0.2
2929/128Exact fraction: 29/128Retained binary64: 0.2265625435/32Exact fraction: 435/32Retained binary64: 13.59375157/128Exact fraction: 157/128Retained binary64: 1.226562524649/16384Exact fraction: 24649/16384Retained binary64: 1.504455566406251740/157Exact fraction: 1740/157Retained binary64: 11.0828025477707594/157Exact fraction: 594/157Retained binary64: 3.7834394904458616384/123245Exact fraction: 16384/123245Retained binary64: 0.132938455921132716384/123245Exact fraction: 16384/123245Retained binary64: 0.1329384559211327268435456/15189330025Exact fraction: 268435456/15189330025Retained binary64: 0.017672633062694943348/785Exact fraction: 348/785Retained binary64: 0.443312101910828041/5Exact fraction: 1/5Retained binary64: 0.2
3015/64Exact fraction: 15/64Retained binary64: 0.234375225/16Exact fraction: 225/16Retained binary64: 14.062579/64Exact fraction: 79/64Retained binary64: 1.2343756241/4096Exact fraction: 6241/4096Retained binary64: 1.523681640625900/79Exact fraction: 900/79Retained binary64: 11.39240506329114294/79Exact fraction: 294/79Retained binary64: 3.7215189873417724096/31205Exact fraction: 4096/31205Retained binary64: 0.13126101586284254096/31205Exact fraction: 4096/31205Retained binary64: 0.131261015862842516777216/973752025Exact fraction: 16777216/973752025Retained binary64: 0.01722945428534538836/79Exact fraction: 36/79Retained binary64: 0.455696202531645561/5Exact fraction: 1/5Retained binary64: 0.2
3131/128Exact fraction: 31/128Retained binary64: 0.2421875465/32Exact fraction: 465/32Retained binary64: 14.53125159/128Exact fraction: 159/128Retained binary64: 1.242187525281/16384Exact fraction: 25281/16384Retained binary64: 1.54302978515625620/53Exact fraction: 620/53Retained binary64: 11.69811320754717194/53Exact fraction: 194/53Retained binary64: 3.66037735849056616384/126405Exact fraction: 16384/126405Retained binary64: 0.129615125983940516384/126405Exact fraction: 16384/126405Retained binary64: 0.1296151259839405268435456/15978224025Exact fraction: 268435456/15978224025Retained binary64: 0.01680008088383277124/265Exact fraction: 124/265Retained binary64: 0.46792452830188681/5Exact fraction: 1/5Retained binary64: 0.2
321/4Exact fraction: 1/4Retained binary64: 0.2515Exact fraction: 15Retained binary64: 15.05/4Exact fraction: 5/4Retained binary64: 1.2525/16Exact fraction: 25/16Retained binary64: 1.562512Exact fraction: 12Retained binary64: 12.018/5Exact fraction: 18/5Retained binary64: 3.616/125Exact fraction: 16/125Retained binary64: 0.12816/125Exact fraction: 16/125Retained binary64: 0.128256/15625Exact fraction: 256/15625Retained binary64: 0.01638412/25Exact fraction: 12/25Retained binary64: 0.481/5Exact fraction: 1/5Retained binary64: 0.2
3333/128Exact fraction: 33/128Retained binary64: 0.2578125495/32Exact fraction: 495/32Retained binary64: 15.46875161/128Exact fraction: 161/128Retained binary64: 1.257812525921/16384Exact fraction: 25921/16384Retained binary64: 1.582092285156251980/161Exact fraction: 1980/161Retained binary64: 12.298136645962733570/161Exact fraction: 570/161Retained binary64: 3.54037267080745316384/129605Exact fraction: 16384/129605Retained binary64: 0.126414875969291316384/129605Exact fraction: 16384/129605Retained binary64: 0.1264148759692913268435456/16797456025Exact fraction: 268435456/16797456025Retained binary64: 0.015980720866331306396/805Exact fraction: 396/805Retained binary64: 0.49192546583850931/5Exact fraction: 1/5Retained binary64: 0.2
3417/64Exact fraction: 17/64Retained binary64: 0.265625255/16Exact fraction: 255/16Retained binary64: 15.937581/64Exact fraction: 81/64Retained binary64: 1.2656256561/4096Exact fraction: 6561/4096Retained binary64: 1.601806640625340/27Exact fraction: 340/27Retained binary64: 12.59259259259259394/27Exact fraction: 94/27Retained binary64: 3.48148148148148144096/32805Exact fraction: 4096/32805Retained binary64: 0.124859015393994814096/32805Exact fraction: 4096/32805Retained binary64: 0.1248590153939948116777216/1076168025Exact fraction: 16777216/1076168025Retained binary64: 0.01558977372515783468/135Exact fraction: 68/135Retained binary64: 0.50370370370370371/5Exact fraction: 1/5Retained binary64: 0.2
3535/128Exact fraction: 35/128Retained binary64: 0.2734375525/32Exact fraction: 525/32Retained binary64: 16.40625163/128Exact fraction: 163/128Retained binary64: 1.273437526569/16384Exact fraction: 26569/16384Retained binary64: 1.621643066406252100/163Exact fraction: 2100/163Retained binary64: 12.883435582822086558/163Exact fraction: 558/163Retained binary64: 3.42331288343558316384/132845Exact fraction: 16384/132845Retained binary64: 0.123331702359893116384/132845Exact fraction: 16384/132845Retained binary64: 0.1233317023598931268435456/17647794025Exact fraction: 268435456/17647794025Retained binary64: 0.01521070880698926384/163Exact fraction: 84/163Retained binary64: 0.51533742331288351/5Exact fraction: 1/5Retained binary64: 0.2
369/32Exact fraction: 9/32Retained binary64: 0.28125135/8Exact fraction: 135/8Retained binary64: 16.87541/32Exact fraction: 41/32Retained binary64: 1.281251681/1024Exact fraction: 1681/1024Retained binary64: 1.6416015625540/41Exact fraction: 540/41Retained binary64: 13.170731707317072138/41Exact fraction: 138/41Retained binary64: 3.36585365853658521024/8405Exact fraction: 1024/8405Retained binary64: 0.121832242712671031024/8405Exact fraction: 1024/8405Retained binary64: 0.121832242712671031048576/70644025Exact fraction: 1048576/70644025Retained binary64: 0.014843095364399183108/205Exact fraction: 108/205Retained binary64: 0.5268292682926831/5Exact fraction: 1/5Retained binary64: 0.2
3737/128Exact fraction: 37/128Retained binary64: 0.2890625555/32Exact fraction: 555/32Retained binary64: 17.34375165/128Exact fraction: 165/128Retained binary64: 1.289062527225/16384Exact fraction: 27225/16384Retained binary64: 1.66168212890625148/11Exact fraction: 148/11Retained binary64: 13.454545454545455182/55Exact fraction: 182/55Retained binary64: 3.30909090909090916384/136125Exact fraction: 16384/136125Retained binary64: 0.1203599632690541816384/136125Exact fraction: 16384/136125Retained binary64: 0.12035996326905418268435456/18530015625Exact fraction: 268435456/18530015625Retained binary64: 0.014486520758128072148/275Exact fraction: 148/275Retained binary64: 0.53818181818181821/5Exact fraction: 1/5Retained binary64: 0.2
3819/64Exact fraction: 19/64Retained binary64: 0.296875285/16Exact fraction: 285/16Retained binary64: 17.812583/64Exact fraction: 83/64Retained binary64: 1.2968756889/4096Exact fraction: 6889/4096Retained binary64: 1.6818847656251140/83Exact fraction: 1140/83Retained binary64: 13.734939759036145270/83Exact fraction: 270/83Retained binary64: 3.2530120481927714096/34445Exact fraction: 4096/34445Retained binary64: 0.118914211061111914096/34445Exact fraction: 4096/34445Retained binary64: 0.1189142110611119116777216/1186458025Exact fraction: 16777216/1186458025Retained binary64: 0.014140589592286671228/415Exact fraction: 228/415Retained binary64: 0.54939759036144581/5Exact fraction: 1/5Retained binary64: 0.2
3939/128Exact fraction: 39/128Retained binary64: 0.3046875585/32Exact fraction: 585/32Retained binary64: 18.28125167/128Exact fraction: 167/128Retained binary64: 1.304687527889/16384Exact fraction: 27889/16384Retained binary64: 1.702209472656252340/167Exact fraction: 2340/167Retained binary64: 14.011976047904191534/167Exact fraction: 534/167Retained binary64: 3.19760479041916216384/139445Exact fraction: 16384/139445Retained binary64: 0.1174943526121409916384/139445Exact fraction: 16384/139445Retained binary64: 0.11749435261214099268435456/19444908025Exact fraction: 268435456/19444908025Retained binary64: 0.013804922895746122468/835Exact fraction: 468/835Retained binary64: 0.56047904191616771/5Exact fraction: 1/5Retained binary64: 0.2
405/16Exact fraction: 5/16Retained binary64: 0.312575/4Exact fraction: 75/4Retained binary64: 18.7521/16Exact fraction: 21/16Retained binary64: 1.3125441/256Exact fraction: 441/256Retained binary64: 1.72265625100/7Exact fraction: 100/7Retained binary64: 14.28571428571428622/7Exact fraction: 22/7Retained binary64: 3.142857142857143256/2205Exact fraction: 256/2205Retained binary64: 0.11609977324263039256/2205Exact fraction: 256/2205Retained binary64: 0.1160997732426303965536/4862025Exact fraction: 65536/4862025Retained binary64: 0.0134791573469901944/7Exact fraction: 4/7Retained binary64: 0.57142857142857141/5Exact fraction: 1/5Retained binary64: 0.2
4141/128Exact fraction: 41/128Retained binary64: 0.3203125615/32Exact fraction: 615/32Retained binary64: 19.21875169/128Exact fraction: 169/128Retained binary64: 1.320312528561/16384Exact fraction: 28561/16384Retained binary64: 1.743225097656252460/169Exact fraction: 2460/169Retained binary64: 14.55621301775148522/169Exact fraction: 522/169Retained binary64: 3.08875739644970416384/142805Exact fraction: 16384/142805Retained binary64: 0.1147298764048877816384/142805Exact fraction: 16384/142805Retained binary64: 0.11472987640488778268435456/20393268025Exact fraction: 268435456/20393268025Retained binary64: 0.013162944539880827492/845Exact fraction: 492/845Retained binary64: 0.58224852071005921/5Exact fraction: 1/5Retained binary64: 0.2
4221/64Exact fraction: 21/64Retained binary64: 0.328125315/16Exact fraction: 315/16Retained binary64: 19.687585/64Exact fraction: 85/64Retained binary64: 1.3281257225/4096Exact fraction: 7225/4096Retained binary64: 1.763916015625252/17Exact fraction: 252/17Retained binary64: 14.823529411764707258/85Exact fraction: 258/85Retained binary64: 3.03529411764705874096/36125Exact fraction: 4096/36125Retained binary64: 0.11338408304498274096/36125Exact fraction: 4096/36125Retained binary64: 0.113384083044982716777216/1305015625Exact fraction: 16777216/1305015625Retained binary64: 0.012855950287951534252/425Exact fraction: 252/425Retained binary64: 0.59294117647058821/5Exact fraction: 1/5Retained binary64: 0.2
4343/128Exact fraction: 43/128Retained binary64: 0.3359375645/32Exact fraction: 645/32Retained binary64: 20.15625171/128Exact fraction: 171/128Retained binary64: 1.335937529241/16384Exact fraction: 29241/16384Retained binary64: 1.78472900390625860/57Exact fraction: 860/57Retained binary64: 15.087719298245615170/57Exact fraction: 170/57Retained binary64: 2.98245614035087716384/146205Exact fraction: 16384/146205Retained binary64: 0.1120618309907321916384/146205Exact fraction: 16384/146205Retained binary64: 0.11206183099073219268435456/21375902025Exact fraction: 268435456/21375902025Retained binary64: 0.012557853964995426172/285Exact fraction: 172/285Retained binary64: 0.60350877192982461/5Exact fraction: 1/5Retained binary64: 0.2
4411/32Exact fraction: 11/32Retained binary64: 0.34375165/8Exact fraction: 165/8Retained binary64: 20.62543/32Exact fraction: 43/32Retained binary64: 1.343751849/1024Exact fraction: 1849/1024Retained binary64: 1.8056640625660/43Exact fraction: 660/43Retained binary64: 15.348837209302326126/43Exact fraction: 126/43Retained binary64: 2.93023255813953481024/9245Exact fraction: 1024/9245Retained binary64: 0.110762574364521361024/9245Exact fraction: 1024/9245Retained binary64: 0.110762574364521361048576/85470025Exact fraction: 1048576/85470025Retained binary64: 0.012268347879856124132/215Exact fraction: 132/215Retained binary64: 0.61395348837209311/5Exact fraction: 1/5Retained binary64: 0.2
4545/128Exact fraction: 45/128Retained binary64: 0.3515625675/32Exact fraction: 675/32Retained binary64: 21.09375173/128Exact fraction: 173/128Retained binary64: 1.351562529929/16384Exact fraction: 29929/16384Retained binary64: 1.826721191406252700/173Exact fraction: 2700/173Retained binary64: 15.606936416184972498/173Exact fraction: 498/173Retained binary64: 2.87861271676300616384/149645Exact fraction: 16384/149645Retained binary64: 0.1094857830198135616384/149645Exact fraction: 16384/149645Retained binary64: 0.10948578301981356268435456/22393626025Exact fraction: 268435456/22393626025Retained binary64: 0.011987136683461694108/173Exact fraction: 108/173Retained binary64: 0.62427745664739891/5Exact fraction: 1/5Retained binary64: 0.2
4623/64Exact fraction: 23/64Retained binary64: 0.359375345/16Exact fraction: 345/16Retained binary64: 21.562587/64Exact fraction: 87/64Retained binary64: 1.3593757569/4096Exact fraction: 7569/4096Retained binary64: 1.847900390625460/29Exact fraction: 460/29Retained binary64: 15.86206896551724282/29Exact fraction: 82/29Retained binary64: 2.82758620689655164096/37845Exact fraction: 4096/37845Retained binary64: 0.108230942000264234096/37845Exact fraction: 4096/37845Retained binary64: 0.1082309420002642316777216/1432244025Exact fraction: 16777216/1432244025Retained binary64: 0.01171393680626456292/145Exact fraction: 92/145Retained binary64: 0.63448275862068971/5Exact fraction: 1/5Retained binary64: 0.2
4747/128Exact fraction: 47/128Retained binary64: 0.3671875705/32Exact fraction: 705/32Retained binary64: 22.03125175/128Exact fraction: 175/128Retained binary64: 1.367187530625/16384Exact fraction: 30625/16384Retained binary64: 1.86920166015625564/35Exact fraction: 564/35Retained binary64: 16.114285714285714486/175Exact fraction: 486/175Retained binary64: 2.77714285714285716384/153125Exact fraction: 16384/153125Retained binary64: 0.1069975510204081616384/153125Exact fraction: 16384/153125Retained binary64: 0.10699755102040816268435456/23447265625Exact fraction: 268435456/23447265625Retained binary64: 0.011448475924364848564/875Exact fraction: 564/875Retained binary64: 0.64457142857142861/5Exact fraction: 1/5Retained binary64: 0.2
483/8Exact fraction: 3/8Retained binary64: 0.37545/2Exact fraction: 45/2Retained binary64: 22.511/8Exact fraction: 11/8Retained binary64: 1.375121/64Exact fraction: 121/64Retained binary64: 1.890625180/11Exact fraction: 180/11Retained binary64: 16.36363636363636330/11Exact fraction: 30/11Retained binary64: 2.72727272727272764/605Exact fraction: 64/605Retained binary64: 0.1057851239669421564/605Exact fraction: 64/605Retained binary64: 0.105785123966942154096/366025Exact fraction: 4096/366025Retained binary64: 0.01119049245270131936/55Exact fraction: 36/55Retained binary64: 0.65454545454545451/5Exact fraction: 1/5Retained binary64: 0.2
4949/128Exact fraction: 49/128Retained binary64: 0.3828125735/32Exact fraction: 735/32Retained binary64: 22.96875177/128Exact fraction: 177/128Retained binary64: 1.382812531329/16384Exact fraction: 31329/16384Retained binary64: 1.91217041015625980/59Exact fraction: 980/59Retained binary64: 16.610169491525422158/59Exact fraction: 158/59Retained binary64: 2.677966101694915216384/156645Exact fraction: 16384/156645Retained binary64: 0.1045931884196750616384/156645Exact fraction: 16384/156645Retained binary64: 0.10459318841967506268435456/24537656025Exact fraction: 268435456/24537656025Retained binary64: 0.01093973506379365196/295Exact fraction: 196/295Retained binary64: 0.66440677966101691/5Exact fraction: 1/5Retained binary64: 0.2
5025/64Exact fraction: 25/64Retained binary64: 0.390625375/16Exact fraction: 375/16Retained binary64: 23.437589/64Exact fraction: 89/64Retained binary64: 1.3906257921/4096Exact fraction: 7921/4096Retained binary64: 1.9338378906251500/89Exact fraction: 1500/89Retained binary64: 16.853932584269664234/89Exact fraction: 234/89Retained binary64: 2.62921348314606764096/39605Exact fraction: 4096/39605Retained binary64: 0.103421285191263734096/39605Exact fraction: 4096/39605Retained binary64: 0.1034212851912637316777216/1568556025Exact fraction: 16777216/1568556025Retained binary64: 0.01069596223061270660/89Exact fraction: 60/89Retained binary64: 0.67415730337078651/5Exact fraction: 1/5Retained binary64: 0.2
5151/128Exact fraction: 51/128Retained binary64: 0.3984375765/32Exact fraction: 765/32Retained binary64: 23.90625179/128Exact fraction: 179/128Retained binary64: 1.398437532041/16384Exact fraction: 32041/16384Retained binary64: 1.955627441406253060/179Exact fraction: 3060/179Retained binary64: 17.094972067039105462/179Exact fraction: 462/179Retained binary64: 2.581005586592178616384/160205Exact fraction: 16384/160205Retained binary64: 0.1022689678848974816384/160205Exact fraction: 16384/160205Retained binary64: 0.10226896788489748268435456/25665642025Exact fraction: 268435456/25665642025Retained binary64: 0.010458941792242191612/895Exact fraction: 612/895Retained binary64: 0.68379888268156421/5Exact fraction: 1/5Retained binary64: 0.2
5213/32Exact fraction: 13/32Retained binary64: 0.40625195/8Exact fraction: 195/8Retained binary64: 24.37545/32Exact fraction: 45/32Retained binary64: 1.406252025/1024Exact fraction: 2025/1024Retained binary64: 1.977539062552/3Exact fraction: 52/3Retained binary64: 17.33333333333333238/15Exact fraction: 38/15Retained binary64: 2.5333333333333331024/10125Exact fraction: 1024/10125Retained binary64: 0.10113580246913581024/10125Exact fraction: 1024/10125Retained binary64: 0.10113580246913581048576/102515625Exact fraction: 1048576/102515625Retained binary64: 0.01022845054107605652/75Exact fraction: 52/75Retained binary64: 0.69333333333333341/5Exact fraction: 1/5Retained binary64: 0.2
5353/128Exact fraction: 53/128Retained binary64: 0.4140625795/32Exact fraction: 795/32Retained binary64: 24.84375181/128Exact fraction: 181/128Retained binary64: 1.414062532761/16384Exact fraction: 32761/16384Retained binary64: 1.999572753906253180/181Exact fraction: 3180/181Retained binary64: 17.569060773480665450/181Exact fraction: 450/181Retained binary64: 2.486187845303867516384/163805Exact fraction: 16384/163805Retained binary64: 0.1000213668691431916384/163805Exact fraction: 16384/163805Retained binary64: 0.10002136686914319268435456/26832078025Exact fraction: 268435456/26832078025Retained binary64: 0.010004273830371734636/905Exact fraction: 636/905Retained binary64: 0.70276243093922651/5Exact fraction: 1/5Retained binary64: 0.2
5427/64Exact fraction: 27/64Retained binary64: 0.421875405/16Exact fraction: 405/16Retained binary64: 25.312591/64Exact fraction: 91/64Retained binary64: 1.4218758281/4096Exact fraction: 8281/4096Retained binary64: 2.0217285156251620/91Exact fraction: 1620/91Retained binary64: 17.802197802197803222/91Exact fraction: 222/91Retained binary64: 2.43956043956043984096/41405Exact fraction: 4096/41405Retained binary64: 0.098925250573602234096/41405Exact fraction: 4096/41405Retained binary64: 0.0989252505736022316777216/1714374025Exact fraction: 16777216/1714374025Retained binary64: 0.009786205201049987324/455Exact fraction: 324/455Retained binary64: 0.71208791208791211/5Exact fraction: 1/5Retained binary64: 0.2
5555/128Exact fraction: 55/128Retained binary64: 0.4296875825/32Exact fraction: 825/32Retained binary64: 25.78125183/128Exact fraction: 183/128Retained binary64: 1.429687533489/16384Exact fraction: 33489/16384Retained binary64: 2.044006347656251100/61Exact fraction: 1100/61Retained binary64: 18.0327868852459146/61Exact fraction: 146/61Retained binary64: 2.393442622950819716384/167445Exact fraction: 16384/167445Retained binary64: 0.0978470542566215716384/167445Exact fraction: 16384/167445Retained binary64: 0.09784705425662157268435456/28037828025Exact fraction: 268435456/28037828025Retained binary64: 0.00957404602669824744/61Exact fraction: 44/61Retained binary64: 0.72131147540983611/5Exact fraction: 1/5Retained binary64: 0.2
567/16Exact fraction: 7/16Retained binary64: 0.4375105/4Exact fraction: 105/4Retained binary64: 26.2523/16Exact fraction: 23/16Retained binary64: 1.4375529/256Exact fraction: 529/256Retained binary64: 2.06640625420/23Exact fraction: 420/23Retained binary64: 18.2608695652173954/23Exact fraction: 54/23Retained binary64: 2.347826086956522256/2645Exact fraction: 256/2645Retained binary64: 0.09678638941398866256/2645Exact fraction: 256/2645Retained binary64: 0.0967863894139886665536/6996025Exact fraction: 65536/6996025Retained binary64: 0.00936760517579625584/115Exact fraction: 84/115Retained binary64: 0.73043478260869571/5Exact fraction: 1/5Retained binary64: 0.2
5757/128Exact fraction: 57/128Retained binary64: 0.4453125855/32Exact fraction: 855/32Retained binary64: 26.71875185/128Exact fraction: 185/128Retained binary64: 1.445312534225/16384Exact fraction: 34225/16384Retained binary64: 2.08892822265625684/37Exact fraction: 684/37Retained binary64: 18.486486486486488426/185Exact fraction: 426/185Retained binary64: 2.302702702702702716384/171125Exact fraction: 16384/171125Retained binary64: 0.0957428780131482816384/171125Exact fraction: 16384/171125Retained binary64: 0.09574287801314828268435456/29283765625Exact fraction: 268435456/29283765625Retained binary64: 0.009166698690240593684/925Exact fraction: 684/925Retained binary64: 0.73945945945945951/5Exact fraction: 1/5Retained binary64: 0.2
5829/64Exact fraction: 29/64Retained binary64: 0.453125435/16Exact fraction: 435/16Retained binary64: 27.187593/64Exact fraction: 93/64Retained binary64: 1.4531258649/4096Exact fraction: 8649/4096Retained binary64: 2.111572265625580/31Exact fraction: 580/31Retained binary64: 18.7096774193548470/31Exact fraction: 70/31Retained binary64: 2.25806451612903254096/43245Exact fraction: 4096/43245Retained binary64: 0.094716152156318654096/43245Exact fraction: 4096/43245Retained binary64: 0.0947161521563186516777216/1870130025Exact fraction: 16777216/1870130025Retained binary64: 0.008971149479298906116/155Exact fraction: 116/155Retained binary64: 0.74838709677419361/5Exact fraction: 1/5Retained binary64: 0.2
5959/128Exact fraction: 59/128Retained binary64: 0.4609375885/32Exact fraction: 885/32Retained binary64: 27.65625187/128Exact fraction: 187/128Retained binary64: 1.460937534969/16384Exact fraction: 34969/16384Retained binary64: 2.134338378906253540/187Exact fraction: 3540/187Retained binary64: 18.93048128342246414/187Exact fraction: 414/187Retained binary64: 2.21390374331550816384/174845Exact fraction: 16384/174845Retained binary64: 0.0937058537561840516384/174845Exact fraction: 16384/174845Retained binary64: 0.09370585375618405268435456/30570774025Exact fraction: 268435456/30570774025Retained binary64: 0.008780787028175353708/935Exact fraction: 708/935Retained binary64: 0.75721925133689841/5Exact fraction: 1/5Retained binary64: 0.2
6015/32Exact fraction: 15/32Retained binary64: 0.46875225/8Exact fraction: 225/8Retained binary64: 28.12547/32Exact fraction: 47/32Retained binary64: 1.468752209/1024Exact fraction: 2209/1024Retained binary64: 2.1572265625900/47Exact fraction: 900/47Retained binary64: 19.148936170212767102/47Exact fraction: 102/47Retained binary64: 2.17021276595744661024/11045Exact fraction: 1024/11045Retained binary64: 0.09271163422363061024/11045Exact fraction: 1024/11045Retained binary64: 0.09271163422363061048576/121992025Exact fraction: 1048576/121992025Retained binary64: 0.00859544712041627336/47Exact fraction: 36/47Retained binary64: 0.76595744680851061/5Exact fraction: 1/5Retained binary64: 0.2
6161/128Exact fraction: 61/128Retained binary64: 0.4765625915/32Exact fraction: 915/32Retained binary64: 28.59375189/128Exact fraction: 189/128Retained binary64: 1.476562535721/16384Exact fraction: 35721/16384Retained binary64: 2.180236816406251220/63Exact fraction: 1220/63Retained binary64: 19.365079365079364134/63Exact fraction: 134/63Retained binary64: 2.12698412698412716384/178605Exact fraction: 16384/178605Retained binary64: 0.091733154167016616384/178605Exact fraction: 16384/178605Retained binary64: 0.0917331541670166268435456/31899746025Exact fraction: 268435456/31899746025Retained binary64: 0.008414971573429636244/315Exact fraction: 244/315Retained binary64: 0.77460317460317461/5Exact fraction: 1/5Retained binary64: 0.2
6231/64Exact fraction: 31/64Retained binary64: 0.484375465/16Exact fraction: 465/16Retained binary64: 29.062595/64Exact fraction: 95/64Retained binary64: 1.4843759025/4096Exact fraction: 9025/4096Retained binary64: 2.203369140625372/19Exact fraction: 372/19Retained binary64: 19.57894736842105198/95Exact fraction: 198/95Retained binary64: 2.08421052631578934096/45125Exact fraction: 4096/45125Retained binary64: 0.090770083102493074096/45125Exact fraction: 4096/45125Retained binary64: 0.0907700831024930716777216/2036265625Exact fraction: 16777216/2036265625Retained binary64: 0.0082392079864335372/475Exact fraction: 372/475Retained binary64: 0.7831578947368421/5Exact fraction: 1/5Retained binary64: 0.2
6363/128Exact fraction: 63/128Retained binary64: 0.4921875945/32Exact fraction: 945/32Retained binary64: 29.53125191/128Exact fraction: 191/128Retained binary64: 1.492187536481/16384Exact fraction: 36481/16384Retained binary64: 2.226623535156253780/191Exact fraction: 3780/191Retained binary64: 19.790575916230367390/191Exact fraction: 390/191Retained binary64: 2.041884816753926716384/182405Exact fraction: 16384/182405Retained binary64: 0.0898220991749129616384/182405Exact fraction: 16384/182405Retained binary64: 0.08982209917491296268435456/33271584025Exact fraction: 268435456/33271584025Retained binary64: 0.0080680095001879756/955Exact fraction: 756/955Retained binary64: 0.79162303664921461/5Exact fraction: 1/5Retained binary64: 0.2
641/2Exact fraction: 1/2Retained binary64: 0.530Exact fraction: 30Retained binary64: 30.03/2Exact fraction: 3/2Retained binary64: 1.59/4Exact fraction: 9/4Retained binary64: 2.2520Exact fraction: 20Retained binary64: 20.02Exact fraction: 2Retained binary64: 2.04/45Exact fraction: 4/45Retained binary64: 0.088888888888888894/45Exact fraction: 4/45Retained binary64: 0.0888888888888888916/2025Exact fraction: 16/2025Retained binary64: 0.0079012345679012354/5Exact fraction: 4/5Retained binary64: 0.81/5Exact fraction: 1/5Retained binary64: 0.2
6565/128Exact fraction: 65/128Retained binary64: 0.5078125975/32Exact fraction: 975/32Retained binary64: 30.46875193/128Exact fraction: 193/128Retained binary64: 1.507812537249/16384Exact fraction: 37249/16384Retained binary64: 2.273498535156253900/193Exact fraction: 3900/193Retained binary64: 20.207253886010363378/193Exact fraction: 378/193Retained binary64: 1.958549222797927516384/186245Exact fraction: 16384/186245Retained binary64: 0.0879701468495798516384/186245Exact fraction: 16384/186245Retained binary64: 0.08797014684957985268435456/34687200025Exact fraction: 268435456/34687200025Retained binary64: 0.007738746736736644156/193Exact fraction: 156/193Retained binary64: 0.80829015544041451/5Exact fraction: 1/5Retained binary64: 0.2
6633/64Exact fraction: 33/64Retained binary64: 0.515625495/16Exact fraction: 495/16Retained binary64: 30.937597/64Exact fraction: 97/64Retained binary64: 1.5156259409/4096Exact fraction: 9409/4096Retained binary64: 2.2971191406251980/97Exact fraction: 1980/97Retained binary64: 20.412371134020617186/97Exact fraction: 186/97Retained binary64: 1.91752577319587644096/47045Exact fraction: 4096/47045Retained binary64: 0.087065575512806894096/47045Exact fraction: 4096/47045Retained binary64: 0.0870655755128068916777216/2213232025Exact fraction: 16777216/2213232025Retained binary64: 0.007580414439376278396/485Exact fraction: 396/485Retained binary64: 0.81649484536082471/5Exact fraction: 1/5Retained binary64: 0.2
6767/128Exact fraction: 67/128Retained binary64: 0.52343751005/32Exact fraction: 1005/32Retained binary64: 31.40625195/128Exact fraction: 195/128Retained binary64: 1.523437538025/16384Exact fraction: 38025/16384Retained binary64: 2.32086181640625268/13Exact fraction: 268/13Retained binary64: 20.615384615384617122/65Exact fraction: 122/65Retained binary64: 1.87692307692307716384/190125Exact fraction: 16384/190125Retained binary64: 0.0861748849441157116384/190125Exact fraction: 16384/190125Retained binary64: 0.08617488494411571268435456/36147515625Exact fraction: 268435456/36147515625Retained binary64: 0.007426110795131581268/325Exact fraction: 268/325Retained binary64: 0.82461538461538461/5Exact fraction: 1/5Retained binary64: 0.2
6817/32Exact fraction: 17/32Retained binary64: 0.53125255/8Exact fraction: 255/8Retained binary64: 31.87549/32Exact fraction: 49/32Retained binary64: 1.531252401/1024Exact fraction: 2401/1024Retained binary64: 2.34472656251020/49Exact fraction: 1020/49Retained binary64: 20.81632653061224490/49Exact fraction: 90/49Retained binary64: 1.8367346938775511024/12005Exact fraction: 1024/12005Retained binary64: 0.085297792586422321024/12005Exact fraction: 1024/12005Retained binary64: 0.085297792586422321048576/144120025Exact fraction: 1048576/144120025Retained binary64: 0.007275713420116323204/245Exact fraction: 204/245Retained binary64: 0.83265306122448981/5Exact fraction: 1/5Retained binary64: 0.2
6969/128Exact fraction: 69/128Retained binary64: 0.53906251035/32Exact fraction: 1035/32Retained binary64: 32.34375197/128Exact fraction: 197/128Retained binary64: 1.539062538809/16384Exact fraction: 38809/16384Retained binary64: 2.368713378906254140/197Exact fraction: 4140/197Retained binary64: 21.01522842639594354/197Exact fraction: 354/197Retained binary64: 1.796954314720812216384/194045Exact fraction: 16384/194045Retained binary64: 0.0844340230358937316384/194045Exact fraction: 16384/194045Retained binary64: 0.08443402303589373268435456/37653462025Exact fraction: 268435456/37653462025Retained binary64: 0.007129104246025834828/985Exact fraction: 828/985Retained binary64: 0.84060913705583761/5Exact fraction: 1/5Retained binary64: 0.2
7035/64Exact fraction: 35/64Retained binary64: 0.546875525/16Exact fraction: 525/16Retained binary64: 32.812599/64Exact fraction: 99/64Retained binary64: 1.5468759801/4096Exact fraction: 9801/4096Retained binary64: 2.392822265625700/33Exact fraction: 700/33Retained binary64: 21.2121212121212158/33Exact fraction: 58/33Retained binary64: 1.75757575757575764096/49005Exact fraction: 4096/49005Retained binary64: 0.083583307825732064096/49005Exact fraction: 4096/49005Retained binary64: 0.0835833078257320616777216/2401490025Exact fraction: 16777216/2401490025Retained binary64: 0.00698616934709108428/33Exact fraction: 28/33Retained binary64: 0.84848484848484851/5Exact fraction: 1/5Retained binary64: 0.2
7171/128Exact fraction: 71/128Retained binary64: 0.55468751065/32Exact fraction: 1065/32Retained binary64: 33.28125199/128Exact fraction: 199/128Retained binary64: 1.554687539601/16384Exact fraction: 39601/16384Retained binary64: 2.417053222656254260/199Exact fraction: 4260/199Retained binary64: 21.407035175879397342/199Exact fraction: 342/199Retained binary64: 1.718592964824120516384/198005Exact fraction: 16384/198005Retained binary64: 0.0827453852175450116384/198005Exact fraction: 16384/198005Retained binary64: 0.08274538521754501268435456/39205980025Exact fraction: 268435456/39205980025Retained binary64: 0.006846798774799916852/995Exact fraction: 852/995Retained binary64: 0.85628140703517591/5Exact fraction: 1/5Retained binary64: 0.2
729/16Exact fraction: 9/16Retained binary64: 0.5625135/4Exact fraction: 135/4Retained binary64: 33.7525/16Exact fraction: 25/16Retained binary64: 1.5625625/256Exact fraction: 625/256Retained binary64: 2.44140625108/5Exact fraction: 108/5Retained binary64: 21.642/25Exact fraction: 42/25Retained binary64: 1.68256/3125Exact fraction: 256/3125Retained binary64: 0.08192256/3125Exact fraction: 256/3125Retained binary64: 0.0819265536/9765625Exact fraction: 65536/9765625Retained binary64: 0.0067108864108/125Exact fraction: 108/125Retained binary64: 0.8641/5Exact fraction: 1/5Retained binary64: 0.2
7373/128Exact fraction: 73/128Retained binary64: 0.57031251095/32Exact fraction: 1095/32Retained binary64: 34.21875201/128Exact fraction: 201/128Retained binary64: 1.570312540401/16384Exact fraction: 40401/16384Retained binary64: 2.465881347656251460/67Exact fraction: 1460/67Retained binary64: 21.791044776119403110/67Exact fraction: 110/67Retained binary64: 1.641791044776119516384/202005Exact fraction: 16384/202005Retained binary64: 0.0811069032944729116384/202005Exact fraction: 16384/202005Retained binary64: 0.08110690329447291268435456/40806020025Exact fraction: 268435456/40806020025Retained binary64: 0.006578329762018981292/335Exact fraction: 292/335Retained binary64: 0.87164179104477611/5Exact fraction: 1/5Retained binary64: 0.2
7437/64Exact fraction: 37/64Retained binary64: 0.578125555/16Exact fraction: 555/16Retained binary64: 34.6875101/64Exact fraction: 101/64Retained binary64: 1.57812510201/4096Exact fraction: 10201/4096Retained binary64: 2.4904785156252220/101Exact fraction: 2220/101Retained binary64: 21.980198019801982162/101Exact fraction: 162/101Retained binary64: 1.6039603960396044096/51005Exact fraction: 4096/51005Retained binary64: 0.080305852367414964096/51005Exact fraction: 4096/51005Retained binary64: 0.0803058523674149616777216/2601510025Exact fraction: 16777216/2601510025Retained binary64: 0.006449029924457047444/505Exact fraction: 444/505Retained binary64: 0.87920792079207921/5Exact fraction: 1/5Retained binary64: 0.2
7575/128Exact fraction: 75/128Retained binary64: 0.58593751125/32Exact fraction: 1125/32Retained binary64: 35.15625203/128Exact fraction: 203/128Retained binary64: 1.585937541209/16384Exact fraction: 41209/16384Retained binary64: 2.515197753906254500/203Exact fraction: 4500/203Retained binary64: 22.167487684729064318/203Exact fraction: 318/203Retained binary64: 1.566502463054187316384/206045Exact fraction: 16384/206045Retained binary64: 0.0795166104491737316384/206045Exact fraction: 16384/206045Retained binary64: 0.07951661044917373268435456/42454542025Exact fraction: 268435456/42454542025Retained binary64: 0.006322891337325644180/203Exact fraction: 180/203Retained binary64: 0.88669950738916261/5Exact fraction: 1/5Retained binary64: 0.2
7619/32Exact fraction: 19/32Retained binary64: 0.59375285/8Exact fraction: 285/8Retained binary64: 35.62551/32Exact fraction: 51/32Retained binary64: 1.593752601/1024Exact fraction: 2601/1024Retained binary64: 2.5400390625380/17Exact fraction: 380/17Retained binary64: 22.35294117647058726/17Exact fraction: 26/17Retained binary64: 1.52941176470588221024/13005Exact fraction: 1024/13005Retained binary64: 0.078738946559015771024/13005Exact fraction: 1024/13005Retained binary64: 0.078738946559015771048576/169130025Exact fraction: 1048576/169130025Retained binary64: 0.006199821705223540576/85Exact fraction: 76/85Retained binary64: 0.89411764705882361/5Exact fraction: 1/5Retained binary64: 0.2
7777/128Exact fraction: 77/128Retained binary64: 0.60156251155/32Exact fraction: 1155/32Retained binary64: 36.09375205/128Exact fraction: 205/128Retained binary64: 1.601562542025/16384Exact fraction: 42025/16384Retained binary64: 2.56500244140625924/41Exact fraction: 924/41Retained binary64: 22.536585365853657306/205Exact fraction: 306/205Retained binary64: 1.492682926829268316384/210125Exact fraction: 16384/210125Retained binary64: 0.0779726353361094616384/210125Exact fraction: 16384/210125Retained binary64: 0.07797263533610946268435456/44152515625Exact fraction: 268435456/44152515625Retained binary64: 0.006079731861257906924/1025Exact fraction: 924/1025Retained binary64: 0.90146341463414641/5Exact fraction: 1/5Retained binary64: 0.2
7839/64Exact fraction: 39/64Retained binary64: 0.609375585/16Exact fraction: 585/16Retained binary64: 36.5625103/64Exact fraction: 103/64Retained binary64: 1.60937510609/4096Exact fraction: 10609/4096Retained binary64: 2.5900878906252340/103Exact fraction: 2340/103Retained binary64: 22.718446601941746150/103Exact fraction: 150/103Retained binary64: 1.45631067961165054096/53045Exact fraction: 4096/53045Retained binary64: 0.077217456876237164096/53045Exact fraction: 4096/53045Retained binary64: 0.0772174568762371616777216/2813772025Exact fraction: 16777216/2813772025Retained binary64: 0.005962535646433545468/515Exact fraction: 468/515Retained binary64: 0.90873786407766991/5Exact fraction: 1/5Retained binary64: 0.2
7979/128Exact fraction: 79/128Retained binary64: 0.61718751185/32Exact fraction: 1185/32Retained binary64: 37.03125207/128Exact fraction: 207/128Retained binary64: 1.617187542849/16384Exact fraction: 42849/16384Retained binary64: 2.615295410156251580/69Exact fraction: 1580/69Retained binary64: 22.8985507246376898/69Exact fraction: 98/69Retained binary64: 1.420289855072463916384/214245Exact fraction: 16384/214245Retained binary64: 0.0764731965740157316384/214245Exact fraction: 16384/214245Retained binary64: 0.07647319657401573268435456/45900920025Exact fraction: 268435456/45900920025Retained binary64: 0.005848149794248051316/345Exact fraction: 316/345Retained binary64: 0.91594202898550731/5Exact fraction: 1/5Retained binary64: 0.2
805/8Exact fraction: 5/8Retained binary64: 0.62575/2Exact fraction: 75/2Retained binary64: 37.513/8Exact fraction: 13/8Retained binary64: 1.625169/64Exact fraction: 169/64Retained binary64: 2.640625300/13Exact fraction: 300/13Retained binary64: 23.07692307692307718/13Exact fraction: 18/13Retained binary64: 1.384615384615384664/845Exact fraction: 64/845Retained binary64: 0.075739644970414264/845Exact fraction: 64/845Retained binary64: 0.07573964497041424096/714025Exact fraction: 4096/714025Retained binary64: 0.00573649382024438912/13Exact fraction: 12/13Retained binary64: 0.92307692307692311/5Exact fraction: 1/5Retained binary64: 0.2
8181/128Exact fraction: 81/128Retained binary64: 0.63281251215/32Exact fraction: 1215/32Retained binary64: 37.96875209/128Exact fraction: 209/128Retained binary64: 1.632812543681/16384Exact fraction: 43681/16384Retained binary64: 2.666076660156254860/209Exact fraction: 4860/209Retained binary64: 23.25358851674641282/209Exact fraction: 282/209Retained binary64: 1.349282296650717616384/218405Exact fraction: 16384/218405Retained binary64: 0.0750165976053661816384/218405Exact fraction: 16384/218405Retained binary64: 0.07501659760536618268435456/47700744025Exact fraction: 268435456/47700744025Retained binary64: 0.005627489916285431972/1045Exact fraction: 972/1045Retained binary64: 0.93014354066985651/5Exact fraction: 1/5Retained binary64: 0.2
8241/64Exact fraction: 41/64Retained binary64: 0.640625615/16Exact fraction: 615/16Retained binary64: 38.4375105/64Exact fraction: 105/64Retained binary64: 1.64062511025/4096Exact fraction: 11025/4096Retained binary64: 2.691650390625164/7Exact fraction: 164/7Retained binary64: 23.42857142857142746/35Exact fraction: 46/35Retained binary64: 1.31428571428571434096/55125Exact fraction: 4096/55125Retained binary64: 0.074303854875283454096/55125Exact fraction: 4096/55125Retained binary64: 0.0743038548752834516777216/3038765625Exact fraction: 16777216/3038765625Retained binary64: 0.005521062849327183164/175Exact fraction: 164/175Retained binary64: 0.93714285714285721/5Exact fraction: 1/5Retained binary64: 0.2
8383/128Exact fraction: 83/128Retained binary64: 0.64843751245/32Exact fraction: 1245/32Retained binary64: 38.90625211/128Exact fraction: 211/128Retained binary64: 1.648437544521/16384Exact fraction: 44521/16384Retained binary64: 2.717346191406254980/211Exact fraction: 4980/211Retained binary64: 23.601895734597157270/211Exact fraction: 270/211Retained binary64: 1.279620853080568616384/222605Exact fraction: 16384/222605Retained binary64: 0.0736012218952853716384/222605Exact fraction: 16384/222605Retained binary64: 0.07360122189528537268435456/49552986025Exact fraction: 268435456/49552986025Retained binary64: 0.005417139864479035996/1055Exact fraction: 996/1055Retained binary64: 0.94407582938388621/5Exact fraction: 1/5Retained binary64: 0.2
8421/32Exact fraction: 21/32Retained binary64: 0.65625315/8Exact fraction: 315/8Retained binary64: 39.37553/32Exact fraction: 53/32Retained binary64: 1.656252809/1024Exact fraction: 2809/1024Retained binary64: 2.74316406251260/53Exact fraction: 1260/53Retained binary64: 23.7735849056603866/53Exact fraction: 66/53Retained binary64: 1.24528301886792451024/14045Exact fraction: 1024/14045Retained binary64: 0.072908508365966531024/14045Exact fraction: 1024/14045Retained binary64: 0.072908508365966531048576/197262025Exact fraction: 1048576/197262025Retained binary64: 0.005315650592150212252/265Exact fraction: 252/265Retained binary64: 0.95094339622641511/5Exact fraction: 1/5Retained binary64: 0.2
8585/128Exact fraction: 85/128Retained binary64: 0.66406251275/32Exact fraction: 1275/32Retained binary64: 39.84375213/128Exact fraction: 213/128Retained binary64: 1.664062545369/16384Exact fraction: 45369/16384Retained binary64: 2.769104003906251700/71Exact fraction: 1700/71Retained binary64: 23.94366197183098486/71Exact fraction: 86/71Retained binary64: 1.211267605633802816384/226845Exact fraction: 16384/226845Retained binary64: 0.0722255284445326216384/226845Exact fraction: 16384/226845Retained binary64: 0.07222552844453262268435456/51458654025Exact fraction: 268435456/51458654025Retained binary64: 0.00521652695909198968/71Exact fraction: 68/71Retained binary64: 0.95774647887323941/5Exact fraction: 1/5Retained binary64: 0.2
8643/64Exact fraction: 43/64Retained binary64: 0.671875645/16Exact fraction: 645/16Retained binary64: 40.3125107/64Exact fraction: 107/64Retained binary64: 1.67187511449/4096Exact fraction: 11449/4096Retained binary64: 2.7951660156252580/107Exact fraction: 2580/107Retained binary64: 24.11214953271028126/107Exact fraction: 126/107Retained binary64: 1.17757009345794384096/57245Exact fraction: 4096/57245Retained binary64: 0.07155210062014154096/57245Exact fraction: 4096/57245Retained binary64: 0.071552100620141516777216/3276990025Exact fraction: 16777216/3276990025Retained binary64: 0.005119703103154853516/535Exact fraction: 516/535Retained binary64: 0.96448598130841121/5Exact fraction: 1/5Retained binary64: 0.2
8787/128Exact fraction: 87/128Retained binary64: 0.67968751305/32Exact fraction: 1305/32Retained binary64: 40.78125215/128Exact fraction: 215/128Retained binary64: 1.679687546225/16384Exact fraction: 46225/16384Retained binary64: 2.821350097656251044/43Exact fraction: 1044/43Retained binary64: 24.27906976744186246/215Exact fraction: 246/215Retained binary64: 1.144186046511627816384/231125Exact fraction: 16384/231125Retained binary64: 0.0708880475932936716384/231125Exact fraction: 16384/231125Retained binary64: 0.07088804759329367268435456/53418765625Exact fraction: 268435456/53418765625Retained binary64: 0.0050251152915890691044/1075Exact fraction: 1044/1075Retained binary64: 0.97116279069767441/5Exact fraction: 1/5Retained binary64: 0.2
8811/16Exact fraction: 11/16Retained binary64: 0.6875165/4Exact fraction: 165/4Retained binary64: 41.2527/16Exact fraction: 27/16Retained binary64: 1.6875729/256Exact fraction: 729/256Retained binary64: 2.84765625220/9Exact fraction: 220/9Retained binary64: 24.44444444444444310/9Exact fraction: 10/9Retained binary64: 1.1111111111111112256/3645Exact fraction: 256/3645Retained binary64: 0.07023319615912209256/3645Exact fraction: 256/3645Retained binary64: 0.0702331961591220965536/13286025Exact fraction: 65536/13286025Retained binary64: 0.00493270184272572144/45Exact fraction: 44/45Retained binary64: 0.97777777777777771/5Exact fraction: 1/5Retained binary64: 0.2
8989/128Exact fraction: 89/128Retained binary64: 0.69531251335/32Exact fraction: 1335/32Retained binary64: 41.71875217/128Exact fraction: 217/128Retained binary64: 1.695312547089/16384Exact fraction: 47089/16384Retained binary64: 2.874084472656255340/217Exact fraction: 5340/217Retained binary64: 24.608294930875577234/217Exact fraction: 234/217Retained binary64: 1.078341013824884816384/235445Exact fraction: 16384/235445Retained binary64: 0.0695873770944381916384/235445Exact fraction: 16384/235445Retained binary64: 0.06958737709443819268435456/55434348025Exact fraction: 268435456/55434348025Retained binary64: 0.0048424030508835411068/1085Exact fraction: 1068/1085Retained binary64: 0.98433179723502311/5Exact fraction: 1/5Retained binary64: 0.2
9045/64Exact fraction: 45/64Retained binary64: 0.703125675/16Exact fraction: 675/16Retained binary64: 42.1875109/64Exact fraction: 109/64Retained binary64: 1.70312511881/4096Exact fraction: 11881/4096Retained binary64: 2.9006347656252700/109Exact fraction: 2700/109Retained binary64: 24.770642201834864114/109Exact fraction: 114/109Retained binary64: 1.04587155963302754096/59405Exact fraction: 4096/59405Retained binary64: 0.06895042504839664096/59405Exact fraction: 4096/59405Retained binary64: 0.068950425048396616777216/3528954025Exact fraction: 16777216/3528954025Retained binary64: 0.004754161114354557108/109Exact fraction: 108/109Retained binary64: 0.99082568807339461/5Exact fraction: 1/5Retained binary64: 0.2
9191/128Exact fraction: 91/128Retained binary64: 0.71093751365/32Exact fraction: 1365/32Retained binary64: 42.65625219/128Exact fraction: 219/128Retained binary64: 1.710937547961/16384Exact fraction: 47961/16384Retained binary64: 2.927307128906251820/73Exact fraction: 1820/73Retained binary64: 24.93150684931506774/73Exact fraction: 74/73Retained binary64: 1.013698630136986416384/239805Exact fraction: 16384/239805Retained binary64: 0.0683221784366464416384/239805Exact fraction: 16384/239805Retained binary64: 0.06832217843664644268435456/57506438025Exact fraction: 268435456/57506438025Retained binary64: 0.004667920066328956364/365Exact fraction: 364/365Retained binary64: 0.99726027397260281/5Exact fraction: 1/5Retained binary64: 0.2
9223/32Exact fraction: 23/32Retained binary64: 0.71875345/8Exact fraction: 345/8Retained binary64: 43.12555/32Exact fraction: 55/32Retained binary64: 1.718753025/1024Exact fraction: 3025/1024Retained binary64: 2.9541015625276/11Exact fraction: 276/11Retained binary64: 25.0909090909090954/55Exact fraction: 54/55Retained binary64: 0.98181818181818181024/15125Exact fraction: 1024/15125Retained binary64: 0.067702479338842981024/15125Exact fraction: 1024/15125Retained binary64: 0.067702479338842981048576/228765625Exact fraction: 1048576/228765625Retained binary64: 0.00458362570862646276/275Exact fraction: 276/275Retained binary64: 1.00363636363636371/5Exact fraction: 1/5Retained binary64: 0.2
9393/128Exact fraction: 93/128Retained binary64: 0.72656251395/32Exact fraction: 1395/32Retained binary64: 43.59375221/128Exact fraction: 221/128Retained binary64: 1.726562548841/16384Exact fraction: 48841/16384Retained binary64: 2.981018066406255580/221Exact fraction: 5580/221Retained binary64: 25.248868778280542210/221Exact fraction: 210/221Retained binary64: 0.950226244343891416384/244205Exact fraction: 16384/244205Retained binary64: 0.0670911733993980516384/244205Exact fraction: 16384/244205Retained binary64: 0.06709117339939805268435456/59636082025Exact fraction: 268435456/59636082025Retained binary64: 0.0045012255481080961116/1105Exact fraction: 1116/1105Retained binary64: 1.00995475113122171/5Exact fraction: 1/5Retained binary64: 0.2
9447/64Exact fraction: 47/64Retained binary64: 0.734375705/16Exact fraction: 705/16Retained binary64: 44.0625111/64Exact fraction: 111/64Retained binary64: 1.73437512321/4096Exact fraction: 12321/4096Retained binary64: 3.008056640625940/37Exact fraction: 940/37Retained binary64: 25.40540540540540734/37Exact fraction: 34/37Retained binary64: 0.9189189189189194096/61605Exact fraction: 4096/61605Retained binary64: 0.066488109731352974096/61605Exact fraction: 4096/61605Retained binary64: 0.0664881097313529716777216/3795176025Exact fraction: 16777216/3795176025Retained binary64: 0.0044206687356484345188/185Exact fraction: 188/185Retained binary64: 1.01621621621621631/5Exact fraction: 1/5Retained binary64: 0.2
9595/128Exact fraction: 95/128Retained binary64: 0.74218751425/32Exact fraction: 1425/32Retained binary64: 44.53125223/128Exact fraction: 223/128Retained binary64: 1.742187549729/16384Exact fraction: 49729/16384Retained binary64: 3.035217285156255700/223Exact fraction: 5700/223Retained binary64: 25.56053811659193198/223Exact fraction: 198/223Retained binary64: 0.887892376681614416384/248645Exact fraction: 16384/248645Retained binary64: 0.0658931408232620916384/248645Exact fraction: 16384/248645Retained binary64: 0.06589314082326209268435456/61824336025Exact fraction: 268435456/61824336025Retained binary64: 0.004341906007554248228/223Exact fraction: 228/223Retained binary64: 1.02242152466367721/5Exact fraction: 1/5Retained binary64: 0.2
963/4Exact fraction: 3/4Retained binary64: 0.7545Exact fraction: 45Retained binary64: 45.07/4Exact fraction: 7/4Retained binary64: 1.7549/16Exact fraction: 49/16Retained binary64: 3.0625180/7Exact fraction: 180/7Retained binary64: 25.7142857142857156/7Exact fraction: 6/7Retained binary64: 0.857142857142857116/245Exact fraction: 16/245Retained binary64: 0.065306122448979616/245Exact fraction: 16/245Retained binary64: 0.0653061224489796256/60025Exact fraction: 256/60025Retained binary64: 0.00426488962932111636/35Exact fraction: 36/35Retained binary64: 1.02857142857142851/5Exact fraction: 1/5Retained binary64: 0.2
9797/128Exact fraction: 97/128Retained binary64: 0.75781251455/32Exact fraction: 1455/32Retained binary64: 45.46875225/128Exact fraction: 225/128Retained binary64: 1.757812550625/16384Exact fraction: 50625/16384Retained binary64: 3.08990478515625388/15Exact fraction: 388/15Retained binary64: 25.86666666666666762/75Exact fraction: 62/75Retained binary64: 0.826666666666666716384/253125Exact fraction: 16384/253125Retained binary64: 0.0647269135802469216384/253125Exact fraction: 16384/253125Retained binary64: 0.06472691358024692268435456/64072265625Exact fraction: 268435456/64072265625Retained binary64: 0.004189573341624752388/375Exact fraction: 388/375Retained binary64: 1.03466666666666661/5Exact fraction: 1/5Retained binary64: 0.2
9849/64Exact fraction: 49/64Retained binary64: 0.765625735/16Exact fraction: 735/16Retained binary64: 45.9375113/64Exact fraction: 113/64Retained binary64: 1.76562512769/4096Exact fraction: 12769/4096Retained binary64: 3.1174316406252940/113Exact fraction: 2940/113Retained binary64: 26.0176991150442590/113Exact fraction: 90/113Retained binary64: 0.79646017699115054096/63845Exact fraction: 4096/63845Retained binary64: 0.064155376301981364096/63845Exact fraction: 4096/63845Retained binary64: 0.0641553763019813616777216/4076184025Exact fraction: 16777216/4076184025Retained binary64: 0.004115912308448832588/565Exact fraction: 588/565Retained binary64: 1.04070796460176981/5Exact fraction: 1/5Retained binary64: 0.2
9999/128Exact fraction: 99/128Retained binary64: 0.77343751485/32Exact fraction: 1485/32Retained binary64: 46.40625227/128Exact fraction: 227/128Retained binary64: 1.773437551529/16384Exact fraction: 51529/16384Retained binary64: 3.145080566406255940/227Exact fraction: 5940/227Retained binary64: 26.16740088105727174/227Exact fraction: 174/227Retained binary64: 0.766519823788546216384/257645Exact fraction: 16384/257645Retained binary64: 0.0635913757301713616384/257645Exact fraction: 16384/257645Retained binary64: 0.06359137573017136268435456/66380946025Exact fraction: 268435456/66380946025Retained binary64: 0.0040438630672558271188/1135Exact fraction: 1188/1135Retained binary64: 1.04669603524229071/5Exact fraction: 1/5Retained binary64: 0.2
10025/32Exact fraction: 25/32Retained binary64: 0.78125375/8Exact fraction: 375/8Retained binary64: 46.87557/32Exact fraction: 57/32Retained binary64: 1.781253249/1024Exact fraction: 3249/1024Retained binary64: 3.1728515625500/19Exact fraction: 500/19Retained binary64: 26.3157894736842114/19Exact fraction: 14/19Retained binary64: 0.73684210526315791024/16245Exact fraction: 1024/16245Retained binary64: 0.063034779932286861024/16245Exact fraction: 1024/16245Retained binary64: 0.063034779932286861048576/263900025Exact fraction: 1048576/263900025Retained binary64: 0.00397338348111183420/19Exact fraction: 20/19Retained binary64: 1.05263157894736841/5Exact fraction: 1/5Retained binary64: 0.2
101101/128Exact fraction: 101/128Retained binary64: 0.78906251515/32Exact fraction: 1515/32Retained binary64: 47.34375229/128Exact fraction: 229/128Retained binary64: 1.789062552441/16384Exact fraction: 52441/16384Retained binary64: 3.200744628906256060/229Exact fraction: 6060/229Retained binary64: 26.46288209606987162/229Exact fraction: 162/229Retained binary64: 0.707423580786026216384/262205Exact fraction: 16384/262205Retained binary64: 0.06248545985011727516384/262205Exact fraction: 16384/262205Retained binary64: 0.062485459850117275268435456/68751462025Exact fraction: 268435456/68751462025Retained binary64: 0.0039044326926806181212/1145Exact fraction: 1212/1145Retained binary64: 1.05851528384279471/5Exact fraction: 1/5Retained binary64: 0.2
10251/64Exact fraction: 51/64Retained binary64: 0.796875765/16Exact fraction: 765/16Retained binary64: 47.8125115/64Exact fraction: 115/64Retained binary64: 1.79687513225/4096Exact fraction: 13225/4096Retained binary64: 3.228759765625612/23Exact fraction: 612/23Retained binary64: 26.60869565217391478/115Exact fraction: 78/115Retained binary64: 0.67826086956521744096/66125Exact fraction: 4096/66125Retained binary64: 0.061943289224952744096/66125Exact fraction: 4096/66125Retained binary64: 0.0619432892249527416777216/4372515625Exact fraction: 16777216/4372515625Retained binary64: 0.0038369710800061465612/575Exact fraction: 612/575Retained binary64: 1.06434782608695651/5Exact fraction: 1/5Retained binary64: 0.2
103103/128Exact fraction: 103/128Retained binary64: 0.80468751545/32Exact fraction: 1545/32Retained binary64: 48.28125231/128Exact fraction: 231/128Retained binary64: 1.804687553361/16384Exact fraction: 53361/16384Retained binary64: 3.256896972656252060/77Exact fraction: 2060/77Retained binary64: 26.75324675324675350/77Exact fraction: 50/77Retained binary64: 0.649350649350649316384/266805Exact fraction: 16384/266805Retained binary64: 0.06140814452502764416384/266805Exact fraction: 16384/266805Retained binary64: 0.061408144525027644268435456/71184908025Exact fraction: 268435456/71184908025Retained binary64: 0.0037709602140066825412/385Exact fraction: 412/385Retained binary64: 1.070129870129871/5Exact fraction: 1/5Retained binary64: 0.2
10413/16Exact fraction: 13/16Retained binary64: 0.8125195/4Exact fraction: 195/4Retained binary64: 48.7529/16Exact fraction: 29/16Retained binary64: 1.8125841/256Exact fraction: 841/256Retained binary64: 3.28515625780/29Exact fraction: 780/29Retained binary64: 26.89655172413793218/29Exact fraction: 18/29Retained binary64: 0.6206896551724138256/4205Exact fraction: 256/4205Retained binary64: 0.06087990487514863256/4205Exact fraction: 256/4205Retained binary64: 0.0608799048751486365536/17682025Exact fraction: 65536/17682025Retained binary64: 0.0037063628176071464156/145Exact fraction: 156/145Retained binary64: 1.07586206896551721/5Exact fraction: 1/5Retained binary64: 0.2
105105/128Exact fraction: 105/128Retained binary64: 0.82031251575/32Exact fraction: 1575/32Retained binary64: 49.21875233/128Exact fraction: 233/128Retained binary64: 1.820312554289/16384Exact fraction: 54289/16384Retained binary64: 3.313537597656256300/233Exact fraction: 6300/233Retained binary64: 27.03862660944206138/233Exact fraction: 138/233Retained binary64: 0.59227467811158816384/271445Exact fraction: 16384/271445Retained binary64: 0.0603584519884322816384/271445Exact fraction: 16384/271445Retained binary64: 0.06035845198843228268435456/73682388025Exact fraction: 268435456/73682388025Retained binary64: 0.0036431427264398846252/233Exact fraction: 252/233Retained binary64: 1.08154506437768251/5Exact fraction: 1/5Retained binary64: 0.2
10653/64Exact fraction: 53/64Retained binary64: 0.828125795/16Exact fraction: 795/16Retained binary64: 49.6875117/64Exact fraction: 117/64Retained binary64: 1.82812513689/4096Exact fraction: 13689/4096Retained binary64: 3.3420410156251060/39Exact fraction: 1060/39Retained binary64: 27.1794871794871822/39Exact fraction: 22/39Retained binary64: 0.56410256410256414096/68445Exact fraction: 4096/68445Retained binary64: 0.0598436701000803554096/68445Exact fraction: 4096/68445Retained binary64: 0.05984367010008035516777216/4684718025Exact fraction: 16777216/4684718025Retained binary64: 0.0035812648510472516212/195Exact fraction: 212/195Retained binary64: 1.0871794871794871/5Exact fraction: 1/5Retained binary64: 0.2
107107/128Exact fraction: 107/128Retained binary64: 0.83593751605/32Exact fraction: 1605/32Retained binary64: 50.15625235/128Exact fraction: 235/128Retained binary64: 1.835937555225/16384Exact fraction: 55225/16384Retained binary64: 3.370666503906251284/47Exact fraction: 1284/47Retained binary64: 27.319148936170212126/235Exact fraction: 126/235Retained binary64: 0.536170212765957416384/276125Exact fraction: 16384/276125Retained binary64: 0.0593354459031235816384/276125Exact fraction: 16384/276125Retained binary64: 0.05933544590312358268435456/76245015625Exact fraction: 268435456/76245015625Retained binary64: 0.00352069514052250541284/1175Exact fraction: 1284/1175Retained binary64: 1.09276595744680851/5Exact fraction: 1/5Retained binary64: 0.2
10827/32Exact fraction: 27/32Retained binary64: 0.84375405/8Exact fraction: 405/8Retained binary64: 50.62559/32Exact fraction: 59/32Retained binary64: 1.843753481/1024Exact fraction: 3481/1024Retained binary64: 3.39941406251620/59Exact fraction: 1620/59Retained binary64: 27.4576271186440730/59Exact fraction: 30/59Retained binary64: 0.50847457627118641024/17405Exact fraction: 1024/17405Retained binary64: 0.058833668486067221024/17405Exact fraction: 1024/17405Retained binary64: 0.058833668486067221048576/302934025Exact fraction: 1048576/302934025Retained binary64: 0.0034614005475284595324/295Exact fraction: 324/295Retained binary64: 1.09830508474576271/5Exact fraction: 1/5Retained binary64: 0.2
109109/128Exact fraction: 109/128Retained binary64: 0.85156251635/32Exact fraction: 1635/32Retained binary64: 51.09375237/128Exact fraction: 237/128Retained binary64: 1.851562556169/16384Exact fraction: 56169/16384Retained binary64: 3.428283691406252180/79Exact fraction: 2180/79Retained binary64: 27.5949367088607638/79Exact fraction: 38/79Retained binary64: 0.481012658227848116384/280845Exact fraction: 16384/280845Retained binary64: 0.0583382292723744416384/280845Exact fraction: 16384/280845Retained binary64: 0.05833822927237444268435456/78873914025Exact fraction: 268435456/78873914025Retained binary64: 0.003403348994636126436/395Exact fraction: 436/395Retained binary64: 1.10379746835443051/5Exact fraction: 1/5Retained binary64: 0.2
11055/64Exact fraction: 55/64Retained binary64: 0.859375825/16Exact fraction: 825/16Retained binary64: 51.5625119/64Exact fraction: 119/64Retained binary64: 1.85937514161/4096Exact fraction: 14161/4096Retained binary64: 3.4572753906253300/119Exact fraction: 3300/119Retained binary64: 27.7310924369747954/119Exact fraction: 54/119Retained binary64: 0.4537815126050424096/70805Exact fraction: 4096/70805Retained binary64: 0.0578490219617258664096/70805Exact fraction: 4096/70805Retained binary64: 0.05784902196172586616777216/5013348025Exact fraction: 16777216/5013348025Retained binary64: 0.0033465093419282415132/119Exact fraction: 132/119Retained binary64: 1.10924369747899171/5Exact fraction: 1/5Retained binary64: 0.2
111111/128Exact fraction: 111/128Retained binary64: 0.86718751665/32Exact fraction: 1665/32Retained binary64: 52.03125239/128Exact fraction: 239/128Retained binary64: 1.867187557121/16384Exact fraction: 57121/16384Retained binary64: 3.486389160156256660/239Exact fraction: 6660/239Retained binary64: 27.86610878661088102/239Exact fraction: 102/239Retained binary64: 0.4267782426778242716384/285605Exact fraction: 16384/285605Retained binary64: 0.0573659424729959216384/285605Exact fraction: 16384/285605Retained binary64: 0.05736594247299592268435456/81570216025Exact fraction: 268435456/81570216025Retained binary64: 0.00329085135581507751332/1195Exact fraction: 1332/1195Retained binary64: 1.1146443514644351/5Exact fraction: 1/5Retained binary64: 0.2
1127/8Exact fraction: 7/8Retained binary64: 0.875105/2Exact fraction: 105/2Retained binary64: 52.515/8Exact fraction: 15/8Retained binary64: 1.875225/64Exact fraction: 225/64Retained binary64: 3.51562528Exact fraction: 28Retained binary64: 28.02/5Exact fraction: 2/5Retained binary64: 0.464/1125Exact fraction: 64/1125Retained binary64: 0.0568888888888888964/1125Exact fraction: 64/1125Retained binary64: 0.056888888888888894096/1265625Exact fraction: 4096/1265625Retained binary64: 0.003236345679012345828/25Exact fraction: 28/25Retained binary64: 1.121/5Exact fraction: 1/5Retained binary64: 0.2
113113/128Exact fraction: 113/128Retained binary64: 0.88281251695/32Exact fraction: 1695/32Retained binary64: 52.96875241/128Exact fraction: 241/128Retained binary64: 1.882812558081/16384Exact fraction: 58081/16384Retained binary64: 3.544982910156256780/241Exact fraction: 6780/241Retained binary64: 28.1327800829875590/241Exact fraction: 90/241Retained binary64: 0.3734439834024896316384/290405Exact fraction: 16384/290405Retained binary64: 0.0564177614021797216384/290405Exact fraction: 16384/290405Retained binary64: 0.05641776140217972268435456/84335064025Exact fraction: 268435456/84335064025Retained binary64: 0.00318296380163327931356/1205Exact fraction: 1356/1205Retained binary64: 1.1253112033195021/5Exact fraction: 1/5Retained binary64: 0.2
11457/64Exact fraction: 57/64Retained binary64: 0.890625855/16Exact fraction: 855/16Retained binary64: 53.4375121/64Exact fraction: 121/64Retained binary64: 1.89062514641/4096Exact fraction: 14641/4096Retained binary64: 3.5744628906253420/121Exact fraction: 3420/121Retained binary64: 28.26446280991735442/121Exact fraction: 42/121Retained binary64: 0.347107438016528944096/73205Exact fraction: 4096/73205Retained binary64: 0.055952462263506594096/73205Exact fraction: 4096/73205Retained binary64: 0.0559524622635065916777216/5358972025Exact fraction: 16777216/5358972025Retained binary64: 0.003130678033349129684/605Exact fraction: 684/605Retained binary64: 1.13057851239669431/5Exact fraction: 1/5Retained binary64: 0.2
115115/128Exact fraction: 115/128Retained binary64: 0.89843751725/32Exact fraction: 1725/32Retained binary64: 53.90625243/128Exact fraction: 243/128Retained binary64: 1.898437559049/16384Exact fraction: 59049/16384Retained binary64: 3.604064941406252300/81Exact fraction: 2300/81Retained binary64: 28.39506172839506326/81Exact fraction: 26/81Retained binary64: 0.3209876543209876416384/295245Exact fraction: 16384/295245Retained binary64: 0.0554928957306643616384/295245Exact fraction: 16384/295245Retained binary64: 0.05549289573066436268435456/87169610025Exact fraction: 268435456/87169610025Retained binary64: 0.00307946147657438792/81Exact fraction: 92/81Retained binary64: 1.13580246913580241/5Exact fraction: 1/5Retained binary64: 0.2
11629/32Exact fraction: 29/32Retained binary64: 0.90625435/8Exact fraction: 435/8Retained binary64: 54.37561/32Exact fraction: 61/32Retained binary64: 1.906253721/1024Exact fraction: 3721/1024Retained binary64: 3.63378906251740/61Exact fraction: 1740/61Retained binary64: 28.52459016393442718/61Exact fraction: 18/61Retained binary64: 0.295081967213114751024/18605Exact fraction: 1024/18605Retained binary64: 0.055038968019349641024/18605Exact fraction: 1024/18605Retained binary64: 0.055038968019349641048576/346146025Exact fraction: 1048576/346146025Retained binary64: 0.0030292880006349923348/305Exact fraction: 348/305Retained binary64: 1.1409836065573771/5Exact fraction: 1/5Retained binary64: 0.2
117117/128Exact fraction: 117/128Retained binary64: 0.91406251755/32Exact fraction: 1755/32Retained binary64: 54.84375245/128Exact fraction: 245/128Retained binary64: 1.914062560025/16384Exact fraction: 60025/16384Retained binary64: 3.663635253906251404/49Exact fraction: 1404/49Retained binary64: 28.65306122448979766/245Exact fraction: 66/245Retained binary64: 0.269387755102040816384/300125Exact fraction: 16384/300125Retained binary64: 0.0545905872553102916384/300125Exact fraction: 16384/300125Retained binary64: 0.05459058725531029268435456/90075015625Exact fraction: 268435456/90075015625Retained binary64: 0.0029801322168796461404/1225Exact fraction: 1404/1225Retained binary64: 1.14612244897959181/5Exact fraction: 1/5Retained binary64: 0.2
11859/64Exact fraction: 59/64Retained binary64: 0.921875885/16Exact fraction: 885/16Retained binary64: 55.3125123/64Exact fraction: 123/64Retained binary64: 1.92187515129/4096Exact fraction: 15129/4096Retained binary64: 3.6936035156251180/41Exact fraction: 1180/41Retained binary64: 28.7804878048780510/41Exact fraction: 10/41Retained binary64: 0.243902439024390244096/75645Exact fraction: 4096/75645Retained binary64: 0.054147663427853794096/75645Exact fraction: 4096/75645Retained binary64: 0.0541476634278537916777216/5722166025Exact fraction: 16777216/5722166025Retained binary64: 0.002931969454696135236/205Exact fraction: 236/205Retained binary64: 1.15121951219512191/5Exact fraction: 1/5Retained binary64: 0.2
119119/128Exact fraction: 119/128Retained binary64: 0.92968751785/32Exact fraction: 1785/32Retained binary64: 55.78125247/128Exact fraction: 247/128Retained binary64: 1.929687561009/16384Exact fraction: 61009/16384Retained binary64: 3.723693847656257140/247Exact fraction: 7140/247Retained binary64: 28.9068825910931254/247Exact fraction: 54/247Retained binary64: 0.2186234817813765316384/305045Exact fraction: 16384/305045Retained binary64: 0.0537101083446704616384/305045Exact fraction: 16384/305045Retained binary64: 0.05371010834467046268435456/93052452025Exact fraction: 268435456/93052452025Retained binary64: 0.0028847757383962391428/1235Exact fraction: 1428/1235Retained binary64: 1.15627530364372481/5Exact fraction: 1/5Retained binary64: 0.2
12015/16Exact fraction: 15/16Retained binary64: 0.9375225/4Exact fraction: 225/4Retained binary64: 56.2531/16Exact fraction: 31/16Retained binary64: 1.9375961/256Exact fraction: 961/256Retained binary64: 3.75390625900/31Exact fraction: 900/31Retained binary64: 29.0322580645161286/31Exact fraction: 6/31Retained binary64: 0.1935483870967742256/4805Exact fraction: 256/4805Retained binary64: 0.05327783558792924256/4805Exact fraction: 256/4805Retained binary64: 0.0532778355879292465536/23088025Exact fraction: 65536/23088025Retained binary64: 0.002838527764934419636/31Exact fraction: 36/31Retained binary64: 1.16129032258064521/5Exact fraction: 1/5Retained binary64: 0.2
121121/128Exact fraction: 121/128Retained binary64: 0.94531251815/32Exact fraction: 1815/32Retained binary64: 56.71875249/128Exact fraction: 249/128Retained binary64: 1.945312562001/16384Exact fraction: 62001/16384Retained binary64: 3.784240722656252420/83Exact fraction: 2420/83Retained binary64: 29.15662650602409714/83Exact fraction: 14/83Retained binary64: 0.168674698795180716384/310005Exact fraction: 16384/310005Retained binary64: 0.052850760471605316384/310005Exact fraction: 16384/310005Retained binary64: 0.0528507604716053268435456/96103100025Exact fraction: 268435456/96103100025Retained binary64: 0.0027932028824269968484/415Exact fraction: 484/415Retained binary64: 1.16626506024096391/5Exact fraction: 1/5Retained binary64: 0.2
12261/64Exact fraction: 61/64Retained binary64: 0.953125915/16Exact fraction: 915/16Retained binary64: 57.1875125/64Exact fraction: 125/64Retained binary64: 1.95312515625/4096Exact fraction: 15625/4096Retained binary64: 3.814697265625732/25Exact fraction: 732/25Retained binary64: 29.2818/125Exact fraction: 18/125Retained binary64: 0.1444096/78125Exact fraction: 4096/78125Retained binary64: 0.05242884096/78125Exact fraction: 4096/78125Retained binary64: 0.052428816777216/6103515625Exact fraction: 16777216/6103515625Retained binary64: 0.00274877906944732/625Exact fraction: 732/625Retained binary64: 1.17121/5Exact fraction: 1/5Retained binary64: 0.2
123123/128Exact fraction: 123/128Retained binary64: 0.96093751845/32Exact fraction: 1845/32Retained binary64: 57.65625251/128Exact fraction: 251/128Retained binary64: 1.960937563001/16384Exact fraction: 63001/16384Retained binary64: 3.845275878906257380/251Exact fraction: 7380/251Retained binary64: 29.4023904382470130/251Exact fraction: 30/251Retained binary64: 0.1195219123505976116384/315005Exact fraction: 16384/315005Retained binary64: 0.05201187282741543516384/315005Exact fraction: 16384/315005Retained binary64: 0.052011872827415435268435456/99228150025Exact fraction: 268435456/99228150025Retained binary64: 0.00270523491501523631476/1255Exact fraction: 1476/1255Retained binary64: 1.17609561752988051/5Exact fraction: 1/5Retained binary64: 0.2
12431/32Exact fraction: 31/32Retained binary64: 0.96875465/8Exact fraction: 465/8Retained binary64: 58.12563/32Exact fraction: 63/32Retained binary64: 1.968753969/1024Exact fraction: 3969/1024Retained binary64: 3.8759765625620/21Exact fraction: 620/21Retained binary64: 29.5238095238095262/21Exact fraction: 2/21Retained binary64: 0.095238095238095231024/19845Exact fraction: 1024/19845Retained binary64: 0.051599899218946841024/19845Exact fraction: 1024/19845Retained binary64: 0.051599899218946841048576/393824025Exact fraction: 1048576/393824025Retained binary64: 0.0026625495994054703124/105Exact fraction: 124/105Retained binary64: 1.1809523809523811/5Exact fraction: 1/5Retained binary64: 0.2
125125/128Exact fraction: 125/128Retained binary64: 0.97656251875/32Exact fraction: 1875/32Retained binary64: 58.59375253/128Exact fraction: 253/128Retained binary64: 1.976562564009/16384Exact fraction: 64009/16384Retained binary64: 3.906799316406257500/253Exact fraction: 7500/253Retained binary64: 29.6442687747035618/253Exact fraction: 18/253Retained binary64: 0.0711462450592885416384/320045Exact fraction: 16384/320045Retained binary64: 0.0511928010123576416384/320045Exact fraction: 16384/320045Retained binary64: 0.05119280101235764268435456/102428802025Exact fraction: 268435456/102428802025Retained binary64: 0.002620702875490845300/253Exact fraction: 300/253Retained binary64: 1.18577075098814231/5Exact fraction: 1/5Retained binary64: 0.2
12663/64Exact fraction: 63/64Retained binary64: 0.984375945/16Exact fraction: 945/16Retained binary64: 59.0625127/64Exact fraction: 127/64Retained binary64: 1.98437516129/4096Exact fraction: 16129/4096Retained binary64: 3.9377441406253780/127Exact fraction: 3780/127Retained binary64: 29.7637795275590566/127Exact fraction: 6/127Retained binary64: 0.0472440944881889764096/80645Exact fraction: 4096/80645Retained binary64: 0.0507905015810031644096/80645Exact fraction: 4096/80645Retained binary64: 0.05079050158100316416777216/6503616025Exact fraction: 16777216/6503616025Retained binary64: 0.0025796750508498847756/635Exact fraction: 756/635Retained binary64: 1.19055118110236211/5Exact fraction: 1/5Retained binary64: 0.2
127127/128Exact fraction: 127/128Retained binary64: 0.99218751905/32Exact fraction: 1905/32Retained binary64: 59.53125255/128Exact fraction: 255/128Retained binary64: 1.992187565025/16384Exact fraction: 65025/16384Retained binary64: 3.96881103515625508/17Exact fraction: 508/17Retained binary64: 29.882352941176472/85Exact fraction: 2/85Retained binary64: 0.02352941176470588216384/325125Exact fraction: 16384/325125Retained binary64: 0.0503929257977700916384/325125Exact fraction: 16384/325125Retained binary64: 0.05039292579777009268435456/105706265625Exact fraction: 268435456/105706265625Retained binary64: 0.002539446970459562508/425Exact fraction: 508/425Retained binary64: 1.19529411764705881/5Exact fraction: 1/5Retained binary64: 0.2
1281Exact fraction: 1Retained binary64: 1.060Exact fraction: 60Retained binary64: 60.02Exact fraction: 2Retained binary64: 2.04Exact fraction: 4Retained binary64: 4.030Exact fraction: 30Retained binary64: 30.00Exact fraction: 0Retained binary64: 0.01/20Exact fraction: 1/20Retained binary64: 0.051/20Exact fraction: 1/20Retained binary64: 0.051/400Exact fraction: 1/400Retained binary64: 0.00256/5Exact fraction: 6/5Retained binary64: 1.21/5Exact fraction: 1/5Retained binary64: 0.2
All three exact reduced-wire controls (3 rows)

Exact results of the stated quasi-one-dimensional cross-section-averaged model only; transverse/end corrections and a full three-dimensional current field have not been computed. s=x/(60L₀), radius=L₀(1+s), positive constant resistivity.

All three exact reduced-wire controls
Rows = x/(60L₀)X = x/L₀Radius/L₀Cross-section area/A₀Accumulated R(s)/R₀V/V₀Cross-section average axial J/J₀Average axial E/E₀Reduced power density/p₀Accumulated reduced power/P₀Conserved I/I₀
00Exact fraction: 0Retained binary64: 0.00Exact fraction: 0Retained binary64: 0.01Exact fraction: 1Retained binary64: 1.01Exact fraction: 1Retained binary64: 1.00Exact fraction: 0Retained binary64: 0.06Exact fraction: 6Retained binary64: 6.01/5Exact fraction: 1/5Retained binary64: 0.21/5Exact fraction: 1/5Retained binary64: 0.21/25Exact fraction: 1/25Retained binary64: 0.040Exact fraction: 0Retained binary64: 0.01/5Exact fraction: 1/5Retained binary64: 0.2
11/2Exact fraction: 1/2Retained binary64: 0.530Exact fraction: 30Retained binary64: 30.03/2Exact fraction: 3/2Retained binary64: 1.59/4Exact fraction: 9/4Retained binary64: 2.2520Exact fraction: 20Retained binary64: 20.02Exact fraction: 2Retained binary64: 2.04/45Exact fraction: 4/45Retained binary64: 0.088888888888888894/45Exact fraction: 4/45Retained binary64: 0.0888888888888888916/2025Exact fraction: 16/2025Retained binary64: 0.0079012345679012354/5Exact fraction: 4/5Retained binary64: 0.81/5Exact fraction: 1/5Retained binary64: 0.2
21Exact fraction: 1Retained binary64: 1.060Exact fraction: 60Retained binary64: 60.02Exact fraction: 2Retained binary64: 2.04Exact fraction: 4Retained binary64: 4.030Exact fraction: 30Retained binary64: 30.00Exact fraction: 0Retained binary64: 0.01/20Exact fraction: 1/20Retained binary64: 0.051/20Exact fraction: 1/20Retained binary64: 0.051/400Exact fraction: 1/400Retained binary64: 0.00256/5Exact fraction: 6/5Retained binary64: 1.21/5Exact fraction: 1/5Retained binary64: 0.2
taperedWire: source scales, exact constants and scope flags (11 rows)

Exact results of the stated quasi-one-dimensional cross-section-averaged model only; transverse/end corrections and a full three-dimensional current field have not been computed. s=x/(60L₀), radius=L₀(1+s), positive constant resistivity.

taperedWire: source scales, exact constants and scope flags
Quantity/source conditionAdmitted value
total R Over R030Exact fraction: 30Retained binary64: 30.0
Conserved I/I₀1/5Exact fraction: 1/5Retained binary64: 0.2
terminal Drop Over V06Exact fraction: 6Retained binary64: 6.0
terminal Power Over P06/5Exact fraction: 6/5Retained binary64: 1.2
integrated Reduced Power Over P06/5Exact fraction: 6/5Retained binary64: 1.2
mean Area Over A07/3Exact fraction: 7/3Retained binary64: 2.3333333333333335
wrong Mean Area R Over R0180/7Exact fraction: 180/7Retained binary64: 25.714285714285715
wrong Mean Area Relative Error-1/7Exact fraction: -1/7Retained binary64: -0.14285714285714285
radius Slope1/60Exact fraction: 1/60Retained binary64: 0.016666666666666666
full Three Dimensional Field Claimedfalse
quantitative Reduced Vs Full3 D Error Boundfalse

Focus a labelled graph or table region and use arrow keys, or scroll sideways. Fixed-size readable diagrams stay within their own regions. Open the complete data disclosures for exact values, retained binary64 numbers and precision qualifications.

Exact controls of the reduced tapered-wire model
s = x/(60L₀)A/(πL₀²)V/V₀Average J/J₀ = average axial E/E₀R(s)/R₀
0161/50
1/29/424/4520
1401/2030
09

An average area does not average the resistance

The resistance integral weights 1/A, not A. For the selected taper the mean area is (7/3)πL₀². Substituting that mean into η₀ℓ/A would give R_wrong/R₀ = 60/(7/3) = 180/7, rather than the exact reduced-model value 30.

The relative error of that mean-area substitution is [(180/7) − 30]/30 = −1/7: it underestimates the reduced-model resistance. This is an exact comparison between two mathematical reductions, not an estimate of the neglected three-dimensional end or transverse corrections.

More generally, several elements may be combined in series when the same current passes through them, and their voltage drops add. Parallel paths share the specified terminal voltage and their currents add. Their effective relations still depend on steady charge constraints and on the absence of significant unmodeled couplings.

A nonuniform resistivity, temperature or cross-section belongs inside the integral under its relevant assumptions. Replacing all nonuniform features by one average number before checking the physical weighting can erase the very feature that controls the response.

10

Follow the electrical energy transfer

The instantaneous rate at which an electric field does work on moving charge per unit volume is J·E. In a passive local isotropic Ohmic model this is σ|E|² = η|J|² ≥ 0. For a steady wire, the corresponding total rate is P = IΔV = I²R = (ΔV)²/R, using the same physical state and passive sign convention.

In the resistive model, carriers receive energy from the field and transfer it through scattering and other interactions to the material and its surroundings. Their average drift kinetic energy need not grow in a steady state. The continuing energy delivery is not explained by imagining each electron using up a fixed store of kinetic energy while crossing the wire.

The electromagnetic field and source maintain the energy flow. A full account of how energy enters a particular volume can use the electromagnetic energy-balance law; the later Maxwell treatment develops that field description. The equality of terminal and local power here is a check on the selected steady model, not a claim of instantaneous propagation throughout a circuit.

Outside a passive Ohmic regime, J·E can contribute to reversible storage, carrier energy, light or other forms of energy as well as heat. Active devices can return electrical energy. Do not relabel every field-to-charge work term as irreversible Joule heat without the accompanying material assumptions.

11

Derive a momentum-relaxation model

A phenomenological Drude model describes the mean drift velocity of negative carriers of charge −q₀ and mass m by m dv/dt = −q₀E − mv/τ_m. The positive τ_m is a momentum-relaxation time in this model. The damping term describes an average loss of directed momentum, not a deterministic rule that every particle stops after exactly τ_m.

For a uniform constant field and a steady drift, v_d = −(q₀τ_m/m)E. If the carrier density is n, then J = −nq₀v_d = (nq₀²τ_m/m)E. Thus σ₀ = nq₀²τ_m/m is positive, and the drift-mobility magnitude is q₀τ_m/m. The two minus signs associated with negative carriers are both needed.

If the field E₀x̂ is switched on at t = 0 with zero initial average drift, the solution is v_x = −v_∞[1 − e^(−t/τ_m)], where v_∞ = q₀τ_mE₀/m is a positive speed scale. The current rises as J_x/J_steady = 1 − e^(−T), with T = t/τ_m. The negative velocity and positive conventional current have separate labels.

The model neglects many details of band structure, collisions, interactions and spatial nonlocality. In a more refined transport description, the momentum-relaxation time need not equal every collision or single-particle lifetime. The equation is useful because its assumptions and response can be tested, not because its one parameter represents every microscopic process.

12

Drift speed is not a signal speed

The mean drift can be small while individual carriers have much larger random thermal or Fermi velocities. Those microscopic velocities largely cancel in the average current. Dividing a wire’s length by its drift speed estimates a carrier transport time under specified conditions; it does not generally estimate when a change at one terminal affects the other.

A circuit’s field adjustment propagates through the electromagnetic environment, with geometry, material response and boundary conditions determining its speed and transient behavior. A finite propagation model is needed when that adjustment cannot be neglected. Neither an instantaneous lumped circuit approximation nor the Drude local step equation proves that signals travel infinitely fast.

Momentum transfer to impurities, lattice vibrations and other excitations can relax directed current. An ideal periodic lattice and a material’s band structure require quantum reasoning, while real scattering processes establish finite resistive response in ordinary regimes. Average drift is not simply a sequence of classical electrons bouncing between stationary nuclei.

The earlier transport-theory lesson supplies the broader kinetic connection: a distribution of carrier states, its driving terms and its collision operator determine a transport coefficient under appropriate scale separation. A one-time-constant Drude equation is a reduced response model within that larger problem.

13

Charge relaxation is another model

Now choose a separate homogeneous electroquasistatic continuum with constant positive conductivity σ, constant scalar permittivity ε and an instantaneous local Ohmic law J = σE. Assume no additional imposed volume-current source or diffusion term. Gauss’s law for this uniform linear material is ε∇·E = ρ_free. Here J is the free conduction current, and its continuity equation uses the free charge density. The specified ε describes the remaining background polarization; the same mobile-carrier response must not also be counted a second time inside ε.

Taking the divergence of J and using charge continuity gives ∂ρ_free/∂t = −σρ_free/ε. Hence ρ_free(t) = ρ_free(0)e^(−t/τ_q), where τ_q = ε/σ is the bulk charge-relaxation time of this continuum approximation. It is not the momentum time τ_m by definition.

If a selected model volume initially holds Q_initial and its outward current deposits the transported charge on the bounding region, the bulk fraction is e^(−S) and the accumulated boundary fraction is 1 − e^(−S), with S = t/τ_q. Their sum is one. Bulk charge relaxes by transfer; it is not destroyed.

The current-onset panel uses T = t/τ_m, while the separate bulk-relaxation panel uses S = t/τ_q. Equal numerical positions on these normalized clocks do not imply equal physical times. The instantaneous Ohmic assumption must itself remain valid on the time scale being discussed; the next return path explains why that qualification matters.

14

Return path: keep inertia when time scales conflict

It would be inconsistent to take a very short ε/σ, apply an instantaneous Ohmic law at that time scale and ignore the finite carrier momentum response used to derive σ. If microscopic inertia, spatial transport or field propagation matter, the first-order bulk exponential is no longer a universal prediction for a real metal.

For a homogeneous longitudinal Drude model, retain τ_m ∂J/∂t + J = σ₀E. Taking a divergence and using ∇·J = −∂ρ_free/∂t and ε∇·E = ρ_free gives τ_m ∂²ρ_free/∂t² + ∂ρ_free/∂t + (σ₀/ε)ρ_free = 0. The second derivative is the term discarded by the instantaneous Ohmic reduction.

This equation can contain damped inertial or plasma-like longitudinal behavior rather than a single universal exponential. When a suitable slow regime separates momentum relaxation from the bulk evolution, the first-order Ohmic relation can emerge as an approximation. The appropriate scale separation and initial data must be checked rather than inferred from dimensional analysis alone.

We do not solve the complete material or electromagnetic transient here. The purpose is to identify which term and assumption change when two apparently familiar time constants conflict. A reduced model remains useful if it carries its validity conditions into every interpretation of its graph.

Two relaxation models, two physical clocks

Distinguish the negative-carrier drift and positive current onset from a separate bulk-to-boundary charge transfer. Each panel has its own normalized clock; matching positions do not assert matching physical times.

Momentum relaxation: negative drift, positive current

  • Signed negative-carrier drift vₓ/v∞ = −(1−e^(−T))
  • Positive conventional current J/J∞ = 1−e^(−T)
Momentum relaxation: negative drift, positive currentAll193 admitted Drude samples on T=t/τ_m, plus four exact logarithmic controls. Negative drift and positive conventional current approach−1 and+1 respectively.0123456-1-0.500.51Drude clock only: T = t/τ_m; no bulk-charge clock identifiedSigned vₓ/v∞; conventional J/J∞T=t/τ_m (Drude momentum clock)

Separate bulk-to-boundary free-charge transfer

  • Bulk free charge/Q_initial = e^(−S)
  • Boundary free charge/Q_initial = 1−e^(−S)
  • Conserved total free charge/Q_initial = 1
Separate bulk-to-boundary free-charge transferAll193 admitted instantaneous-Ohmic bulk samples on S=t/τ_q, plus four exact controls. Bulk and boundary fractions sum to the retained constant1. No equal-time overlay or bulk velocity is implied.012345600.250.50.751Separate clock S = t/τ_q; charge moves to the boundary, not awayBulk, boundary and conserved total Q/Q_initialS=t/τ_q, τ_q=ε/σ (separate continuum clock)
  • Separate physical models/clocks: Drude T=t/τ_m and homogeneous instantaneous-Ohmic bulk S=t/τ_q with τ_q=ε/σ. ε is background permittivity and σ/free current describe the mobile response without duplication. Exact log controls use retained binary64 x-coordinates. The bulk control schema also retains an algebraic signedDrift value; it is not used as a bulk velocity or an equal-clock assertion.
  • Drude step: prescribed uniform field+E₀x̂, negative carriers, zero initial average drift. Signed drift vₓ/v∞=−(1−exp(−T)) and conventional current J/J∞=1−exp(−T), with positive v∞=q₀τ_mE₀/m. σ₀=nq₀²τ_m/m.
  • Separate bulk relaxation: compatible free charge/free current, homogeneous positive instantaneous σ and background ε, no imposed volume source or diffusion, electroquasistatic validity required. Bulk exp(−S) transfers to boundary1−exp(−S); the total is1, so charge is not destroyed.
  • The retained time-curve rise values use stable −expm1(−T) or −expm1(−S); approximate data keep their original12-significant-digit displays and full stored binary64 values. Exact log controls use their own retained approximate horizontal coordinates. The renderer never recomputes an exponential.
  • Matching horizontal numbers do not identify τ_m with τ_q or physical times. Retaining Drude inertia gives τ_mρ̈+ρ̇+(σ₀/ε)ρ=0. The instantaneous-Ohmic exponential is a qualified slow-limit model, not a universal microscopic metal-decay prediction.
  • Every plotted value is read directly from the admitted model. Complete exact fractions,12-significant-digit approximate displays, original function-specific arithmetic qualifications and retained binary64 values remain in the data disclosures. No empirical material constant or numerical solver is introduced.

Complete labelled data equivalents retain every admitted scalar, record, source condition and null. Original manuscript tables remain separate.

All 193 retained Drude step samples (193 rows)

Separate physical models/clocks: Drude T=t/τ_m and homogeneous instantaneous-Ohmic bulk S=t/τ_q with τ_q=ε/σ. ε is background permittivity and σ/free current describe the mobile response without duplication. Exact log controls use retained binary64 x-coordinates. The bulk control schema also retains an algebraic signedDrift value; it is not used as a bulk velocity or an equal-clock assertion.

All 193 retained Drude step samples
RowT = t/τ_m (Drude clock)Signed vₓ/v∞ magnitudeConventional J/J∞Remaining fraction to steady current
00Exact fraction: 0Retained binary64: 0.00Retained binary64: 0.0binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0Retained binary64: 0.0binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Retained binary64: 1.0binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
11/32Exact fraction: 1/32Retained binary64: 0.03125-0.0307667655237Retained binary64: -0.03076676552365592binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0307667655237Retained binary64: 0.03076676552365592binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.969233234476Retained binary64: 0.9692332344763441binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
21/16Exact fraction: 1/16Retained binary64: 0.0625-0.0605869371865Retained binary64: -0.06058693718652421binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0605869371865Retained binary64: 0.06058693718652421binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.939413062813Retained binary64: 0.9394130628134758binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
33/32Exact fraction: 3/32Retained binary64: 0.09375-0.08948963862Retained binary64: -0.08948963861996587binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.08948963862Retained binary64: 0.08948963861996587binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.91051036138Retained binary64: 0.9105103613800342binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
41/8Exact fraction: 1/8Retained binary64: 0.125-0.117503097415Retained binary64: -0.11750309741540459binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.117503097415Retained binary64: 0.11750309741540459binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.882496902585Retained binary64: 0.8824969025845955binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
55/32Exact fraction: 5/32Retained binary64: 0.15625-0.144654672693Retained binary64: -0.14465467269257745binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.144654672693Retained binary64: 0.14465467269257745binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.855345327307Retained binary64: 0.8553453273074225binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
63/16Exact fraction: 3/16Retained binary64: 0.1875-0.17097088182Retained binary64: -0.17097088181959966binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.17097088182Retained binary64: 0.17097088181959966binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.82902911818Retained binary64: 0.8290291181804004binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
77/32Exact fraction: 7/32Retained binary64: 0.21875-0.196477426311Retained binary64: -0.19647742631093928binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.196477426311Retained binary64: 0.19647742631093928binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.803522573689Retained binary64: 0.8035225736890608binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
81/4Exact fraction: 1/4Retained binary64: 0.25-0.221199216929Retained binary64: -0.22119921692859512binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.221199216929Retained binary64: 0.22119921692859512binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.778800783071Retained binary64: 0.7788007830714049binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
99/32Exact fraction: 9/32Retained binary64: 0.28125-0.245160398011Retained binary64: -0.24516039801099265binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.245160398011Retained binary64: 0.24516039801099265binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.754839601989Retained binary64: 0.7548396019890073binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
105/16Exact fraction: 5/16Retained binary64: 0.3125-0.268384371053Retained binary64: -0.2683843710533582binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.268384371053Retained binary64: 0.2683843710533582binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.731615628947Retained binary64: 0.7316156289466418binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
1111/32Exact fraction: 11/32Retained binary64: 0.34375-0.290893817563Retained binary64: -0.2908938175626015binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.290893817563Retained binary64: 0.2908938175626015binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.709106182437Retained binary64: 0.7091061824373984binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
123/8Exact fraction: 3/8Retained binary64: 0.375-0.312710721209Retained binary64: -0.3127107212090278binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.312710721209Retained binary64: 0.3127107212090278binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.687289278791Retained binary64: 0.6872892787909722binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
1313/32Exact fraction: 13/32Retained binary64: 0.40625-0.333856389297Retained binary64: -0.33385638929651223binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.333856389297Retained binary64: 0.33385638929651223binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.666143610703Retained binary64: 0.6661436107034878binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
147/16Exact fraction: 7/16Retained binary64: 0.4375-0.354351473572Retained binary64: -0.354351473572108binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.354351473572Retained binary64: 0.354351473572108binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.645648526428Retained binary64: 0.645648526427892binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
1515/32Exact fraction: 15/32Retained binary64: 0.46875-0.374215990395Retained binary64: -0.37421599039540887binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.374215990395Retained binary64: 0.37421599039540887binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.625784009605Retained binary64: 0.6257840096045911binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
161/2Exact fraction: 1/2Retained binary64: 0.5-0.393469340287Retained binary64: -0.39346934028736663binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.393469340287Retained binary64: 0.39346934028736663binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.606530659713Retained binary64: 0.6065306597126334binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
1717/32Exact fraction: 17/32Retained binary64: 0.53125-0.412130326878Retained binary64: -0.4121303268776535binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.412130326878Retained binary64: 0.4121303268776535binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.587869673122Retained binary64: 0.5878696731223465binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
189/16Exact fraction: 9/16Retained binary64: 0.5625-0.430217175269Retained binary64: -0.430217175269077binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.430217175269Retained binary64: 0.430217175269077binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.569782824731Retained binary64: 0.569782824730923binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
1919/32Exact fraction: 19/32Retained binary64: 0.59375-0.447747549837Retained binary64: -0.4477475498369796binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.447747549837Retained binary64: 0.4477475498369796binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.552252450163Retained binary64: 0.5522524501630204binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
205/8Exact fraction: 5/8Retained binary64: 0.625-0.464738571481Retained binary64: -0.4647385714810098binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.464738571481Retained binary64: 0.4647385714810098binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.535261428519Retained binary64: 0.5352614285189903binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
2121/32Exact fraction: 21/32Retained binary64: 0.65625-0.481206834346Retained binary64: -0.48120683434611067binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.481206834346Retained binary64: 0.48120683434611067binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.518793165654Retained binary64: 0.5187931656538893binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
2211/16Exact fraction: 11/16Retained binary64: 0.6875-0.497168422029Retained binary64: -0.49716842202905903binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.497168422029Retained binary64: 0.49716842202905903binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.502831577971Retained binary64: 0.5028315779709409binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
2323/32Exact fraction: 23/32Retained binary64: 0.71875-0.512638923286Retained binary64: -0.5126389232863808binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.512638923286Retained binary64: 0.5126389232863808binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.487361076714Retained binary64: 0.4873610767136191binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
243/4Exact fraction: 3/4Retained binary64: 0.75-0.527633447259Retained binary64: -0.5276334472589853binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.527633447259Retained binary64: 0.5276334472589853binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.472366552741Retained binary64: 0.4723665527410147binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
2525/32Exact fraction: 25/32Retained binary64: 0.78125-0.542166638228Retained binary64: -0.5421666382283857binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.542166638228Retained binary64: 0.5421666382283857binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.457833361772Retained binary64: 0.45783336177161427binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
2613/16Exact fraction: 13/16Retained binary64: 0.8125-0.556252689919Retained binary64: -0.5562526899189202binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.556252689919Retained binary64: 0.5562526899189202binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.443747310081Retained binary64: 0.44374731008107987binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
2727/32Exact fraction: 27/32Retained binary64: 0.84375-0.56990535936Retained binary64: -0.5699053593599377binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.56990535936Retained binary64: 0.5699053593599377binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.43009464064Retained binary64: 0.43009464064006225binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
287/8Exact fraction: 7/8Retained binary64: 0.875-0.583137980321Retained binary64: -0.5831379803214916binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.583137980321Retained binary64: 0.5831379803214916binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.416862019679Retained binary64: 0.4168620196785084binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
2929/32Exact fraction: 29/32Retained binary64: 0.90625-0.595963476337Retained binary64: -0.5959634763366579binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.595963476337Retained binary64: 0.5959634763366579binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.404036523663Retained binary64: 0.4040365236633421binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
3015/16Exact fraction: 15/16Retained binary64: 0.9375-0.608394373323Retained binary64: -0.608394373323201binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.608394373323Retained binary64: 0.608394373323201binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.391605626677Retained binary64: 0.391605626676799binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
3131/32Exact fraction: 31/32Retained binary64: 0.96875-0.620442811817Retained binary64: -0.6204428118169104binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.620442811817Retained binary64: 0.6204428118169104binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.379557188183Retained binary64: 0.3795571881830896binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
321Exact fraction: 1Retained binary64: 1.0-0.632120558829Retained binary64: -0.6321205588285577binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.632120558829Retained binary64: 0.6321205588285577binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.367879441171Retained binary64: 0.36787944117144233binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
3333/32Exact fraction: 33/32Retained binary64: 1.03125-0.643439019336Retained binary64: -0.6434390193360531binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.643439019336Retained binary64: 0.6434390193360531binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.356560980664Retained binary64: 0.356560980663947binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
3417/16Exact fraction: 17/16Retained binary64: 1.0625-0.654409247423Retained binary64: -0.6544092474230254binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.654409247423Retained binary64: 0.6544092474230254binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.345590752577Retained binary64: 0.3455907525769745binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
3535/32Exact fraction: 35/32Retained binary64: 1.09375-0.665041957075Retained binary64: -0.665041957074705binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.665041957075Retained binary64: 0.665041957074705binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.334958042925Retained binary64: 0.33495804292529496binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
369/8Exact fraction: 9/8Retained binary64: 1.125-0.675347532642Retained binary64: -0.6753475326416503binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.675347532642Retained binary64: 0.6753475326416503binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.324652467358Retained binary64: 0.32465246735834974binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
3737/32Exact fraction: 37/32Retained binary64: 1.15625-0.685336038982Retained binary64: -0.685336038981541binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.685336038982Retained binary64: 0.685336038981541binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.314663961018Retained binary64: 0.314663961018459binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
3819/16Exact fraction: 19/16Retained binary64: 1.1875-0.695017231289Retained binary64: -0.6950172312889407binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.695017231289Retained binary64: 0.6950172312889407binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.304982768711Retained binary64: 0.3049827687110593binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
3939/32Exact fraction: 39/32Retained binary64: 1.21875-0.704400564623Retained binary64: -0.7044005646226292binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.704400564623Retained binary64: 0.7044005646226292binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.295599435377Retained binary64: 0.29559943537737077binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
405/4Exact fraction: 5/4Retained binary64: 1.25-0.71349520314Retained binary64: -0.7134952031398099binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.71349520314Retained binary64: 0.7134952031398099binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.28650479686Retained binary64: 0.2865047968601901binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
4141/32Exact fraction: 41/32Retained binary64: 1.28125-0.722310029046Retained binary64: -0.7223100290462101binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.722310029046Retained binary64: 0.7223100290462101binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.277689970954Retained binary64: 0.27768997095378994binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
4221/16Exact fraction: 21/16Retained binary64: 1.3125-0.730853651271Retained binary64: -0.7308536512708161binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.730853651271Retained binary64: 0.7308536512708161binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.269146348729Retained binary64: 0.26914634872918386binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
4343/32Exact fraction: 43/32Retained binary64: 1.34375-0.739134413874Retained binary64: -0.739134413873715binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.739134413874Retained binary64: 0.739134413873715binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.260865586126Retained binary64: 0.26086558612628497binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
4411/8Exact fraction: 11/8Retained binary64: 1.375-0.747160404195Retained binary64: -0.7471604041952535binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.747160404195Retained binary64: 0.7471604041952535binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.252839595805Retained binary64: 0.25283959580474646binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
4545/32Exact fraction: 45/32Retained binary64: 1.40625-0.754939460754Retained binary64: -0.7549394607544742binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.754939460754Retained binary64: 0.7549394607544742binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.245060539246Retained binary64: 0.2450605392455259binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
4623/16Exact fraction: 23/16Retained binary64: 1.4375-0.762479180905Retained binary64: -0.7624791809045419binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.762479180905Retained binary64: 0.7624791809045419binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.237520819095Retained binary64: 0.23752081909545814binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
4747/32Exact fraction: 47/32Retained binary64: 1.46875-0.769786928253Retained binary64: -0.7697869282526385binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.769786928253Retained binary64: 0.7697869282526385binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.230213071747Retained binary64: 0.23021307174736147binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
483/2Exact fraction: 3/2Retained binary64: 1.5-0.776869839852Retained binary64: -0.7768698398515702binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.776869839852Retained binary64: 0.7768698398515702binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.223130160148Retained binary64: 0.22313016014842982binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
4949/32Exact fraction: 49/32Retained binary64: 1.53125-0.78373483317Retained binary64: -0.7837348331701127binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.78373483317Retained binary64: 0.7837348331701127binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.21626516683Retained binary64: 0.2162651668298873binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
5025/16Exact fraction: 25/16Retained binary64: 1.5625-0.790388612849Retained binary64: -0.7903886128489022binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.790388612849Retained binary64: 0.7903886128489022binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.209611387151Retained binary64: 0.2096113871510978binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
5151/32Exact fraction: 51/32Retained binary64: 1.59375-0.796837677248Retained binary64: -0.7968376772484682binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.796837677248Retained binary64: 0.7968376772484682binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.203162322752Retained binary64: 0.20316232275153173binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
5213/8Exact fraction: 13/8Retained binary64: 1.625-0.803088324796Retained binary64: -0.8030883247958059binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.803088324796Retained binary64: 0.8030883247958059binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.196911675204Retained binary64: 0.19691167520419406binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
5353/32Exact fraction: 53/32Retained binary64: 1.65625-0.809146660136Retained binary64: -0.8091466601356837binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.809146660136Retained binary64: 0.8091466601356837binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.190853339864Retained binary64: 0.19085333986431632binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
5427/16Exact fraction: 27/16Retained binary64: 1.6875-0.815018600093Retained binary64: -0.8150186000926958binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.815018600093Retained binary64: 0.8150186000926958binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.184981399907Retained binary64: 0.18498139990730428binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
5555/32Exact fraction: 55/32Retained binary64: 1.71875-0.82070987945Retained binary64: -0.8207098794498814binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.82070987945Retained binary64: 0.8207098794498814binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.17929012055Retained binary64: 0.17929012055011864binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
567/4Exact fraction: 7/4Retained binary64: 1.75-0.82622605655Retained binary64: -0.8262260565495548binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.82622605655Retained binary64: 0.8262260565495548binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.17377394345Retained binary64: 0.17377394345044514binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
5757/32Exact fraction: 57/32Retained binary64: 1.78125-0.831572518722Retained binary64: -0.8315725187218157binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.831572518722Retained binary64: 0.8315725187218157binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.168427481278Retained binary64: 0.16842748127818424binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
5829/16Exact fraction: 29/16Retained binary64: 1.8125-0.836754487546Retained binary64: -0.8367544875460415binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.836754487546Retained binary64: 0.8367544875460415binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.163245512454Retained binary64: 0.1632455124539584binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
5959/32Exact fraction: 59/32Retained binary64: 1.84375-0.841777023951Retained binary64: -0.8417770239505016binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.841777023951Retained binary64: 0.8417770239505016binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.158222976049Retained binary64: 0.15822297604949842binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
6015/8Exact fraction: 15/8Retained binary64: 1.875-0.846645033155Retained binary64: -0.8466450331550716binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.846645033155Retained binary64: 0.8466450331550716binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.153354966845Retained binary64: 0.15335496684492847binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
6161/32Exact fraction: 61/32Retained binary64: 1.90625-0.851363269462Retained binary64: -0.8513632694618775binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.851363269462Retained binary64: 0.8513632694618775binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.148636730538Retained binary64: 0.1486367305381225binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
6231/16Exact fraction: 31/16Retained binary64: 1.9375-0.855936340899Retained binary64: -0.8559363408985468binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.855936340899Retained binary64: 0.8559363408985468binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.144063659101Retained binary64: 0.14406365910145327binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
6363/32Exact fraction: 63/32Retained binary64: 1.96875-0.860368713719Retained binary64: -0.860368713718601binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.860368713719Retained binary64: 0.860368713718601binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.139631286281Retained binary64: 0.13963128628139898binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
642Exact fraction: 2Retained binary64: 2.0-0.864664716763Retained binary64: -0.8646647167633873binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.864664716763Retained binary64: 0.8646647167633873binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.135335283237Retained binary64: 0.1353352832366127binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
6565/32Exact fraction: 65/32Retained binary64: 2.03125-0.86882854569Retained binary64: -0.8688285456898057binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.86882854569Retained binary64: 0.8688285456898057binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.13117145431Retained binary64: 0.13117145431019428binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
6633/16Exact fraction: 33/16Retained binary64: 2.0625-0.872864267068Retained binary64: -0.8728642670679644binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.872864267068Retained binary64: 0.8728642670679644binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.127135732932Retained binary64: 0.1271357329320356binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
6767/32Exact fraction: 67/32Retained binary64: 2.09375-0.876775822353Retained binary64: -0.8767758223527625binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.876775822353Retained binary64: 0.8767758223527625binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.123224177647Retained binary64: 0.12322417764723749binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
6817/8Exact fraction: 17/8Retained binary64: 2.125-0.880567031733Retained binary64: -0.8805670317332803binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.880567031733Retained binary64: 0.8805670317332803binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.119432968267Retained binary64: 0.11943296826671962binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
6969/32Exact fraction: 69/32Retained binary64: 2.15625-0.884241597864Retained binary64: -0.8842415978637368binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.884241597864Retained binary64: 0.8842415978637368binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.115758402136Retained binary64: 0.11575840213626322binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
7035/16Exact fraction: 35/16Retained binary64: 2.1875-0.88780310948Retained binary64: -0.8878031094796562binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.88780310948Retained binary64: 0.8878031094796562binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.11219689052Retained binary64: 0.11219689052034373binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
7171/32Exact fraction: 71/32Retained binary64: 2.21875-0.891255044903Retained binary64: -0.891255044902779binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.891255044903Retained binary64: 0.891255044902779binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.108744955097Retained binary64: 0.10874495509722103binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
729/4Exact fraction: 9/4Retained binary64: 2.25-0.894600775438Retained binary64: -0.8946007754381357binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.894600775438Retained binary64: 0.8946007754381357binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.105399224562Retained binary64: 0.10539922456186433binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
7373/32Exact fraction: 73/32Retained binary64: 2.28125-0.897843568667Retained binary64: -0.8978435686666058binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.897843568667Retained binary64: 0.8978435686666058binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.102156431333Retained binary64: 0.1021564313333943binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
7437/16Exact fraction: 37/16Retained binary64: 2.3125-0.900986591636Retained binary64: -0.9009865916361737binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.900986591636Retained binary64: 0.9009865916361737binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0990134083638Retained binary64: 0.0990134083638263binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
7575/32Exact fraction: 75/32Retained binary64: 2.34375-0.904032913955Retained binary64: -0.9040329139550015binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.904032913955Retained binary64: 0.9040329139550015binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.095967086045Retained binary64: 0.09596708604499847binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
7619/8Exact fraction: 19/8Retained binary64: 2.375-0.906985510789Retained binary64: -0.9069855107893365binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.906985510789Retained binary64: 0.9069855107893365binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0930144892107Retained binary64: 0.09301448921066349binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
7777/32Exact fraction: 77/32Retained binary64: 2.40625-0.909847265769Retained binary64: -0.9098472657691836binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.909847265769Retained binary64: 0.9098472657691836binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0901527342308Retained binary64: 0.09015273423081638binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
7839/16Exact fraction: 39/16Retained binary64: 2.4375-0.912620973805Retained binary64: -0.9126209738045796binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.912620973805Retained binary64: 0.9126209738045796binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0873790261954Retained binary64: 0.08737902619542039binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
7979/32Exact fraction: 79/32Retained binary64: 2.46875-0.915309343815Retained binary64: -0.9153093438152196binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.915309343815Retained binary64: 0.9153093438152196binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0846906561848Retained binary64: 0.0846906561847805binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
805/2Exact fraction: 5/2Retained binary64: 2.5-0.917915001376Retained binary64: -0.9179150013761012binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.917915001376Retained binary64: 0.9179150013761012binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0820849986239Retained binary64: 0.0820849986238988binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
8181/32Exact fraction: 81/32Retained binary64: 2.53125-0.920440491282Retained binary64: -0.9204404912817723binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.920440491282Retained binary64: 0.9204404912817723binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0795595087182Retained binary64: 0.07955950871822769binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
8241/16Exact fraction: 41/16Retained binary64: 2.5625-0.922888280032Retained binary64: -0.9228882800316833binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.922888280032Retained binary64: 0.9228882800316833binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0771117199683Retained binary64: 0.07711171996831671binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
8383/32Exact fraction: 83/32Retained binary64: 2.59375-0.925260758239Retained binary64: -0.9252607582390743binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.925260758239Retained binary64: 0.9252607582390743binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0747392417609Retained binary64: 0.0747392417609257binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
8421/8Exact fraction: 21/8Retained binary64: 2.625-0.927560242966Retained binary64: -0.9275602429657486binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.927560242966Retained binary64: 0.9275602429657486binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0724397570343Retained binary64: 0.07243975703425146binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
8585/32Exact fraction: 85/32Retained binary64: 2.65625-0.929788979985Retained binary64: -0.929788979985012binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.929788979985Retained binary64: 0.929788979985012binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.070211020015Retained binary64: 0.07021102001498804binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
8643/16Exact fraction: 43/16Retained binary64: 2.6875-0.931949145975Retained binary64: -0.9319491459749898binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.931949145975Retained binary64: 0.9319491459749898binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.068050854025Retained binary64: 0.0680508540250102binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
8787/32Exact fraction: 87/32Retained binary64: 2.71875-0.934042850644Retained binary64: -0.9340428506444618binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.934042850644Retained binary64: 0.9340428506444618binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0659571493555Retained binary64: 0.06595714935553817binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
8811/4Exact fraction: 11/4Retained binary64: 2.75-0.936072138793Retained binary64: -0.9360721387932924binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.936072138793Retained binary64: 0.9360721387932924binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0639278612067Retained binary64: 0.06392786120670757binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
8989/32Exact fraction: 89/32Retained binary64: 2.78125-0.938038992309Retained binary64: -0.938038992309468binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.938038992309Retained binary64: 0.938038992309468binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0619610076905Retained binary64: 0.061961007690531984binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
9045/16Exact fraction: 45/16Retained binary64: 2.8125-0.939945332105Retained binary64: -0.939945332104692binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.939945332105Retained binary64: 0.939945332104692binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0600546678953Retained binary64: 0.060054667895307945binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
9191/32Exact fraction: 91/32Retained binary64: 2.84375-0.94179301999Retained binary64: -0.9417930199904281binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.94179301999Retained binary64: 0.9417930199904281binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0582069800096Retained binary64: 0.058206980009571974binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
9223/8Exact fraction: 23/8Retained binary64: 2.875-0.943583860496Retained binary64: -0.9435838604962227binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.943583860496Retained binary64: 0.9435838604962227binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0564161395038Retained binary64: 0.05641613950377735binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
9393/32Exact fraction: 93/32Retained binary64: 2.90625-0.945319602632Retained binary64: -0.9453196026320853binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.945319602632Retained binary64: 0.9453196026320853binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0546803973679Retained binary64: 0.05468039736791477binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
9447/16Exact fraction: 47/16Retained binary64: 2.9375-0.947001941597Retained binary64: -0.9470019415966442binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.947001941597Retained binary64: 0.9470019415966442binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0529980584034Retained binary64: 0.0529980584033558binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
9595/32Exact fraction: 95/32Retained binary64: 2.96875-0.948632520433Retained binary64: -0.9486325204327493binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.948632520433Retained binary64: 0.9486325204327493binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0513674795673Retained binary64: 0.05136747956725073binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
963Exact fraction: 3Retained binary64: 3.0-0.950212931632Retained binary64: -0.950212931632136binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.950212931632Retained binary64: 0.950212931632136binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0497870683679Retained binary64: 0.049787068367863944binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
9797/32Exact fraction: 97/32Retained binary64: 3.03125-0.951744718691Retained binary64: -0.9517447186907203binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.951744718691Retained binary64: 0.9517447186907203binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0482552813093Retained binary64: 0.04825528130927965binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
9849/16Exact fraction: 49/16Retained binary64: 3.0625-0.953229377616Retained binary64: -0.953229377616041binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.953229377616Retained binary64: 0.953229377616041binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.046770622384Retained binary64: 0.04677062238395898binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
9999/32Exact fraction: 99/32Retained binary64: 3.09375-0.954668358388Retained binary64: -0.9546683583883238binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.954668358388Retained binary64: 0.9546683583883238binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0453316416117Retained binary64: 0.045331641611676264binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
10025/8Exact fraction: 25/8Retained binary64: 3.125-0.956063066377Retained binary64: -0.9560630663765926binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.956063066377Retained binary64: 0.9560630663765926binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0439369336234Retained binary64: 0.04393693362340742binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
101101/32Exact fraction: 101/32Retained binary64: 3.15625-0.957414863711Retained binary64: -0.9574148637112124binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.957414863711Retained binary64: 0.9574148637112124binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0425851362888Retained binary64: 0.04258513628878761binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
10251/16Exact fraction: 51/16Retained binary64: 3.1875-0.958725070614Retained binary64: -0.9587250706142024binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.958725070614Retained binary64: 0.9587250706142024binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0412749293858Retained binary64: 0.04127492938579755binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
103103/32Exact fraction: 103/32Retained binary64: 3.21875-0.959994966689Retained binary64: -0.9599949666886207binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.959994966689Retained binary64: 0.9599949666886207binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0400050333114Retained binary64: 0.04000503331137926binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
10413/4Exact fraction: 13/4Retained binary64: 3.25-0.961225792168Retained binary64: -0.961225792168278binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.961225792168Retained binary64: 0.961225792168278binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0387742078317Retained binary64: 0.03877420783172201binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
105105/32Exact fraction: 105/32Retained binary64: 3.28125-0.962418749129Retained binary64: -0.962418749129002binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.962418749129Retained binary64: 0.962418749129002binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.037581250871Retained binary64: 0.037581250870997916binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
10653/16Exact fraction: 53/16Retained binary64: 3.3125-0.963575002663Retained binary64: -0.9635750026626357binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.963575002663Retained binary64: 0.9635750026626357binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0364249973374Retained binary64: 0.036424997337364234binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
107107/32Exact fraction: 107/32Retained binary64: 3.34375-0.964695682015Retained binary64: -0.9646956820149143binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.964695682015Retained binary64: 0.9646956820149143binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0353043179851Retained binary64: 0.03530431798508576binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
10827/8Exact fraction: 27/8Retained binary64: 3.375-0.965781881688Retained binary64: -0.965781881688334binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.965781881688Retained binary64: 0.965781881688334binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0342181183117Retained binary64: 0.03421811831166603binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
109109/32Exact fraction: 109/32Retained binary64: 3.40625-0.966834662511Retained binary64: -0.9668346625110897binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.966834662511Retained binary64: 0.9668346625110897binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0331653374889Retained binary64: 0.03316533748891029binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
11055/16Exact fraction: 55/16Retained binary64: 3.4375-0.967855052673Retained binary64: -0.9678550526731239binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.967855052673Retained binary64: 0.9678550526731239binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0321449473269Retained binary64: 0.03214494732687607binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
111111/32Exact fraction: 111/32Retained binary64: 3.46875-0.96884404873Retained binary64: -0.9688440487303002binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.96884404873Retained binary64: 0.9688440487303002binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0311559512697Retained binary64: 0.031155951269699805binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
1127/2Exact fraction: 7/2Retained binary64: 3.5-0.969802616578Retained binary64: -0.9698026165776815binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.969802616578Retained binary64: 0.9698026165776815binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0301973834223Retained binary64: 0.0301973834223185binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
113113/32Exact fraction: 113/32Retained binary64: 3.53125-0.970731692393Retained binary64: -0.9707316923928639binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.970731692393Retained binary64: 0.9707316923928639binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0292683076071Retained binary64: 0.029268307607136092binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
11457/16Exact fraction: 57/16Retained binary64: 3.5625-0.97163218355Retained binary64: -0.9716321835502869binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.97163218355Retained binary64: 0.9716321835502869binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0283678164497Retained binary64: 0.0283678164497131binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
115115/32Exact fraction: 115/32Retained binary64: 3.59375-0.972504969507Retained binary64: -0.9725049695074133binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.972504969507Retained binary64: 0.9725049695074133binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0274950304926Retained binary64: 0.02749503049258667binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
11629/8Exact fraction: 29/8Retained binary64: 3.625-0.973350902664Retained binary64: -0.9733509026636445binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.973350902664Retained binary64: 0.9733509026636445binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0266490973364Retained binary64: 0.026649097336355485binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
117117/32Exact fraction: 117/32Retained binary64: 3.65625-0.974170809193Retained binary64: -0.9741708091928093binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.974170809193Retained binary64: 0.9741708091928093binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0258291908072Retained binary64: 0.025829190807190754binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
11859/16Exact fraction: 59/16Retained binary64: 3.6875-0.97496548985Retained binary64: -0.9749654898500398binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.97496548985Retained binary64: 0.9749654898500398binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.02503451015Retained binary64: 0.025034510149960148binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
119119/32Exact fraction: 119/32Retained binary64: 3.71875-0.975735720754Retained binary64: -0.9757357207538233binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.975735720754Retained binary64: 0.9757357207538233binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0242642792462Retained binary64: 0.02426427924617674binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
12015/4Exact fraction: 15/4Retained binary64: 3.75-0.976482254144Retained binary64: -0.9764822541439909binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.976482254144Retained binary64: 0.9764822541439909binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.023517745856Retained binary64: 0.023517745856009107binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
121121/32Exact fraction: 121/32Retained binary64: 3.78125-0.977205819116Retained binary64: -0.9772058191163877binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.977205819116Retained binary64: 0.9772058191163877binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0227941808836Retained binary64: 0.022794180883612344binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
12261/16Exact fraction: 61/16Retained binary64: 3.8125-0.977907122335Retained binary64: -0.9779071223349376binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.977907122335Retained binary64: 0.9779071223349376binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0220928776651Retained binary64: 0.022092877665062443binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
123123/32Exact fraction: 123/32Retained binary64: 3.84375-0.978586848722Retained binary64: -0.9785868487218014binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.978586848722Retained binary64: 0.9785868487218014binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0214131512782Retained binary64: 0.021413151278198653binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
12431/8Exact fraction: 31/8Retained binary64: 3.875-0.979245662126Retained binary64: -0.9792456621263003binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.979245662126Retained binary64: 0.9792456621263003binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0207543378737Retained binary64: 0.020754337873699742binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
125125/32Exact fraction: 125/32Retained binary64: 3.90625-0.979884205973Retained binary64: -0.9798842059732591binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.979884205973Retained binary64: 0.9798842059732591binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0201157940267Retained binary64: 0.02011579402674089binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
12663/16Exact fraction: 63/16Retained binary64: 3.9375-0.980503103891Retained binary64: -0.980503103891402binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.980503103891Retained binary64: 0.980503103891402binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0194968961086Retained binary64: 0.019496896108597995binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
127127/32Exact fraction: 127/32Retained binary64: 3.96875-0.981102960322Retained binary64: -0.9811029603224143binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.981102960322Retained binary64: 0.9811029603224143binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0188970396776Retained binary64: 0.018897039677585683binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
1284Exact fraction: 4Retained binary64: 4.0-0.981684361111Retained binary64: -0.9816843611112658binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.981684361111Retained binary64: 0.9816843611112658binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0183156388887Retained binary64: 0.01831563888873418binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
129129/32Exact fraction: 129/32Retained binary64: 4.03125-0.982247874078Retained binary64: -0.9822478740783714binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.982247874078Retained binary64: 0.9822478740783714binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0177521259216Retained binary64: 0.01775212592162854binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
13065/16Exact fraction: 65/16Retained binary64: 4.0625-0.982794049574Retained binary64: -0.9827940495741486binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.982794049574Retained binary64: 0.9827940495741486binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0172059504259Retained binary64: 0.017205950425851383binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
131131/32Exact fraction: 131/32Retained binary64: 4.09375-0.983323421017Retained binary64: -0.9833234210165125binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.983323421017Retained binary64: 0.9833234210165125binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0166765789835Retained binary64: 0.016676578983487567binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
13233/8Exact fraction: 33/8Retained binary64: 4.125-0.983836505412Retained binary64: -0.9838365054118341binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.983836505412Retained binary64: 0.9838365054118341binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0161634945882Retained binary64: 0.016163494588165874binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
133133/32Exact fraction: 133/32Retained binary64: 4.15625-0.98433380386Retained binary64: -0.9843338038598711binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.98433380386Retained binary64: 0.9843338038598711binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0156661961401Retained binary64: 0.015666196140128895binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
13467/16Exact fraction: 67/16Retained binary64: 4.1875-0.984815802043Retained binary64: -0.9848158020431621binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.984815802043Retained binary64: 0.9848158020431621binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0151841979568Retained binary64: 0.015184197956837946binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
135135/32Exact fraction: 135/32Retained binary64: 4.21875-0.985282970701Retained binary64: -0.9852829707013648binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.985282970701Retained binary64: 0.9852829707013648binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0147170292986Retained binary64: 0.014717029298635137binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
13617/4Exact fraction: 17/4Retained binary64: 4.25-0.985735766091Retained binary64: -0.9857357660910008binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.985735766091Retained binary64: 0.9857357660910008binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.014264233909Retained binary64: 0.014264233908999256binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
137137/32Exact fraction: 137/32Retained binary64: 4.28125-0.986174630431Retained binary64: -0.9861746304310535binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.986174630431Retained binary64: 0.9861746304310535binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0138253695689Retained binary64: 0.013825369568946493binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
13869/16Exact fraction: 69/16Retained binary64: 4.3125-0.986599992335Retained binary64: -0.9865999923348592binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.986599992335Retained binary64: 0.9865999923348592binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0134000076651Retained binary64: 0.013400007665140828binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
139139/32Exact fraction: 139/32Retained binary64: 4.34375-0.987012267229Retained binary64: -0.9870122672287077binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.987012267229Retained binary64: 0.9870122672287077binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0129877327713Retained binary64: 0.012987732771292249binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
14035/8Exact fraction: 35/8Retained binary64: 4.375-0.987411857758Retained binary64: -0.987411857757566binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.987411857758Retained binary64: 0.987411857757566binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0125881422424Retained binary64: 0.012588142242433998binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
141141/32Exact fraction: 141/32Retained binary64: 4.40625-0.987799154178Retained binary64: -0.9877991541783174binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.987799154178Retained binary64: 0.9877991541783174binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0122008458217Retained binary64: 0.012200845821682604binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
14271/16Exact fraction: 71/16Retained binary64: 4.4375-0.988174534741Retained binary64: -0.9881745347409033binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.988174534741Retained binary64: 0.9881745347409033binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0118254652591Retained binary64: 0.011825465259096618binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
143143/32Exact fraction: 143/32Retained binary64: 4.46875-0.988538366058Retained binary64: -0.9885383660577381binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.988538366058Retained binary64: 0.9885383660577381binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0114616339423Retained binary64: 0.011461633942261852binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
1449/2Exact fraction: 9/2Retained binary64: 4.5-0.988891003462Retained binary64: -0.9888910034617577binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.988891003462Retained binary64: 0.9888910034617577binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0111089965382Retained binary64: 0.011108996538242306binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
145145/32Exact fraction: 145/32Retained binary64: 4.53125-0.989232791353Retained binary64: -0.9892327913534529binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.989232791353Retained binary64: 0.9892327913534529binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0107672086465Retained binary64: 0.010767208646547101binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
14673/16Exact fraction: 73/16Retained binary64: 4.5625-0.989564063537Retained binary64: -0.9895640635372255binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.989564063537Retained binary64: 0.9895640635372255binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0104359364628Retained binary64: 0.010435936462774504binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
147147/32Exact fraction: 147/32Retained binary64: 4.59375-0.989885143547Retained binary64: -0.9898851435473954binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.989885143547Retained binary64: 0.9898851435473954binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0101148564526Retained binary64: 0.010114856452604551binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
14837/8Exact fraction: 37/8Retained binary64: 4.625-0.990196344964Retained binary64: -0.9901963449641782binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.990196344964Retained binary64: 0.9901963449641782binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00980365503582Retained binary64: 0.009803655035821828binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
149149/32Exact fraction: 149/32Retained binary64: 4.65625-0.99049797172Retained binary64: -0.9904979717199401binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.99049797172Retained binary64: 0.9904979717199401binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00950202828006Retained binary64: 0.00950202828005989binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
15075/16Exact fraction: 75/16Retained binary64: 4.6875-0.990790318396Retained binary64: -0.9907903183960318binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.990790318396Retained binary64: 0.9907903183960318binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00920968160397Retained binary64: 0.00920968160396814binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
151151/32Exact fraction: 151/32Retained binary64: 4.71875-0.99107367051Retained binary64: -0.9910736705104887binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.99107367051Retained binary64: 0.9910736705104887binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00892632948951Retained binary64: 0.008926329489511324binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
15219/4Exact fraction: 19/4Retained binary64: 4.75-0.991348304797Retained binary64: -0.9913483047968794binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.991348304797Retained binary64: 0.9913483047968794binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00865169520312Retained binary64: 0.008651695203120634binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
153153/32Exact fraction: 153/32Retained binary64: 4.78125-0.991614489475Retained binary64: -0.9916144894745759binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.991614489475Retained binary64: 0.9916144894745759binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00838551052542Retained binary64: 0.008385510525424083binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
15477/16Exact fraction: 77/16Retained binary64: 4.8125-0.991872484511Retained binary64: -0.9918724845107078binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.991872484511Retained binary64: 0.9918724845107078binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00812751548929Retained binary64: 0.008127515489292211binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
155155/32Exact fraction: 155/32Retained binary64: 4.84375-0.992122541874Retained binary64: -0.9921225418740567binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.992122541874Retained binary64: 0.9921225418740567binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00787745812594Retained binary64: 0.007877458125943277binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
15639/8Exact fraction: 39/8Retained binary64: 4.875-0.992364905781Retained binary64: -0.99236490578114binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.992364905781Retained binary64: 0.99236490578114binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00763509421886Retained binary64: 0.007635094218859962binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
157157/32Exact fraction: 157/32Retained binary64: 4.90625-0.992599812935Retained binary64: -0.9925998129347228binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.992599812935Retained binary64: 0.9925998129347228binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00740018706528Retained binary64: 0.0074001870652772766binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
15879/16Exact fraction: 79/16Retained binary64: 4.9375-0.992827492755Retained binary64: -0.9928274927549913binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.992827492755Retained binary64: 0.9928274927549913binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00717250724501Retained binary64: 0.007172507245008699binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
159159/32Exact fraction: 159/32Retained binary64: 4.96875-0.993048167604Retained binary64: -0.9930481676036153binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.993048167604Retained binary64: 0.9930481676036153binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00695183239638Retained binary64: 0.006951832396384793binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
1605Exact fraction: 5Retained binary64: 5.0-0.993262053001Retained binary64: -0.9932620530009145binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.993262053001Retained binary64: 0.9932620530009145binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00673794699909Retained binary64: 0.006737946999085467binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
161161/32Exact fraction: 161/32Retained binary64: 5.03125-0.993469357836Retained binary64: -0.9934693578363463binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.993469357836Retained binary64: 0.9934693578363463binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00653064216365Retained binary64: 0.006530642163653784binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
16281/16Exact fraction: 81/16Retained binary64: 5.0625-0.993670284573Retained binary64: -0.9936702845725143binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.993670284573Retained binary64: 0.9936702845725143binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00632971542749Retained binary64: 0.006329715427485747binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
163163/32Exact fraction: 163/32Retained binary64: 5.09375-0.993865029443Retained binary64: -0.9938650294429032binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.993865029443Retained binary64: 0.9938650294429032binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0061349705571Retained binary64: 0.006134970557096825binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
16441/8Exact fraction: 41/8Retained binary64: 5.125-0.994053782644Retained binary64: -0.994053782643528binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.994053782644Retained binary64: 0.994053782643528binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00594621735647Retained binary64: 0.005946217356472094binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
165165/32Exact fraction: 165/32Retained binary64: 5.15625-0.994236728519Retained binary64: -0.9942367285186872binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.994236728519Retained binary64: 0.9942367285186872binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00576327148131Retained binary64: 0.005763271481312824binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
16683/16Exact fraction: 83/16Retained binary64: 5.1875-0.994414045741Retained binary64: -0.9944140457410019binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.994414045741Retained binary64: 0.9944140457410019binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.005585954259Retained binary64: 0.0055859542589981binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
167167/32Exact fraction: 167/32Retained binary64: 5.21875-0.994585907486Retained binary64: -0.9945859074859144binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.994585907486Retained binary64: 0.9945859074859144binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00541409251409Retained binary64: 0.0054140925140856375binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
16821/4Exact fraction: 21/4Retained binary64: 5.25-0.994752481601Retained binary64: -0.9947524816008186binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.994752481601Retained binary64: 0.9947524816008186binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00524751839918Retained binary64: 0.005247518399181385binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
169169/32Exact fraction: 169/32Retained binary64: 5.28125-0.994913930769Retained binary64: -0.9949139307689873binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.994913930769Retained binary64: 0.9949139307689873binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00508606923101Retained binary64: 0.005086069231012701binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
17085/16Exact fraction: 85/16Retained binary64: 5.3125-0.995070412668Retained binary64: -0.995070412668455binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.995070412668Retained binary64: 0.995070412668455binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00492958733155Retained binary64: 0.004929587331545052binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
171171/32Exact fraction: 171/32Retained binary64: 5.34375-0.995222080126Retained binary64: -0.995222080126013binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.995222080126Retained binary64: 0.995222080126013binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00477791987399Retained binary64: 0.00477791987398702binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
17243/8Exact fraction: 43/8Retained binary64: 5.375-0.995369081266Retained binary64: -0.9953690812664667binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.995369081266Retained binary64: 0.9953690812664667binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00463091873353Retained binary64: 0.004630918733533246binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
173173/32Exact fraction: 173/32Retained binary64: 5.40625-0.995511559657Retained binary64: -0.9955115596573004binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.995511559657Retained binary64: 0.9955115596573004binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0044884403427Retained binary64: 0.004488440342699523binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
17487/16Exact fraction: 87/16Retained binary64: 5.4375-0.995649654449Retained binary64: -0.9956496544488912binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.995649654449Retained binary64: 0.9956496544488912binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00435034555111Retained binary64: 0.004350345551108769binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
175175/32Exact fraction: 175/32Retained binary64: 5.46875-0.99578350051Retained binary64: -0.9957835005104091binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.99578350051Retained binary64: 0.9957835005104091binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00421649948959Retained binary64: 0.004216499489590926binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
17611/2Exact fraction: 11/2Retained binary64: 5.5-0.995913228562Retained binary64: -0.995913228561536binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.995913228562Retained binary64: 0.995913228561536binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00408677143846Retained binary64: 0.004086771438464067binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
177177/32Exact fraction: 177/32Retained binary64: 5.53125-0.9960389653Retained binary64: -0.9960389653001319binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.9960389653Retained binary64: 0.9960389653001319binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00396103469987Retained binary64: 0.003961034699868069binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
17889/16Exact fraction: 89/16Retained binary64: 5.5625-0.996160833526Retained binary64: -0.9961608335259738binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996160833526Retained binary64: 0.9961608335259738binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00383916647403Retained binary64: 0.0038391664740261636binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
179179/32Exact fraction: 179/32Retained binary64: 5.59375-0.996278952261Retained binary64: -0.9962789522606865binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996278952261Retained binary64: 0.9962789522606865binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00372104773931Retained binary64: 0.0037210477393135196binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
18045/8Exact fraction: 45/8Retained binary64: 5.625-0.996393436864Retained binary64: -0.9963934368639843binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996393436864Retained binary64: 0.9963934368639843binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00360656313602Retained binary64: 0.0036065631360157305binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
181181/32Exact fraction: 181/32Retained binary64: 5.65625-0.996504399146Retained binary64: -0.9965043991463364binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996504399146Retained binary64: 0.9965043991463364binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00349560085366Retained binary64: 0.0034956008536636734binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
18291/16Exact fraction: 91/16Retained binary64: 5.6875-0.996611947478Retained binary64: -0.9966119474781653binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996611947478Retained binary64: 0.9966119474781653binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00338805252183Retained binary64: 0.0033880525218347116binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
183183/32Exact fraction: 183/32Retained binary64: 5.71875-0.996716186896Retained binary64: -0.9967161868956864binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996716186896Retained binary64: 0.9967161868956864binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00328381310431Retained binary64: 0.003283813104313592binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
18423/4Exact fraction: 23/4Retained binary64: 5.75-0.996817219203Retained binary64: -0.9968172192034903binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996817219203Retained binary64: 0.9968172192034903binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00318278079651Retained binary64: 0.003182780796509667binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
185185/32Exact fraction: 185/32Retained binary64: 5.78125-0.996915143074Retained binary64: -0.9969151430739698binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996915143074Retained binary64: 0.9969151430739698binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00308485692603Retained binary64: 0.0030848569260302594binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
18693/16Exact fraction: 93/16Retained binary64: 5.8125-0.997010054144Retained binary64: -0.9970100541436869binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997010054144Retained binary64: 0.9970100541436869binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00298994585631Retained binary64: 0.0029899458563130603binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
187187/32Exact fraction: 187/32Retained binary64: 5.84375-0.997102045107Retained binary64: -0.9971020451067766binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997102045107Retained binary64: 0.9971020451067766binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00289795489322Retained binary64: 0.00289795489322345binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
18847/8Exact fraction: 47/8Retained binary64: 5.875-0.997191205805Retained binary64: -0.9971912058054745binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997191205805Retained binary64: 0.9971912058054745binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00280879419453Retained binary64: 0.0028087941945255128binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
189189/32Exact fraction: 189/32Retained binary64: 5.90625-0.997277623318Retained binary64: -0.9972776233178616binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997277623318Retained binary64: 0.9972776233178616binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00272237668214Retained binary64: 0.00272237668213834binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
19095/16Exact fraction: 95/16Retained binary64: 5.9375-0.997361382043Retained binary64: -0.997361382042908binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997361382043Retained binary64: 0.997361382042908binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00263861795709Retained binary64: 0.0026386179570919216binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
191191/32Exact fraction: 191/32Retained binary64: 5.96875-0.997442563783Retained binary64: -0.9974425637829004binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997442563783Retained binary64: 0.9974425637829004binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0025574362171Retained binary64: 0.0025574362170995664binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
1926Exact fraction: 6Retained binary64: 6.0-0.997521247823Retained binary64: -0.9975212478233336binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997521247823Retained binary64: 0.9975212478233336binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.00247875217667Retained binary64: 0.0024787521766663585binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained
All four exact logarithmic Drude controls (4 rows)

Separate physical models/clocks: Drude T=t/τ_m and homogeneous instantaneous-Ohmic bulk S=t/τ_q with τ_q=ε/σ. ε is background permittivity and σ/free current describe the mobile response without duplication. Exact log controls use retained binary64 x-coordinates. The bulk control schema also retains an algebraic signedDrift value; it is not used as a bulk velocity or an equal-clock assertion.

All four exact logarithmic Drude controls
RowExact symbolic clockRetained binary64 clock coordinateExact remaining fractionExact rise or boundary fractionAlgebraic signed drift controlTwo physical clocks identified? (false)
000Retained binary64: 0.0binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.00Exact fraction: 0Retained binary64: 0.00Exact fraction: 0Retained binary64: 0.0false
1ln20.69314718056Retained binary64: 0.6931471805599453binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1/2Exact fraction: 1/2Retained binary64: 0.51/2Exact fraction: 1/2Retained binary64: 0.5-1/2Exact fraction: -1/2Retained binary64: -0.5false
2ln41.38629436112Retained binary64: 1.3862943611198906binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1/4Exact fraction: 1/4Retained binary64: 0.253/4Exact fraction: 3/4Retained binary64: 0.75-3/4Exact fraction: -3/4Retained binary64: -0.75false
3ln82.07944154168Retained binary64: 2.0794415416798357binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1/8Exact fraction: 1/8Retained binary64: 0.1257/8Exact fraction: 7/8Retained binary64: 0.875-7/8Exact fraction: -7/8Retained binary64: -0.875false
drudeStep: source scales, exact constants and scope flags (5 rows)

Separate physical models/clocks: Drude T=t/τ_m and homogeneous instantaneous-Ohmic bulk S=t/τ_q with τ_q=ε/σ. ε is background permittivity and σ/free current describe the mobile response without duplication. Exact log controls use retained binary64 x-coordinates. The bulk control schema also retains an algebraic signedDrift value; it is not used as a bulk velocity or an equal-clock assertion.

drudeStep: source scales, exact constants and scope flags
Quantity/source conditionAdmitted value
clockT=t/tau_m
signed Velocity Directionnegative-x
conventional Current Directionpositive-x
zero Initial Drifttrue
sigma0nq0^2tau_m/m
All 193 retained bulk-to-boundary samples (193 rows)

Separate physical models/clocks: Drude T=t/τ_m and homogeneous instantaneous-Ohmic bulk S=t/τ_q with τ_q=ε/σ. ε is background permittivity and σ/free current describe the mobile response without duplication. Exact log controls use retained binary64 x-coordinates. The bulk control schema also retains an algebraic signedDrift value; it is not used as a bulk velocity or an equal-clock assertion.

All 193 retained bulk-to-boundary samples
RowS = t/τ_q (separate bulk clock)Bulk free charge/Q_initialBoundary free charge/Q_initialConserved total free charge/Q_initial
00Exact fraction: 0Retained binary64: 0.01Retained binary64: 1.0binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0Retained binary64: 0.0binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
11/32Exact fraction: 1/32Retained binary64: 0.031250.969233234476Retained binary64: 0.9692332344763441binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0307667655237Retained binary64: 0.03076676552365592binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
21/16Exact fraction: 1/16Retained binary64: 0.06250.939413062813Retained binary64: 0.9394130628134758binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.0605869371865Retained binary64: 0.06058693718652421binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
33/32Exact fraction: 3/32Retained binary64: 0.093750.91051036138Retained binary64: 0.9105103613800342binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.08948963862Retained binary64: 0.08948963861996587binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
41/8Exact fraction: 1/8Retained binary64: 0.1250.882496902585Retained binary64: 0.8824969025845955binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.117503097415Retained binary64: 0.11750309741540459binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
55/32Exact fraction: 5/32Retained binary64: 0.156250.855345327307Retained binary64: 0.8553453273074225binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.144654672693Retained binary64: 0.14465467269257745binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
63/16Exact fraction: 3/16Retained binary64: 0.18750.82902911818Retained binary64: 0.8290291181804004binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.17097088182Retained binary64: 0.17097088181959966binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
77/32Exact fraction: 7/32Retained binary64: 0.218750.803522573689Retained binary64: 0.8035225736890608binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.196477426311Retained binary64: 0.19647742631093928binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
81/4Exact fraction: 1/4Retained binary64: 0.250.778800783071Retained binary64: 0.7788007830714049binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.221199216929Retained binary64: 0.22119921692859512binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
99/32Exact fraction: 9/32Retained binary64: 0.281250.754839601989Retained binary64: 0.7548396019890073binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.245160398011Retained binary64: 0.24516039801099265binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
105/16Exact fraction: 5/16Retained binary64: 0.31250.731615628947Retained binary64: 0.7316156289466418binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.268384371053Retained binary64: 0.2683843710533582binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
1111/32Exact fraction: 11/32Retained binary64: 0.343750.709106182437Retained binary64: 0.7091061824373984binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.290893817563Retained binary64: 0.2908938175626015binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
123/8Exact fraction: 3/8Retained binary64: 0.3750.687289278791Retained binary64: 0.6872892787909722binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.312710721209Retained binary64: 0.3127107212090278binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
1313/32Exact fraction: 13/32Retained binary64: 0.406250.666143610703Retained binary64: 0.6661436107034878binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.333856389297Retained binary64: 0.33385638929651223binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
147/16Exact fraction: 7/16Retained binary64: 0.43750.645648526428Retained binary64: 0.645648526427892binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.354351473572Retained binary64: 0.354351473572108binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
1515/32Exact fraction: 15/32Retained binary64: 0.468750.625784009605Retained binary64: 0.6257840096045911binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.374215990395Retained binary64: 0.37421599039540887binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
161/2Exact fraction: 1/2Retained binary64: 0.50.606530659713Retained binary64: 0.6065306597126334binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.393469340287Retained binary64: 0.39346934028736663binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
1717/32Exact fraction: 17/32Retained binary64: 0.531250.587869673122Retained binary64: 0.5878696731223465binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.412130326878Retained binary64: 0.4121303268776535binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
189/16Exact fraction: 9/16Retained binary64: 0.56250.569782824731Retained binary64: 0.569782824730923binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.430217175269Retained binary64: 0.430217175269077binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
1919/32Exact fraction: 19/32Retained binary64: 0.593750.552252450163Retained binary64: 0.5522524501630204binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.447747549837Retained binary64: 0.4477475498369796binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
205/8Exact fraction: 5/8Retained binary64: 0.6250.535261428519Retained binary64: 0.5352614285189903binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.464738571481Retained binary64: 0.4647385714810098binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
2121/32Exact fraction: 21/32Retained binary64: 0.656250.518793165654Retained binary64: 0.5187931656538893binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.481206834346Retained binary64: 0.48120683434611067binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
2211/16Exact fraction: 11/16Retained binary64: 0.68750.502831577971Retained binary64: 0.5028315779709409binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.497168422029Retained binary64: 0.49716842202905903binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
2323/32Exact fraction: 23/32Retained binary64: 0.718750.487361076714Retained binary64: 0.4873610767136191binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.512638923286Retained binary64: 0.5126389232863808binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
243/4Exact fraction: 3/4Retained binary64: 0.750.472366552741Retained binary64: 0.4723665527410147binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.527633447259Retained binary64: 0.5276334472589853binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
2525/32Exact fraction: 25/32Retained binary64: 0.781250.457833361772Retained binary64: 0.45783336177161427binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.542166638228Retained binary64: 0.5421666382283857binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
2613/16Exact fraction: 13/16Retained binary64: 0.81250.443747310081Retained binary64: 0.44374731008107987binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.556252689919Retained binary64: 0.5562526899189202binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
2727/32Exact fraction: 27/32Retained binary64: 0.843750.43009464064Retained binary64: 0.43009464064006225binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.56990535936Retained binary64: 0.5699053593599377binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
287/8Exact fraction: 7/8Retained binary64: 0.8750.416862019679Retained binary64: 0.4168620196785084binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.583137980321Retained binary64: 0.5831379803214916binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
2929/32Exact fraction: 29/32Retained binary64: 0.906250.404036523663Retained binary64: 0.4040365236633421binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.595963476337Retained binary64: 0.5959634763366579binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
3015/16Exact fraction: 15/16Retained binary64: 0.93750.391605626677Retained binary64: 0.391605626676799binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.608394373323Retained binary64: 0.608394373323201binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
3131/32Exact fraction: 31/32Retained binary64: 0.968750.379557188183Retained binary64: 0.3795571881830896binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.620442811817Retained binary64: 0.6204428118169104binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
321Exact fraction: 1Retained binary64: 1.00.367879441171Retained binary64: 0.36787944117144233binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.632120558829Retained binary64: 0.6321205588285577binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
3333/32Exact fraction: 33/32Retained binary64: 1.031250.356560980664Retained binary64: 0.356560980663947binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.643439019336Retained binary64: 0.6434390193360531binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
3417/16Exact fraction: 17/16Retained binary64: 1.06250.345590752577Retained binary64: 0.3455907525769745binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.654409247423Retained binary64: 0.6544092474230254binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
3535/32Exact fraction: 35/32Retained binary64: 1.093750.334958042925Retained binary64: 0.33495804292529496binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.665041957075Retained binary64: 0.665041957074705binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
369/8Exact fraction: 9/8Retained binary64: 1.1250.324652467358Retained binary64: 0.32465246735834974binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.675347532642Retained binary64: 0.6753475326416503binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
3737/32Exact fraction: 37/32Retained binary64: 1.156250.314663961018Retained binary64: 0.314663961018459binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.685336038982Retained binary64: 0.685336038981541binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
3819/16Exact fraction: 19/16Retained binary64: 1.18750.304982768711Retained binary64: 0.3049827687110593binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.695017231289Retained binary64: 0.6950172312889407binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
3939/32Exact fraction: 39/32Retained binary64: 1.218750.295599435377Retained binary64: 0.29559943537737077binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.704400564623Retained binary64: 0.7044005646226292binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
405/4Exact fraction: 5/4Retained binary64: 1.250.28650479686Retained binary64: 0.2865047968601901binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.71349520314Retained binary64: 0.7134952031398099binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
4141/32Exact fraction: 41/32Retained binary64: 1.281250.277689970954Retained binary64: 0.27768997095378994binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.722310029046Retained binary64: 0.7223100290462101binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
4221/16Exact fraction: 21/16Retained binary64: 1.31250.269146348729Retained binary64: 0.26914634872918386binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.730853651271Retained binary64: 0.7308536512708161binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
4343/32Exact fraction: 43/32Retained binary64: 1.343750.260865586126Retained binary64: 0.26086558612628497binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.739134413874Retained binary64: 0.739134413873715binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
4411/8Exact fraction: 11/8Retained binary64: 1.3750.252839595805Retained binary64: 0.25283959580474646binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.747160404195Retained binary64: 0.7471604041952535binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
4545/32Exact fraction: 45/32Retained binary64: 1.406250.245060539246Retained binary64: 0.2450605392455259binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.754939460754Retained binary64: 0.7549394607544742binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
4623/16Exact fraction: 23/16Retained binary64: 1.43750.237520819095Retained binary64: 0.23752081909545814binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.762479180905Retained binary64: 0.7624791809045419binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
4747/32Exact fraction: 47/32Retained binary64: 1.468750.230213071747Retained binary64: 0.23021307174736147binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.769786928253Retained binary64: 0.7697869282526385binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
483/2Exact fraction: 3/2Retained binary64: 1.50.223130160148Retained binary64: 0.22313016014842982binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.776869839852Retained binary64: 0.7768698398515702binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
4949/32Exact fraction: 49/32Retained binary64: 1.531250.21626516683Retained binary64: 0.2162651668298873binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.78373483317Retained binary64: 0.7837348331701127binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
5025/16Exact fraction: 25/16Retained binary64: 1.56250.209611387151Retained binary64: 0.2096113871510978binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.790388612849Retained binary64: 0.7903886128489022binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
5151/32Exact fraction: 51/32Retained binary64: 1.593750.203162322752Retained binary64: 0.20316232275153173binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.796837677248Retained binary64: 0.7968376772484682binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
5213/8Exact fraction: 13/8Retained binary64: 1.6250.196911675204Retained binary64: 0.19691167520419406binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.803088324796Retained binary64: 0.8030883247958059binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
5353/32Exact fraction: 53/32Retained binary64: 1.656250.190853339864Retained binary64: 0.19085333986431632binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.809146660136Retained binary64: 0.8091466601356837binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
5427/16Exact fraction: 27/16Retained binary64: 1.68750.184981399907Retained binary64: 0.18498139990730428binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.815018600093Retained binary64: 0.8150186000926958binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
5555/32Exact fraction: 55/32Retained binary64: 1.718750.17929012055Retained binary64: 0.17929012055011864binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.82070987945Retained binary64: 0.8207098794498814binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
567/4Exact fraction: 7/4Retained binary64: 1.750.17377394345Retained binary64: 0.17377394345044514binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.82622605655Retained binary64: 0.8262260565495548binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
5757/32Exact fraction: 57/32Retained binary64: 1.781250.168427481278Retained binary64: 0.16842748127818424binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.831572518722Retained binary64: 0.8315725187218157binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
5829/16Exact fraction: 29/16Retained binary64: 1.81250.163245512454Retained binary64: 0.1632455124539584binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.836754487546Retained binary64: 0.8367544875460415binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
5959/32Exact fraction: 59/32Retained binary64: 1.843750.158222976049Retained binary64: 0.15822297604949842binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.841777023951Retained binary64: 0.8417770239505016binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
6015/8Exact fraction: 15/8Retained binary64: 1.8750.153354966845Retained binary64: 0.15335496684492847binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.846645033155Retained binary64: 0.8466450331550716binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
6161/32Exact fraction: 61/32Retained binary64: 1.906250.148636730538Retained binary64: 0.1486367305381225binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.851363269462Retained binary64: 0.8513632694618775binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
6231/16Exact fraction: 31/16Retained binary64: 1.93750.144063659101Retained binary64: 0.14406365910145327binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.855936340899Retained binary64: 0.8559363408985468binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
6363/32Exact fraction: 63/32Retained binary64: 1.968750.139631286281Retained binary64: 0.13963128628139898binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.860368713719Retained binary64: 0.860368713718601binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
642Exact fraction: 2Retained binary64: 2.00.135335283237Retained binary64: 0.1353352832366127binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.864664716763Retained binary64: 0.8646647167633873binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
6565/32Exact fraction: 65/32Retained binary64: 2.031250.13117145431Retained binary64: 0.13117145431019428binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.86882854569Retained binary64: 0.8688285456898057binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
6633/16Exact fraction: 33/16Retained binary64: 2.06250.127135732932Retained binary64: 0.1271357329320356binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.872864267068Retained binary64: 0.8728642670679644binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
6767/32Exact fraction: 67/32Retained binary64: 2.093750.123224177647Retained binary64: 0.12322417764723749binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.876775822353Retained binary64: 0.8767758223527625binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
6817/8Exact fraction: 17/8Retained binary64: 2.1250.119432968267Retained binary64: 0.11943296826671962binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.880567031733Retained binary64: 0.8805670317332803binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
6969/32Exact fraction: 69/32Retained binary64: 2.156250.115758402136Retained binary64: 0.11575840213626322binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.884241597864Retained binary64: 0.8842415978637368binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
7035/16Exact fraction: 35/16Retained binary64: 2.18750.11219689052Retained binary64: 0.11219689052034373binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.88780310948Retained binary64: 0.8878031094796562binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
7171/32Exact fraction: 71/32Retained binary64: 2.218750.108744955097Retained binary64: 0.10874495509722103binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.891255044903Retained binary64: 0.891255044902779binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
729/4Exact fraction: 9/4Retained binary64: 2.250.105399224562Retained binary64: 0.10539922456186433binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.894600775438Retained binary64: 0.8946007754381357binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
7373/32Exact fraction: 73/32Retained binary64: 2.281250.102156431333Retained binary64: 0.1021564313333943binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.897843568667Retained binary64: 0.8978435686666058binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
7437/16Exact fraction: 37/16Retained binary64: 2.31250.0990134083638Retained binary64: 0.0990134083638263binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.900986591636Retained binary64: 0.9009865916361737binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
7575/32Exact fraction: 75/32Retained binary64: 2.343750.095967086045Retained binary64: 0.09596708604499847binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.904032913955Retained binary64: 0.9040329139550015binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
7619/8Exact fraction: 19/8Retained binary64: 2.3750.0930144892107Retained binary64: 0.09301448921066349binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.906985510789Retained binary64: 0.9069855107893365binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
7777/32Exact fraction: 77/32Retained binary64: 2.406250.0901527342308Retained binary64: 0.09015273423081638binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.909847265769Retained binary64: 0.9098472657691836binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
7839/16Exact fraction: 39/16Retained binary64: 2.43750.0873790261954Retained binary64: 0.08737902619542039binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.912620973805Retained binary64: 0.9126209738045796binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
7979/32Exact fraction: 79/32Retained binary64: 2.468750.0846906561848Retained binary64: 0.0846906561847805binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.915309343815Retained binary64: 0.9153093438152196binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
805/2Exact fraction: 5/2Retained binary64: 2.50.0820849986239Retained binary64: 0.0820849986238988binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.917915001376Retained binary64: 0.9179150013761012binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
8181/32Exact fraction: 81/32Retained binary64: 2.531250.0795595087182Retained binary64: 0.07955950871822769binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.920440491282Retained binary64: 0.9204404912817723binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
8241/16Exact fraction: 41/16Retained binary64: 2.56250.0771117199683Retained binary64: 0.07711171996831671binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.922888280032Retained binary64: 0.9228882800316833binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
8383/32Exact fraction: 83/32Retained binary64: 2.593750.0747392417609Retained binary64: 0.0747392417609257binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.925260758239Retained binary64: 0.9252607582390743binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
8421/8Exact fraction: 21/8Retained binary64: 2.6250.0724397570343Retained binary64: 0.07243975703425146binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.927560242966Retained binary64: 0.9275602429657486binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
8585/32Exact fraction: 85/32Retained binary64: 2.656250.070211020015Retained binary64: 0.07021102001498804binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.929788979985Retained binary64: 0.929788979985012binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
8643/16Exact fraction: 43/16Retained binary64: 2.68750.068050854025Retained binary64: 0.0680508540250102binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.931949145975Retained binary64: 0.9319491459749898binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
8787/32Exact fraction: 87/32Retained binary64: 2.718750.0659571493555Retained binary64: 0.06595714935553817binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.934042850644Retained binary64: 0.9340428506444618binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
8811/4Exact fraction: 11/4Retained binary64: 2.750.0639278612067Retained binary64: 0.06392786120670757binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.936072138793Retained binary64: 0.9360721387932924binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
8989/32Exact fraction: 89/32Retained binary64: 2.781250.0619610076905Retained binary64: 0.061961007690531984binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.938038992309Retained binary64: 0.938038992309468binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
9045/16Exact fraction: 45/16Retained binary64: 2.81250.0600546678953Retained binary64: 0.060054667895307945binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.939945332105Retained binary64: 0.939945332104692binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
9191/32Exact fraction: 91/32Retained binary64: 2.843750.0582069800096Retained binary64: 0.058206980009571974binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.94179301999Retained binary64: 0.9417930199904281binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
9223/8Exact fraction: 23/8Retained binary64: 2.8750.0564161395038Retained binary64: 0.05641613950377735binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.943583860496Retained binary64: 0.9435838604962227binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
9393/32Exact fraction: 93/32Retained binary64: 2.906250.0546803973679Retained binary64: 0.05468039736791477binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.945319602632Retained binary64: 0.9453196026320853binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
9447/16Exact fraction: 47/16Retained binary64: 2.93750.0529980584034Retained binary64: 0.0529980584033558binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.947001941597Retained binary64: 0.9470019415966442binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
9595/32Exact fraction: 95/32Retained binary64: 2.968750.0513674795673Retained binary64: 0.05136747956725073binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.948632520433Retained binary64: 0.9486325204327493binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
963Exact fraction: 3Retained binary64: 3.00.0497870683679Retained binary64: 0.049787068367863944binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.950212931632Retained binary64: 0.950212931632136binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
9797/32Exact fraction: 97/32Retained binary64: 3.031250.0482552813093Retained binary64: 0.04825528130927965binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.951744718691Retained binary64: 0.9517447186907203binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
9849/16Exact fraction: 49/16Retained binary64: 3.06250.046770622384Retained binary64: 0.04677062238395898binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.953229377616Retained binary64: 0.953229377616041binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
9999/32Exact fraction: 99/32Retained binary64: 3.093750.0453316416117Retained binary64: 0.045331641611676264binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.954668358388Retained binary64: 0.9546683583883238binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
10025/8Exact fraction: 25/8Retained binary64: 3.1250.0439369336234Retained binary64: 0.04393693362340742binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.956063066377Retained binary64: 0.9560630663765926binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
101101/32Exact fraction: 101/32Retained binary64: 3.156250.0425851362888Retained binary64: 0.04258513628878761binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.957414863711Retained binary64: 0.9574148637112124binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
10251/16Exact fraction: 51/16Retained binary64: 3.18750.0412749293858Retained binary64: 0.04127492938579755binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.958725070614Retained binary64: 0.9587250706142024binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
103103/32Exact fraction: 103/32Retained binary64: 3.218750.0400050333114Retained binary64: 0.04000503331137926binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.959994966689Retained binary64: 0.9599949666886207binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
10413/4Exact fraction: 13/4Retained binary64: 3.250.0387742078317Retained binary64: 0.03877420783172201binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.961225792168Retained binary64: 0.961225792168278binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
105105/32Exact fraction: 105/32Retained binary64: 3.281250.037581250871Retained binary64: 0.037581250870997916binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.962418749129Retained binary64: 0.962418749129002binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
10653/16Exact fraction: 53/16Retained binary64: 3.31250.0364249973374Retained binary64: 0.036424997337364234binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.963575002663Retained binary64: 0.9635750026626357binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
107107/32Exact fraction: 107/32Retained binary64: 3.343750.0353043179851Retained binary64: 0.03530431798508576binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.964695682015Retained binary64: 0.9646956820149143binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
10827/8Exact fraction: 27/8Retained binary64: 3.3750.0342181183117Retained binary64: 0.03421811831166603binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.965781881688Retained binary64: 0.965781881688334binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
109109/32Exact fraction: 109/32Retained binary64: 3.406250.0331653374889Retained binary64: 0.03316533748891029binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.966834662511Retained binary64: 0.9668346625110897binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
11055/16Exact fraction: 55/16Retained binary64: 3.43750.0321449473269Retained binary64: 0.03214494732687607binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.967855052673Retained binary64: 0.9678550526731239binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
111111/32Exact fraction: 111/32Retained binary64: 3.468750.0311559512697Retained binary64: 0.031155951269699805binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.96884404873Retained binary64: 0.9688440487303002binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
1127/2Exact fraction: 7/2Retained binary64: 3.50.0301973834223Retained binary64: 0.0301973834223185binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.969802616578Retained binary64: 0.9698026165776815binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
113113/32Exact fraction: 113/32Retained binary64: 3.531250.0292683076071Retained binary64: 0.029268307607136092binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.970731692393Retained binary64: 0.9707316923928639binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
11457/16Exact fraction: 57/16Retained binary64: 3.56250.0283678164497Retained binary64: 0.0283678164497131binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.97163218355Retained binary64: 0.9716321835502869binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
115115/32Exact fraction: 115/32Retained binary64: 3.593750.0274950304926Retained binary64: 0.02749503049258667binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.972504969507Retained binary64: 0.9725049695074133binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
11629/8Exact fraction: 29/8Retained binary64: 3.6250.0266490973364Retained binary64: 0.026649097336355485binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.973350902664Retained binary64: 0.9733509026636445binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
117117/32Exact fraction: 117/32Retained binary64: 3.656250.0258291908072Retained binary64: 0.025829190807190754binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.974170809193Retained binary64: 0.9741708091928093binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
11859/16Exact fraction: 59/16Retained binary64: 3.68750.02503451015Retained binary64: 0.025034510149960148binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.97496548985Retained binary64: 0.9749654898500398binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
119119/32Exact fraction: 119/32Retained binary64: 3.718750.0242642792462Retained binary64: 0.02426427924617674binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.975735720754Retained binary64: 0.9757357207538233binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
12015/4Exact fraction: 15/4Retained binary64: 3.750.023517745856Retained binary64: 0.023517745856009107binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.976482254144Retained binary64: 0.9764822541439909binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
121121/32Exact fraction: 121/32Retained binary64: 3.781250.0227941808836Retained binary64: 0.022794180883612344binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.977205819116Retained binary64: 0.9772058191163877binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
12261/16Exact fraction: 61/16Retained binary64: 3.81250.0220928776651Retained binary64: 0.022092877665062443binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.977907122335Retained binary64: 0.9779071223349376binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
123123/32Exact fraction: 123/32Retained binary64: 3.843750.0214131512782Retained binary64: 0.021413151278198653binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.978586848722Retained binary64: 0.9785868487218014binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
12431/8Exact fraction: 31/8Retained binary64: 3.8750.0207543378737Retained binary64: 0.020754337873699742binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.979245662126Retained binary64: 0.9792456621263003binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
125125/32Exact fraction: 125/32Retained binary64: 3.906250.0201157940267Retained binary64: 0.02011579402674089binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.979884205973Retained binary64: 0.9798842059732591binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
12663/16Exact fraction: 63/16Retained binary64: 3.93750.0194968961086Retained binary64: 0.019496896108597995binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.980503103891Retained binary64: 0.980503103891402binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
127127/32Exact fraction: 127/32Retained binary64: 3.968750.0188970396776Retained binary64: 0.018897039677585683binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.981102960322Retained binary64: 0.9811029603224143binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
1284Exact fraction: 4Retained binary64: 4.00.0183156388887Retained binary64: 0.01831563888873418binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.981684361111Retained binary64: 0.9816843611112658binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
129129/32Exact fraction: 129/32Retained binary64: 4.031250.0177521259216Retained binary64: 0.01775212592162854binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.982247874078Retained binary64: 0.9822478740783714binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
13065/16Exact fraction: 65/16Retained binary64: 4.06250.0172059504259Retained binary64: 0.017205950425851383binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.982794049574Retained binary64: 0.9827940495741486binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
131131/32Exact fraction: 131/32Retained binary64: 4.093750.0166765789835Retained binary64: 0.016676578983487567binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.983323421017Retained binary64: 0.9833234210165125binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
13233/8Exact fraction: 33/8Retained binary64: 4.1250.0161634945882Retained binary64: 0.016163494588165874binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.983836505412Retained binary64: 0.9838365054118341binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
133133/32Exact fraction: 133/32Retained binary64: 4.156250.0156661961401Retained binary64: 0.015666196140128895binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.98433380386Retained binary64: 0.9843338038598711binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
13467/16Exact fraction: 67/16Retained binary64: 4.18750.0151841979568Retained binary64: 0.015184197956837946binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.984815802043Retained binary64: 0.9848158020431621binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
135135/32Exact fraction: 135/32Retained binary64: 4.218750.0147170292986Retained binary64: 0.014717029298635137binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.985282970701Retained binary64: 0.9852829707013648binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
13617/4Exact fraction: 17/4Retained binary64: 4.250.014264233909Retained binary64: 0.014264233908999256binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.985735766091Retained binary64: 0.9857357660910008binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
137137/32Exact fraction: 137/32Retained binary64: 4.281250.0138253695689Retained binary64: 0.013825369568946493binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.986174630431Retained binary64: 0.9861746304310535binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
13869/16Exact fraction: 69/16Retained binary64: 4.31250.0134000076651Retained binary64: 0.013400007665140828binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.986599992335Retained binary64: 0.9865999923348592binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
139139/32Exact fraction: 139/32Retained binary64: 4.343750.0129877327713Retained binary64: 0.012987732771292249binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.987012267229Retained binary64: 0.9870122672287077binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
14035/8Exact fraction: 35/8Retained binary64: 4.3750.0125881422424Retained binary64: 0.012588142242433998binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.987411857758Retained binary64: 0.987411857757566binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
141141/32Exact fraction: 141/32Retained binary64: 4.406250.0122008458217Retained binary64: 0.012200845821682604binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.987799154178Retained binary64: 0.9877991541783174binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
14271/16Exact fraction: 71/16Retained binary64: 4.43750.0118254652591Retained binary64: 0.011825465259096618binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.988174534741Retained binary64: 0.9881745347409033binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
143143/32Exact fraction: 143/32Retained binary64: 4.468750.0114616339423Retained binary64: 0.011461633942261852binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.988538366058Retained binary64: 0.9885383660577381binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
1449/2Exact fraction: 9/2Retained binary64: 4.50.0111089965382Retained binary64: 0.011108996538242306binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.988891003462Retained binary64: 0.9888910034617577binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
145145/32Exact fraction: 145/32Retained binary64: 4.531250.0107672086465Retained binary64: 0.010767208646547101binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.989232791353Retained binary64: 0.9892327913534529binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
14673/16Exact fraction: 73/16Retained binary64: 4.56250.0104359364628Retained binary64: 0.010435936462774504binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.989564063537Retained binary64: 0.9895640635372255binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
147147/32Exact fraction: 147/32Retained binary64: 4.593750.0101148564526Retained binary64: 0.010114856452604551binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.989885143547Retained binary64: 0.9898851435473954binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
14837/8Exact fraction: 37/8Retained binary64: 4.6250.00980365503582Retained binary64: 0.009803655035821828binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.990196344964Retained binary64: 0.9901963449641782binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
149149/32Exact fraction: 149/32Retained binary64: 4.656250.00950202828006Retained binary64: 0.00950202828005989binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.99049797172Retained binary64: 0.9904979717199401binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
15075/16Exact fraction: 75/16Retained binary64: 4.68750.00920968160397Retained binary64: 0.00920968160396814binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.990790318396Retained binary64: 0.9907903183960318binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
151151/32Exact fraction: 151/32Retained binary64: 4.718750.00892632948951Retained binary64: 0.008926329489511324binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.99107367051Retained binary64: 0.9910736705104887binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
15219/4Exact fraction: 19/4Retained binary64: 4.750.00865169520312Retained binary64: 0.008651695203120634binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.991348304797Retained binary64: 0.9913483047968794binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
153153/32Exact fraction: 153/32Retained binary64: 4.781250.00838551052542Retained binary64: 0.008385510525424083binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.991614489475Retained binary64: 0.9916144894745759binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
15477/16Exact fraction: 77/16Retained binary64: 4.81250.00812751548929Retained binary64: 0.008127515489292211binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.991872484511Retained binary64: 0.9918724845107078binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
155155/32Exact fraction: 155/32Retained binary64: 4.843750.00787745812594Retained binary64: 0.007877458125943277binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.992122541874Retained binary64: 0.9921225418740567binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
15639/8Exact fraction: 39/8Retained binary64: 4.8750.00763509421886Retained binary64: 0.007635094218859962binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.992364905781Retained binary64: 0.99236490578114binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
157157/32Exact fraction: 157/32Retained binary64: 4.906250.00740018706528Retained binary64: 0.0074001870652772766binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.992599812935Retained binary64: 0.9925998129347228binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
15879/16Exact fraction: 79/16Retained binary64: 4.93750.00717250724501Retained binary64: 0.007172507245008699binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.992827492755Retained binary64: 0.9928274927549913binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
159159/32Exact fraction: 159/32Retained binary64: 4.968750.00695183239638Retained binary64: 0.006951832396384793binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.993048167604Retained binary64: 0.9930481676036153binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
1605Exact fraction: 5Retained binary64: 5.00.00673794699909Retained binary64: 0.006737946999085467binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.993262053001Retained binary64: 0.9932620530009145binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
161161/32Exact fraction: 161/32Retained binary64: 5.031250.00653064216365Retained binary64: 0.006530642163653784binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.993469357836Retained binary64: 0.9934693578363463binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
16281/16Exact fraction: 81/16Retained binary64: 5.06250.00632971542749Retained binary64: 0.006329715427485747binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.993670284573Retained binary64: 0.9936702845725143binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
163163/32Exact fraction: 163/32Retained binary64: 5.093750.0061349705571Retained binary64: 0.006134970557096825binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.993865029443Retained binary64: 0.9938650294429032binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
16441/8Exact fraction: 41/8Retained binary64: 5.1250.00594621735647Retained binary64: 0.005946217356472094binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.994053782644Retained binary64: 0.994053782643528binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
165165/32Exact fraction: 165/32Retained binary64: 5.156250.00576327148131Retained binary64: 0.005763271481312824binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.994236728519Retained binary64: 0.9942367285186872binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
16683/16Exact fraction: 83/16Retained binary64: 5.18750.005585954259Retained binary64: 0.0055859542589981binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.994414045741Retained binary64: 0.9944140457410019binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
167167/32Exact fraction: 167/32Retained binary64: 5.218750.00541409251409Retained binary64: 0.0054140925140856375binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.994585907486Retained binary64: 0.9945859074859144binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
16821/4Exact fraction: 21/4Retained binary64: 5.250.00524751839918Retained binary64: 0.005247518399181385binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.994752481601Retained binary64: 0.9947524816008186binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
169169/32Exact fraction: 169/32Retained binary64: 5.281250.00508606923101Retained binary64: 0.005086069231012701binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.994913930769Retained binary64: 0.9949139307689873binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
17085/16Exact fraction: 85/16Retained binary64: 5.31250.00492958733155Retained binary64: 0.004929587331545052binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.995070412668Retained binary64: 0.995070412668455binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
171171/32Exact fraction: 171/32Retained binary64: 5.343750.00477791987399Retained binary64: 0.00477791987398702binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.995222080126Retained binary64: 0.995222080126013binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
17243/8Exact fraction: 43/8Retained binary64: 5.3750.00463091873353Retained binary64: 0.004630918733533246binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.995369081266Retained binary64: 0.9953690812664667binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
173173/32Exact fraction: 173/32Retained binary64: 5.406250.0044884403427Retained binary64: 0.004488440342699523binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.995511559657Retained binary64: 0.9955115596573004binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
17487/16Exact fraction: 87/16Retained binary64: 5.43750.00435034555111Retained binary64: 0.004350345551108769binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.995649654449Retained binary64: 0.9956496544488912binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
175175/32Exact fraction: 175/32Retained binary64: 5.468750.00421649948959Retained binary64: 0.004216499489590926binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.99578350051Retained binary64: 0.9957835005104091binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
17611/2Exact fraction: 11/2Retained binary64: 5.50.00408677143846Retained binary64: 0.004086771438464067binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.995913228562Retained binary64: 0.995913228561536binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
177177/32Exact fraction: 177/32Retained binary64: 5.531250.00396103469987Retained binary64: 0.003961034699868069binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.9960389653Retained binary64: 0.9960389653001319binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
17889/16Exact fraction: 89/16Retained binary64: 5.56250.00383916647403Retained binary64: 0.0038391664740261636binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996160833526Retained binary64: 0.9961608335259738binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
179179/32Exact fraction: 179/32Retained binary64: 5.593750.00372104773931Retained binary64: 0.0037210477393135196binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996278952261Retained binary64: 0.9962789522606865binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
18045/8Exact fraction: 45/8Retained binary64: 5.6250.00360656313602Retained binary64: 0.0036065631360157305binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996393436864Retained binary64: 0.9963934368639843binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
181181/32Exact fraction: 181/32Retained binary64: 5.656250.00349560085366Retained binary64: 0.0034956008536636734binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996504399146Retained binary64: 0.9965043991463364binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
18291/16Exact fraction: 91/16Retained binary64: 5.68750.00338805252183Retained binary64: 0.0033880525218347116binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996611947478Retained binary64: 0.9966119474781653binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
183183/32Exact fraction: 183/32Retained binary64: 5.718750.00328381310431Retained binary64: 0.003283813104313592binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996716186896Retained binary64: 0.9967161868956864binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
18423/4Exact fraction: 23/4Retained binary64: 5.750.00318278079651Retained binary64: 0.003182780796509667binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996817219203Retained binary64: 0.9968172192034903binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
185185/32Exact fraction: 185/32Retained binary64: 5.781250.00308485692603Retained binary64: 0.0030848569260302594binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.996915143074Retained binary64: 0.9969151430739698binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
18693/16Exact fraction: 93/16Retained binary64: 5.81250.00298994585631Retained binary64: 0.0029899458563130603binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997010054144Retained binary64: 0.9970100541436869binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
187187/32Exact fraction: 187/32Retained binary64: 5.843750.00289795489322Retained binary64: 0.00289795489322345binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997102045107Retained binary64: 0.9971020451067766binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
18847/8Exact fraction: 47/8Retained binary64: 5.8750.00280879419453Retained binary64: 0.0028087941945255128binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997191205805Retained binary64: 0.9971912058054745binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
189189/32Exact fraction: 189/32Retained binary64: 5.906250.00272237668214Retained binary64: 0.00272237668213834binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997277623318Retained binary64: 0.9972776233178616binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
19095/16Exact fraction: 95/16Retained binary64: 5.93750.00263861795709Retained binary64: 0.0026386179570919216binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997361382043Retained binary64: 0.997361382042908binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
191191/32Exact fraction: 191/32Retained binary64: 5.968750.0025574362171Retained binary64: 0.0025574362170995664binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997442563783Retained binary64: 0.9974425637829004binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
1926Exact fraction: 6Retained binary64: 6.00.00247875217667Retained binary64: 0.0024787521766663585binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained0.997521247823Retained binary64: 0.9975212478233336binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.0
All four exact logarithmic bulk-charge controls (4 rows)

Separate physical models/clocks: Drude T=t/τ_m and homogeneous instantaneous-Ohmic bulk S=t/τ_q with τ_q=ε/σ. ε is background permittivity and σ/free current describe the mobile response without duplication. Exact log controls use retained binary64 x-coordinates. The bulk control schema also retains an algebraic signedDrift value; it is not used as a bulk velocity or an equal-clock assertion.

All four exact logarithmic bulk-charge controls
RowExact symbolic clockRetained binary64 clock coordinateExact remaining fractionExact rise or boundary fractionAlgebraic signedDrift; not a bulk velocityTwo physical clocks identified? (false)
000Retained binary64: 0.0binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1Exact fraction: 1Retained binary64: 1.00Exact fraction: 0Retained binary64: 0.00Exact fraction: 0Retained binary64: 0.0false
1ln20.69314718056Retained binary64: 0.6931471805599453binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1/2Exact fraction: 1/2Retained binary64: 0.51/2Exact fraction: 1/2Retained binary64: 0.5-1/2Exact fraction: -1/2Retained binary64: -0.5false
2ln41.38629436112Retained binary64: 1.3862943611198906binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1/4Exact fraction: 1/4Retained binary64: 0.253/4Exact fraction: 3/4Retained binary64: 0.75-3/4Exact fraction: -3/4Retained binary64: -0.75false
3ln82.07944154168Retained binary64: 2.0794415416798357binary64;12significantdigitdisplay; mathematicalzero normalized; originalanalyticformula retained1/8Exact fraction: 1/8Retained binary64: 0.1257/8Exact fraction: 7/8Retained binary64: 0.875-7/8Exact fraction: -7/8Retained binary64: -0.875false
bulkChargeRelaxation: source scales, exact constants and scope flags (6 rows)

Separate physical models/clocks: Drude T=t/τ_m and homogeneous instantaneous-Ohmic bulk S=t/τ_q with τ_q=ε/σ. ε is background permittivity and σ/free current describe the mobile response without duplication. Exact log controls use retained binary64 x-coordinates. The bulk control schema also retains an algebraic signedDrift value; it is not used as a bulk velocity or an equal-clock assertion.

bulkChargeRelaxation: source scales, exact constants and scope flags
Quantity/source conditionAdmitted value
clockS=t/tau_q,tau_q=epsilon/sigma
separate Continuum Casetrue
same Physical Time As Drude Axisfalse
charge Destroyedfalse
instant Ohm Assumptions Requiredtrue
inertial Consistencytau_m rho_ddot+rho_dot+(sigma0/epsilon)rho=0; no generalmicroscopicexponential claim

Focus a labelled graph or table region and use arrow keys, or scroll sideways. Fixed-size readable diagrams stay within their own regions. Open the complete data disclosures for exact values, retained binary64 numbers and precision qualifications.

15

Return path: frequency-dependent conductivity

Use peak complex amplitudes with physical fields given by the real part of a phasor times e^(−iωt). For ω > 0, insert that convention into the Drude equation. It gives σ(ω) = σ₀/(1 − iωτ_m). Changing the time convention would change the sign of the reported imaginary part, so the convention belongs beside the formula.

Let Z = ωτ_m. Then Re[σ/σ₀] = 1/(1 + Z²) and Im[σ/σ₀] = Z/(1 + Z²). At Z = 1 the coefficient is (1 + i)/2. Its positive phasor argument corresponds to a physical current that lags the applied cosine field with this e^(−iωt) convention.

The cycle-averaged electrical work density is (1/2)Re(σ)|E_phasor|² for peak amplitude and ω > 0. The real part is nonnegative in this passive model; the imaginary part records the reactive phase response. With RMS amplitudes the corresponding amplitude factor is expressed differently. A constant DC field instead uses J·E directly; it does not acquire a fictitious one-half merely by naming the zero-frequency coefficient.

When ωτ_m is small, the conductivity approaches its DC value within this model. When it is large, inertia changes the phase and response. The curve is not a universal optical-conductivity law for every material: interband transitions, collective effects, frequency-dependent scattering and other physics can change it.

The phasor sign fixes the reactive sign

Compare the real and imaginary parts of the Drude conductivity using physical fields proportional to the real part of a phasor times exp(−iωt). The one-half power factor applies to a nonzero-frequency cycle with peak amplitudes; constant DC power is stated separately.

Dissipative and reactive response with DC kept separate

  • Re(σ/σ₀) = 1/(1+Z²): dissipative coefficient
  • Im(σ/σ₀) = Z/(1+Z²): declared e^(−iωt) convention
Dissipative and reactive response with DC kept separateAll257 admitted rational samples and four exact controls of real and imaginary conductivity versus Z. The conductivity coefficients include Z=0, but no AC cycle-average power is assigned to constant DC.0123400.250.50.751Z = 1: phasor arg +π/4 means physical current lag for e^(−iωt)Constant DC: power coefficient = 1; AC cycle coefficient at DC = nullFor ω > 0 only: peak-cycle power = (1/2) Re(σ) |E_phasor|²Re(σ/σ₀) and Im(σ/σ₀)Z=ωτ_m
  • Physical fields are Re(phasor exp(−iωt)). Peak-amplitude cycle averages require ω>0. At DC, the cycle coefficient is null/unassigned and constant power uses coefficient1, not1/2. No magnitude/phase curve is claimed.
  • σ/σ₀=1/(1−iZ)=(1+iZ)/(1+Z²), Z=ωτ_m. The real component is nonnegative; the imaginary component is positive for positive Z under the declared negative-time-exponent convention.
  • At Z=1 both components equal1/2. The phasor argument+π/4 corresponds to physical current lag for exp(−iωt), not a lead. No native magnitude or phase curve is inferred.
  • For ω>0 and peak field amplitude, average power=(1/2)Re(σ)|E_phasor|². At constant DC, use J·E directly: the normalized constant-DC power coefficient is1. The AC cycle coefficient at Z=0 remains null/unassigned in every retained source row.
  • Every plotted value is read directly from the admitted model. Complete exact fractions,12-significant-digit approximate displays, original function-specific arithmetic qualifications and retained binary64 values remain in the data disclosures. No empirical material constant or numerical solver is introduced.

Complete labelled data equivalents retain every admitted scalar, record, source condition and null. Original manuscript tables remain separate.

Exact normalized Drude conductivity controls for e^(−iωt)

All 257 exact AC response records, including the DC null (257 rows)

Physical fields are Re(phasor exp(−iωt)). Peak-amplitude cycle averages require ω>0. At DC, the cycle coefficient is null/unassigned and constant power uses coefficient1, not1/2. No magnitude/phase curve is claimed.

All 257 exact AC response records, including the DC null
RowZ = ωτ_mRe(σ/σ₀)Im(σ/σ₀)Nonzero-frequency peak-cycle power coefficient (null at DC)Constant DC point?
0DC: no nonzero-frequency cycle-average coefficient is assigned here. Constant DC power has coefficient1.0Exact fraction: 0Retained binary64: 0.01Exact fraction: 1Retained binary64: 1.00Exact fraction: 0Retained binary64: 0.0nulltrue
11/64Exact fraction: 1/64Retained binary64: 0.0156254096/4097Exact fraction: 4096/4097Retained binary64: 0.999755918965096464/4097Exact fraction: 64/4097Retained binary64: 0.0156211862338296312048/4097Exact fraction: 2048/4097Retained binary64: 0.4998779594825482false
21/32Exact fraction: 1/32Retained binary64: 0.031251024/1025Exact fraction: 1024/1025Retained binary64: 0.999024390243902432/1025Exact fraction: 32/1025Retained binary64: 0.03121951219512195512/1025Exact fraction: 512/1025Retained binary64: 0.4995121951219512false
33/64Exact fraction: 3/64Retained binary64: 0.0468754096/4105Exact fraction: 4096/4105Retained binary64: 0.9978075517661389192/4105Exact fraction: 192/4105Retained binary64: 0.046772228989037762048/4105Exact fraction: 2048/4105Retained binary64: 0.49890377588306944false
41/16Exact fraction: 1/16Retained binary64: 0.0625256/257Exact fraction: 256/257Retained binary64: 0.996108949416342416/257Exact fraction: 16/257Retained binary64: 0.0622568093385214128/257Exact fraction: 128/257Retained binary64: 0.4980544747081712false
55/64Exact fraction: 5/64Retained binary64: 0.0781254096/4121Exact fraction: 4096/4121Retained binary64: 0.993933511283669320/4121Exact fraction: 320/4121Retained binary64: 0.077651055569036652048/4121Exact fraction: 2048/4121Retained binary64: 0.4969667556418345false
63/32Exact fraction: 3/32Retained binary64: 0.093751024/1033Exact fraction: 1024/1033Retained binary64: 0.991287512100677796/1033Exact fraction: 96/1033Retained binary64: 0.09293320425943853512/1033Exact fraction: 512/1033Retained binary64: 0.49564375605033884false
77/64Exact fraction: 7/64Retained binary64: 0.1093754096/4145Exact fraction: 4096/4145Retained binary64: 0.9881785283474065448/4145Exact fraction: 448/4145Retained binary64: 0.108082026537997592048/4145Exact fraction: 2048/4145Retained binary64: 0.49408926417370325false
81/8Exact fraction: 1/8Retained binary64: 0.12564/65Exact fraction: 64/65Retained binary64: 0.98461538461538478/65Exact fraction: 8/65Retained binary64: 0.1230769230769230832/65Exact fraction: 32/65Retained binary64: 0.49230769230769234false
99/64Exact fraction: 9/64Retained binary64: 0.1406254096/4177Exact fraction: 4096/4177Retained binary64: 0.9806080919320086576/4177Exact fraction: 576/4177Retained binary64: 0.13789801292793872048/4177Exact fraction: 2048/4177Retained binary64: 0.4903040459660043false
105/32Exact fraction: 5/32Retained binary64: 0.156251024/1049Exact fraction: 1024/1049Retained binary64: 0.9761677788369876160/1049Exact fraction: 160/1049Retained binary64: 0.15252621544327932512/1049Exact fraction: 512/1049Retained binary64: 0.4880838894184938false
1111/64Exact fraction: 11/64Retained binary64: 0.1718754096/4217Exact fraction: 4096/4217Retained binary64: 0.9713066160777805704/4217Exact fraction: 704/4217Retained binary64: 0.16694332463836852048/4217Exact fraction: 2048/4217Retained binary64: 0.48565330803889023false
123/16Exact fraction: 3/16Retained binary64: 0.1875256/265Exact fraction: 256/265Retained binary64: 0.966037735849056648/265Exact fraction: 48/265Retained binary64: 0.1811320754716981128/265Exact fraction: 128/265Retained binary64: 0.4830188679245283false
1313/64Exact fraction: 13/64Retained binary64: 0.2031254096/4265Exact fraction: 4096/4265Retained binary64: 0.9603751465416178832/4265Exact fraction: 832/4265Retained binary64: 0.195076201641266132048/4265Exact fraction: 2048/4265Retained binary64: 0.4801875732708089false
147/32Exact fraction: 7/32Retained binary64: 0.218751024/1073Exact fraction: 1024/1073Retained binary64: 0.9543336439888164224/1073Exact fraction: 224/1073Retained binary64: 0.2087604846225536512/1073Exact fraction: 512/1073Retained binary64: 0.4771668219944082false
1515/64Exact fraction: 15/64Retained binary64: 0.2343754096/4321Exact fraction: 4096/4321Retained binary64: 0.9479287202036566960/4321Exact fraction: 960/4321Retained binary64: 0.2221707937977322048/4321Exact fraction: 2048/4321Retained binary64: 0.4739643601018283false
161/4Exact fraction: 1/4Retained binary64: 0.2516/17Exact fraction: 16/17Retained binary64: 0.94117647058823534/17Exact fraction: 4/17Retained binary64: 0.235294117647058828/17Exact fraction: 8/17Retained binary64: 0.47058823529411764false
1717/64Exact fraction: 17/64Retained binary64: 0.2656254096/4385Exact fraction: 4096/4385Retained binary64: 0.93409350057012541088/4385Exact fraction: 1088/4385Retained binary64: 0.248118586088939582048/4385Exact fraction: 2048/4385Retained binary64: 0.4670467502850627false
189/32Exact fraction: 9/32Retained binary64: 0.281251024/1105Exact fraction: 1024/1105Retained binary64: 0.9266968325791856288/1105Exact fraction: 288/1105Retained binary64: 0.26063348416289595512/1105Exact fraction: 512/1105Retained binary64: 0.4633484162895928false
1919/64Exact fraction: 19/64Retained binary64: 0.2968754096/4457Exact fraction: 4096/4457Retained binary64: 0.91900381422481491216/4457Exact fraction: 1216/4457Retained binary64: 0.27282925734799192048/4457Exact fraction: 2048/4457Retained binary64: 0.45950190711240746false
205/16Exact fraction: 5/16Retained binary64: 0.3125256/281Exact fraction: 256/281Retained binary64: 0.911032028469750880/281Exact fraction: 80/281Retained binary64: 0.2846975088967972128/281Exact fraction: 128/281Retained binary64: 0.4555160142348754false
2121/64Exact fraction: 21/64Retained binary64: 0.3281254096/4537Exact fraction: 4096/4537Retained binary64: 0.90279920652413491344/4537Exact fraction: 1344/4537Retained binary64: 0.296230989640731752048/4537Exact fraction: 2048/4537Retained binary64: 0.45139960326206746false
2211/32Exact fraction: 11/32Retained binary64: 0.343751024/1145Exact fraction: 1024/1145Retained binary64: 0.8943231441048035352/1145Exact fraction: 352/1145Retained binary64: 0.3074235807860262512/1145Exact fraction: 512/1145Retained binary64: 0.44716157205240176false
2323/64Exact fraction: 23/64Retained binary64: 0.3593754096/4625Exact fraction: 4096/4625Retained binary64: 0.88562162162162161472/4625Exact fraction: 1472/4625Retained binary64: 0.31827027027027032048/4625Exact fraction: 2048/4625Retained binary64: 0.4428108108108108false
243/8Exact fraction: 3/8Retained binary64: 0.37564/73Exact fraction: 64/73Retained binary64: 0.876712328767123224/73Exact fraction: 24/73Retained binary64: 0.328767123287671232/73Exact fraction: 32/73Retained binary64: 0.4383561643835616false
2525/64Exact fraction: 25/64Retained binary64: 0.3906254096/4721Exact fraction: 4096/4721Retained binary64: 0.86761279389959761600/4721Exact fraction: 1600/4721Retained binary64: 0.33891124761703032048/4721Exact fraction: 2048/4721Retained binary64: 0.4338063969497988false
2613/32Exact fraction: 13/32Retained binary64: 0.406251024/1193Exact fraction: 1024/1193Retained binary64: 0.8583403185247276416/1193Exact fraction: 416/1193Retained binary64: 0.34870075440067055512/1193Exact fraction: 512/1193Retained binary64: 0.4291701592623638false
2727/64Exact fraction: 27/64Retained binary64: 0.4218754096/4825Exact fraction: 4096/4825Retained binary64: 0.84891191709844561728/4825Exact fraction: 1728/4825Retained binary64: 0.35813471502590672048/4825Exact fraction: 2048/4825Retained binary64: 0.4244559585492228false
287/16Exact fraction: 7/16Retained binary64: 0.4375256/305Exact fraction: 256/305Retained binary64: 0.839344262295082112/305Exact fraction: 112/305Retained binary64: 0.36721311475409835128/305Exact fraction: 128/305Retained binary64: 0.419672131147541false
2929/64Exact fraction: 29/64Retained binary64: 0.4531254096/4937Exact fraction: 4096/4937Retained binary64: 0.82965363581122141856/4937Exact fraction: 1856/4937Retained binary64: 0.37593680372695972048/4937Exact fraction: 2048/4937Retained binary64: 0.4148268179056107false
3015/32Exact fraction: 15/32Retained binary64: 0.468751024/1249Exact fraction: 1024/1249Retained binary64: 0.8198558847077662480/1249Exact fraction: 480/1249Retained binary64: 0.3843074459567654512/1249Exact fraction: 512/1249Retained binary64: 0.4099279423538831false
3131/64Exact fraction: 31/64Retained binary64: 0.4843754096/5057Exact fraction: 4096/5057Retained binary64: 0.80996638323116481984/5057Exact fraction: 1984/5057Retained binary64: 0.39232746687759542048/5057Exact fraction: 2048/5057Retained binary64: 0.4049831916155824false
321/2Exact fraction: 1/2Retained binary64: 0.54/5Exact fraction: 4/5Retained binary64: 0.82/5Exact fraction: 2/5Retained binary64: 0.42/5Exact fraction: 2/5Retained binary64: 0.4false
3333/64Exact fraction: 33/64Retained binary64: 0.5156254096/5185Exact fraction: 4096/5185Retained binary64: 0.78997107039537132112/5185Exact fraction: 2112/5185Retained binary64: 0.40732883317261332048/5185Exact fraction: 2048/5185Retained binary64: 0.39498553519768564false
3417/32Exact fraction: 17/32Retained binary64: 0.531251024/1313Exact fraction: 1024/1313Retained binary64: 0.7798933739527799544/1313Exact fraction: 544/1313Retained binary64: 0.4143183549124143512/1313Exact fraction: 512/1313Retained binary64: 0.38994668697638996false
3535/64Exact fraction: 35/64Retained binary64: 0.5468754096/5321Exact fraction: 4096/5321Retained binary64: 0.76978011651945122240/5321Exact fraction: 2240/5321Retained binary64: 0.42097350122157492048/5321Exact fraction: 2048/5321Retained binary64: 0.3848900582597256false
369/16Exact fraction: 9/16Retained binary64: 0.5625256/337Exact fraction: 256/337Retained binary64: 0.7596439169139466144/337Exact fraction: 144/337Retained binary64: 0.42729970326409494128/337Exact fraction: 128/337Retained binary64: 0.3798219584569733false
3737/64Exact fraction: 37/64Retained binary64: 0.5781254096/5465Exact fraction: 4096/5465Retained binary64: 0.74949679780420862368/5465Exact fraction: 2368/5465Retained binary64: 0.43330283623055812048/5465Exact fraction: 2048/5465Retained binary64: 0.3747483989021043false
3819/32Exact fraction: 19/32Retained binary64: 0.593751024/1385Exact fraction: 1024/1385Retained binary64: 0.7393501805054151608/1385Exact fraction: 608/1385Retained binary64: 0.4389891696750903512/1385Exact fraction: 512/1385Retained binary64: 0.36967509025270756false
3939/64Exact fraction: 39/64Retained binary64: 0.6093754096/5617Exact fraction: 4096/5617Retained binary64: 0.72921488338970992496/5617Exact fraction: 2496/5617Retained binary64: 0.444365319565604432048/5617Exact fraction: 2048/5617Retained binary64: 0.36460744169485493false
405/8Exact fraction: 5/8Retained binary64: 0.62564/89Exact fraction: 64/89Retained binary64: 0.719101123595505640/89Exact fraction: 40/89Retained binary64: 0.44943820224719132/89Exact fraction: 32/89Retained binary64: 0.3595505617977528false
4141/64Exact fraction: 41/64Retained binary64: 0.6406254096/5777Exact fraction: 4096/5777Retained binary64: 0.70901852172407822624/5777Exact fraction: 2624/5777Retained binary64: 0.45421499047948762048/5777Exact fraction: 2048/5777Retained binary64: 0.3545092608620391false
4221/32Exact fraction: 21/32Retained binary64: 0.656251024/1465Exact fraction: 1024/1465Retained binary64: 0.6989761092150171672/1465Exact fraction: 672/1465Retained binary64: 0.45870307167235497512/1465Exact fraction: 512/1465Retained binary64: 0.34948805460750854false
4343/64Exact fraction: 43/64Retained binary64: 0.6718754096/5945Exact fraction: 4096/5945Retained binary64: 0.6889823380992432752/5945Exact fraction: 2752/5945Retained binary64: 0.462910008410428952048/5945Exact fraction: 2048/5945Retained binary64: 0.3444911690496215false
4411/16Exact fraction: 11/16Retained binary64: 0.6875256/377Exact fraction: 256/377Retained binary64: 0.6790450928381963176/377Exact fraction: 176/377Retained binary64: 0.46684350132625996128/377Exact fraction: 128/377Retained binary64: 0.3395225464190981false
4545/64Exact fraction: 45/64Retained binary64: 0.7031254096/6121Exact fraction: 4096/6121Retained binary64: 0.66917170396993952880/6121Exact fraction: 2880/6121Retained binary64: 0.470511354353863762048/6121Exact fraction: 2048/6121Retained binary64: 0.33458585198496976false
4623/32Exact fraction: 23/32Retained binary64: 0.718751024/1553Exact fraction: 1024/1553Retained binary64: 0.6593689632968448736/1553Exact fraction: 736/1553Retained binary64: 0.4739214423696072512/1553Exact fraction: 512/1553Retained binary64: 0.3296844816484224false
4747/64Exact fraction: 47/64Retained binary64: 0.7343754096/6305Exact fraction: 4096/6305Retained binary64: 0.64964314036478993008/6305Exact fraction: 3008/6305Retained binary64: 0.477081681205392572048/6305Exact fraction: 2048/6305Retained binary64: 0.32482157018239494false
483/4Exact fraction: 3/4Retained binary64: 0.7516/25Exact fraction: 16/25Retained binary64: 0.6412/25Exact fraction: 12/25Retained binary64: 0.488/25Exact fraction: 8/25Retained binary64: 0.32false
4949/64Exact fraction: 49/64Retained binary64: 0.7656254096/6497Exact fraction: 4096/6497Retained binary64: 0.63044482068647073136/6497Exact fraction: 3136/6497Retained binary64: 0.482684315838079142048/6497Exact fraction: 2048/6497Retained binary64: 0.31522241034323534false
5025/32Exact fraction: 25/32Retained binary64: 0.781251024/1649Exact fraction: 1024/1649Retained binary64: 0.6209824135839903800/1649Exact fraction: 800/1649Retained binary64: 0.4851425106124924512/1649Exact fraction: 512/1649Retained binary64: 0.31049120679199516false
5151/64Exact fraction: 51/64Retained binary64: 0.7968754096/6697Exact fraction: 4096/6697Retained binary64: 0.61161714200388233264/6697Exact fraction: 3264/6697Retained binary64: 0.487382410034343742048/6697Exact fraction: 2048/6697Retained binary64: 0.30580857100194114false
5213/16Exact fraction: 13/16Retained binary64: 0.8125256/425Exact fraction: 256/425Retained binary64: 0.6023529411764705208/425Exact fraction: 208/425Retained binary64: 0.4894117647058824128/425Exact fraction: 128/425Retained binary64: 0.30117647058823527false
5353/64Exact fraction: 53/64Retained binary64: 0.8281254096/6905Exact fraction: 4096/6905Retained binary64: 0.5931933381607533392/6905Exact fraction: 3392/6905Retained binary64: 0.491238233164373652048/6905Exact fraction: 2048/6905Retained binary64: 0.2965966690803765false
5427/32Exact fraction: 27/32Retained binary64: 0.843751024/1753Exact fraction: 1024/1753Retained binary64: 0.5841414717626925864/1753Exact fraction: 864/1753Retained binary64: 0.49286936679977184512/1753Exact fraction: 512/1753Retained binary64: 0.29207073588134624false
5555/64Exact fraction: 55/64Retained binary64: 0.8593754096/7121Exact fraction: 4096/7121Retained binary64: 0.5752001123437723520/7121Exact fraction: 3520/7121Retained binary64: 0.494312596545429032048/7121Exact fraction: 2048/7121Retained binary64: 0.287600056171886false
567/8Exact fraction: 7/8Retained binary64: 0.87564/113Exact fraction: 64/113Retained binary64: 0.566371681415929256/113Exact fraction: 56/113Retained binary64: 0.4955752212389380532/113Exact fraction: 32/113Retained binary64: 0.2831858407079646false
5757/64Exact fraction: 57/64Retained binary64: 0.8906254096/7345Exact fraction: 4096/7345Retained binary64: 0.55765827093260723648/7345Exact fraction: 3648/7345Retained binary64: 0.49666439754935332048/7345Exact fraction: 2048/7345Retained binary64: 0.2788291354663036false
5829/32Exact fraction: 29/32Retained binary64: 0.906251024/1865Exact fraction: 1024/1865Retained binary64: 0.5490616621983915928/1865Exact fraction: 928/1865Retained binary64: 0.4975871313672922512/1865Exact fraction: 512/1865Retained binary64: 0.27453083109919574false
5959/64Exact fraction: 59/64Retained binary64: 0.9218754096/7577Exact fraction: 4096/7577Retained binary64: 0.54058334433152973776/7577Exact fraction: 3776/7577Retained binary64: 0.498350270555628862048/7577Exact fraction: 2048/7577Retained binary64: 0.27029167216576483false
6015/16Exact fraction: 15/16Retained binary64: 0.9375256/481Exact fraction: 256/481Retained binary64: 0.5322245322245323240/481Exact fraction: 240/481Retained binary64: 0.498960498960499128/481Exact fraction: 128/481Retained binary64: 0.2661122661122661false
6161/64Exact fraction: 61/64Retained binary64: 0.9531254096/7817Exact fraction: 4096/7817Retained binary64: 0.52398618395804023904/7817Exact fraction: 3904/7817Retained binary64: 0.499424331585007042048/7817Exact fraction: 2048/7817Retained binary64: 0.2619930919790201false
6231/32Exact fraction: 31/32Retained binary64: 0.968751024/1985Exact fraction: 1024/1985Retained binary64: 0.5158690176322418992/1985Exact fraction: 992/1985Retained binary64: 0.49974811083123427512/1985Exact fraction: 512/1985Retained binary64: 0.2579345088161209false
6363/64Exact fraction: 63/64Retained binary64: 0.9843754096/8065Exact fraction: 4096/8065Retained binary64: 0.50787352758834474032/8065Exact fraction: 4032/8065Retained binary64: 0.49993800371977682048/8065Exact fraction: 2048/8065Retained binary64: 0.25393676379417235false
641Exact fraction: 1Retained binary64: 1.01/2Exact fraction: 1/2Retained binary64: 0.51/2Exact fraction: 1/2Retained binary64: 0.51/4Exact fraction: 1/4Retained binary64: 0.25false
6565/64Exact fraction: 65/64Retained binary64: 1.0156254096/8321Exact fraction: 4096/8321Retained binary64: 0.492248527821175344160/8321Exact fraction: 4160/8321Retained binary64: 0.49993991106838122048/8321Exact fraction: 2048/8321Retained binary64: 0.24612426391058767false
6633/32Exact fraction: 33/32Retained binary64: 1.031251024/2113Exact fraction: 1024/2113Retained binary64: 0.48461902508282061056/2113Exact fraction: 1056/2113Retained binary64: 0.4997633696166588512/2113Exact fraction: 512/2113Retained binary64: 0.2423095125414103false
6767/64Exact fraction: 67/64Retained binary64: 1.0468754096/8585Exact fraction: 4096/8585Retained binary64: 0.477111240535818274288/8585Exact fraction: 4288/8585Retained binary64: 0.499475829935934762048/8585Exact fraction: 2048/8585Retained binary64: 0.23855562026790914false
6817/16Exact fraction: 17/16Retained binary64: 1.0625256/545Exact fraction: 256/545Retained binary64: 0.46972477064220186272/545Exact fraction: 272/545Retained binary64: 0.4990825688073395128/545Exact fraction: 128/545Retained binary64: 0.23486238532110093false
6969/64Exact fraction: 69/64Retained binary64: 1.0781254096/8857Exact fraction: 4096/8857Retained binary64: 0.462459071920514834416/8857Exact fraction: 4416/8857Retained binary64: 0.49858868691430512048/8857Exact fraction: 2048/8857Retained binary64: 0.23122953596025742false
7035/32Exact fraction: 35/32Retained binary64: 1.093751024/2249Exact fraction: 1024/2249Retained binary64: 0.45531347265451311120/2249Exact fraction: 1120/2249Retained binary64: 0.4979991107158737512/2249Exact fraction: 512/2249Retained binary64: 0.22765673632725655false
7171/64Exact fraction: 71/64Retained binary64: 1.1093754096/9137Exact fraction: 4096/9137Retained binary64: 0.448287183977235444544/9137Exact fraction: 4544/9137Retained binary64: 0.497318594724745542048/9137Exact fraction: 2048/9137Retained binary64: 0.22414359198861772false
729/8Exact fraction: 9/8Retained binary64: 1.12564/145Exact fraction: 64/145Retained binary64: 0.441379310344827672/145Exact fraction: 72/145Retained binary64: 0.49655172413793132/145Exact fraction: 32/145Retained binary64: 0.2206896551724138false
7373/64Exact fraction: 73/64Retained binary64: 1.1406254096/9425Exact fraction: 4096/9425Retained binary64: 0.434588859416445654672/9425Exact fraction: 4672/9425Retained binary64: 0.49570291777188332048/9425Exact fraction: 2048/9425Retained binary64: 0.21729442970822282false
7437/32Exact fraction: 37/32Retained binary64: 1.156251024/2393Exact fraction: 1024/2393Retained binary64: 0.42791475135812791184/2393Exact fraction: 1184/2393Retained binary64: 0.4947764312578354512/2393Exact fraction: 512/2393Retained binary64: 0.21395737567906395false
7575/64Exact fraction: 75/64Retained binary64: 1.1718754096/9721Exact fraction: 4096/9721Retained binary64: 0.421355827589754154800/9721Exact fraction: 4800/9721Retained binary64: 0.49377636045674312048/9721Exact fraction: 2048/9721Retained binary64: 0.21067791379487708false
7619/16Exact fraction: 19/16Retained binary64: 1.1875256/617Exact fraction: 256/617Retained binary64: 0.41491085899513774304/617Exact fraction: 304/617Retained binary64: 0.49270664505672607128/617Exact fraction: 128/617Retained binary64: 0.20745542949756887false
7777/64Exact fraction: 77/64Retained binary64: 1.2031254096/10025Exact fraction: 4096/10025Retained binary64: 0.40857855361596014928/10025Exact fraction: 4928/10025Retained binary64: 0.4915710723192022048/10025Exact fraction: 2048/10025Retained binary64: 0.20428927680798006false
7839/32Exact fraction: 39/32Retained binary64: 1.218751024/2545Exact fraction: 1024/2545Retained binary64: 0.40235756385068761248/2545Exact fraction: 1248/2545Retained binary64: 0.4903732809430255512/2545Exact fraction: 512/2545Retained binary64: 0.2011787819253438false
7979/64Exact fraction: 79/64Retained binary64: 1.2343754096/10337Exact fraction: 4096/10337Retained binary64: 0.39624649317983945056/10337Exact fraction: 5056/10337Retained binary64: 0.489116765018864252048/10337Exact fraction: 2048/10337Retained binary64: 0.1981232465899197false
805/4Exact fraction: 5/4Retained binary64: 1.2516/41Exact fraction: 16/41Retained binary64: 0.390243902439024420/41Exact fraction: 20/41Retained binary64: 0.48780487804878058/41Exact fraction: 8/41Retained binary64: 0.1951219512195122false
8181/64Exact fraction: 81/64Retained binary64: 1.2656254096/10657Exact fraction: 4096/10657Retained binary64: 0.384348315661067865184/10657Exact fraction: 5184/10657Retained binary64: 0.486440837008538962048/10657Exact fraction: 2048/10657Retained binary64: 0.19217415783053393false
8241/32Exact fraction: 41/32Retained binary64: 1.281251024/2705Exact fraction: 1024/2705Retained binary64: 0.37855822550831791312/2705Exact fraction: 1312/2705Retained binary64: 0.48502772643253234512/2705Exact fraction: 512/2705Retained binary64: 0.18927911275415896false
8383/64Exact fraction: 83/64Retained binary64: 1.2968754096/10985Exact fraction: 4096/10985Retained binary64: 0.37287209831588535312/10985Exact fraction: 5312/10985Retained binary64: 0.483568502503413772048/10985Exact fraction: 2048/10985Retained binary64: 0.18643604915794265false
8421/16Exact fraction: 21/16Retained binary64: 1.3125256/697Exact fraction: 256/697Retained binary64: 0.3672883787661406336/697Exact fraction: 336/697Retained binary64: 0.48206599713055953128/697Exact fraction: 128/697Retained binary64: 0.1836441893830703false
8585/64Exact fraction: 85/64Retained binary64: 1.3281254096/11321Exact fraction: 4096/11321Retained binary64: 0.361805494214292035440/11321Exact fraction: 5440/11321Retained binary64: 0.48052292200335662048/11321Exact fraction: 2048/11321Retained binary64: 0.18090274710714602false
8643/32Exact fraction: 43/32Retained binary64: 1.343751024/2873Exact fraction: 1024/2873Retained binary64: 0.356421858684302151376/2873Exact fraction: 1376/2873Retained binary64: 0.47894187260703097512/2873Exact fraction: 512/2873Retained binary64: 0.17821092934215108false
8787/64Exact fraction: 87/64Retained binary64: 1.3593754096/11665Exact fraction: 4096/11665Retained binary64: 0.35113587655379345568/11665Exact fraction: 5568/11665Retained binary64: 0.47732533219031292048/11665Exact fraction: 2048/11665Retained binary64: 0.1755679382768967false
8811/8Exact fraction: 11/8Retained binary64: 1.37564/185Exact fraction: 64/185Retained binary64: 0.3459459459459459788/185Exact fraction: 88/185Retained binary64: 0.475675675675675732/185Exact fraction: 32/185Retained binary64: 0.17297297297297298false
8989/64Exact fraction: 89/64Retained binary64: 1.3906254096/12017Exact fraction: 4096/12017Retained binary64: 0.340850461845718545696/12017Exact fraction: 5696/12017Retained binary64: 0.47399517350420242048/12017Exact fraction: 2048/12017Retained binary64: 0.17042523092285927false
9045/32Exact fraction: 45/32Retained binary64: 1.406251024/3049Exact fraction: 1024/3049Retained binary64: 0.33584781895703511440/3049Exact fraction: 1440/3049Retained binary64: 0.4722859954083306512/3049Exact fraction: 512/3049Retained binary64: 0.16792390947851754false
9191/64Exact fraction: 91/64Retained binary64: 1.4218754096/12377Exact fraction: 4096/12377Retained binary64: 0.33093641431687815824/12377Exact fraction: 5824/12377Retained binary64: 0.4705502141068112048/12377Exact fraction: 2048/12377Retained binary64: 0.16546820715843905false
9223/16Exact fraction: 23/16Retained binary64: 1.4375256/785Exact fraction: 256/785Retained binary64: 0.32611464968152865368/785Exact fraction: 368/785Retained binary64: 0.46878980891719746128/785Exact fraction: 128/785Retained binary64: 0.16305732484076432false
9393/64Exact fraction: 93/64Retained binary64: 1.4531254096/12745Exact fraction: 4096/12745Retained binary64: 0.321380933699495952/12745Exact fraction: 5952/12745Retained binary64: 0.46700666928207142048/12745Exact fraction: 2048/12745Retained binary64: 0.160690466849745false
9447/32Exact fraction: 47/32Retained binary64: 1.468751024/3233Exact fraction: 1024/3233Retained binary64: 0.31673368388493661504/3233Exact fraction: 1504/3233Retained binary64: 0.46520259820600063512/3233Exact fraction: 512/3233Retained binary64: 0.1583668419424683false
9595/64Exact fraction: 95/64Retained binary64: 1.4843754096/13121Exact fraction: 4096/13121Retained binary64: 0.31217132840484726080/13121Exact fraction: 6080/13121Retained binary64: 0.463379315600945032048/13121Exact fraction: 2048/13121Retained binary64: 0.1560856642024236false
963/2Exact fraction: 3/2Retained binary64: 1.54/13Exact fraction: 4/13Retained binary64: 0.30769230769230776/13Exact fraction: 6/13Retained binary64: 0.461538461538461562/13Exact fraction: 2/13Retained binary64: 0.15384615384615385false
9797/64Exact fraction: 97/64Retained binary64: 1.5156254096/13505Exact fraction: 4096/13505Retained binary64: 0.30329507589781566208/13505Exact fraction: 6208/13505Retained binary64: 0.45968159940762682048/13505Exact fraction: 2048/13505Retained binary64: 0.1516475379489078false
9849/32Exact fraction: 49/32Retained binary64: 1.531251024/3425Exact fraction: 1024/3425Retained binary64: 0.298978102189781051568/3425Exact fraction: 1568/3425Retained binary64: 0.4578102189781022512/3425Exact fraction: 512/3425Retained binary64: 0.14948905109489052false
9999/64Exact fraction: 99/64Retained binary64: 1.5468754096/13897Exact fraction: 4096/13897Retained binary64: 0.29473987191480186336/13897Exact fraction: 6336/13897Retained binary64: 0.455925739368208962048/13897Exact fraction: 2048/13897Retained binary64: 0.1473699359574009false
10025/16Exact fraction: 25/16Retained binary64: 1.5625256/881Exact fraction: 256/881Retained binary64: 0.2905788876276958400/881Exact fraction: 400/881Retained binary64: 0.4540295119182747128/881Exact fraction: 128/881Retained binary64: 0.1452894438138479false
101101/64Exact fraction: 101/64Retained binary64: 1.5781254096/14297Exact fraction: 4096/14297Retained binary64: 0.28649367000069956464/14297Exact fraction: 6464/14297Retained binary64: 0.45212282296985382048/14297Exact fraction: 2048/14297Retained binary64: 0.14324683500034974false
10251/32Exact fraction: 51/32Retained binary64: 1.593751024/3625Exact fraction: 1024/3625Retained binary64: 0.282482758620689671632/3625Exact fraction: 1632/3625Retained binary64: 0.45020689655172413512/3625Exact fraction: 512/3625Retained binary64: 0.14124137931034483false
103103/64Exact fraction: 103/64Retained binary64: 1.6093754096/14705Exact fraction: 4096/14705Retained binary64: 0.2785447126827616592/14705Exact fraction: 6592/14705Retained binary64: 0.448282896973818462048/14705Exact fraction: 2048/14705Retained binary64: 0.1392723563413805false
10413/8Exact fraction: 13/8Retained binary64: 1.62564/233Exact fraction: 64/233Retained binary64: 0.27467811158798283104/233Exact fraction: 104/233Retained binary64: 0.4463519313304721332/233Exact fraction: 32/233Retained binary64: 0.13733905579399142false
105105/64Exact fraction: 105/64Retained binary64: 1.6406254096/15121Exact fraction: 4096/15121Retained binary64: 0.27088155545268176720/15121Exact fraction: 6720/15121Retained binary64: 0.444415051914555942048/15121Exact fraction: 2048/15121Retained binary64: 0.13544077772634086false
10653/32Exact fraction: 53/32Retained binary64: 1.656251024/3833Exact fraction: 1024/3833Retained binary64: 0.267153665536133571696/3833Exact fraction: 1696/3833Retained binary64: 0.44247325854422126512/3833Exact fraction: 512/3833Retained binary64: 0.13357683276806678false
107107/64Exact fraction: 107/64Retained binary64: 1.6718754096/15545Exact fraction: 4096/15545Retained binary64: 0.263493084593116776848/15545Exact fraction: 6848/15545Retained binary64: 0.440527500804117072048/15545Exact fraction: 2048/15545Retained binary64: 0.13174654229655838false
10827/16Exact fraction: 27/16Retained binary64: 1.6875256/985Exact fraction: 256/985Retained binary64: 0.2598984771573604432/985Exact fraction: 432/985Retained binary64: 0.4385786802030457128/985Exact fraction: 128/985Retained binary64: 0.1299492385786802false
109109/64Exact fraction: 109/64Retained binary64: 1.7031254096/15977Exact fraction: 4096/15977Retained binary64: 0.25636852976153226976/15977Exact fraction: 6976/15977Retained binary64: 0.43662765225010952048/15977Exact fraction: 2048/15977Retained binary64: 0.1281842648807661false
11055/32Exact fraction: 55/32Retained binary64: 1.718751024/4049Exact fraction: 1024/4049Retained binary64: 0.25290195109903681760/4049Exact fraction: 1760/4049Retained binary64: 0.4346752284514695512/4049Exact fraction: 512/4049Retained binary64: 0.1264509755495184false
111111/64Exact fraction: 111/64Retained binary64: 1.7343754096/16417Exact fraction: 4096/16417Retained binary64: 0.249497472132545537104/16417Exact fraction: 7104/16417Retained binary64: 0.43272217822988372048/16417Exact fraction: 2048/16417Retained binary64: 0.12474873606627276false
1127/4Exact fraction: 7/4Retained binary64: 1.7516/65Exact fraction: 16/65Retained binary64: 0.2461538461538461728/65Exact fraction: 28/65Retained binary64: 0.43076923076923088/65Exact fraction: 8/65Retained binary64: 0.12307692307692308false
113113/64Exact fraction: 113/64Retained binary64: 1.7656254096/16865Exact fraction: 4096/16865Retained binary64: 0.242869848799288477232/16865Exact fraction: 7232/16865Retained binary64: 0.42881707678624372048/16865Exact fraction: 2048/16865Retained binary64: 0.12143492439964423false
11457/32Exact fraction: 57/32Retained binary64: 1.781251024/4273Exact fraction: 1024/4273Retained binary64: 0.239644278024806921824/4273Exact fraction: 1824/4273Retained binary64: 0.42686637023168733512/4273Exact fraction: 512/4273Retained binary64: 0.11982213901240346false
115115/64Exact fraction: 115/64Retained binary64: 1.7968754096/17321Exact fraction: 4096/17321Retained binary64: 0.236475954044223787360/17321Exact fraction: 7360/17321Retained binary64: 0.42491772992321462048/17321Exact fraction: 2048/17321Retained binary64: 0.11823797702211189false
11629/16Exact fraction: 29/16Retained binary64: 1.8125256/1097Exact fraction: 256/1097Retained binary64: 0.2333637192342753464/1097Exact fraction: 464/1097Retained binary64: 0.422971741112124128/1097Exact fraction: 128/1097Retained binary64: 0.11668185961713765false
117117/64Exact fraction: 117/64Retained binary64: 1.8281254096/17785Exact fraction: 4096/17785Retained binary64: 0.23030643800955867488/17785Exact fraction: 7488/17785Retained binary64: 0.421028956986224332048/17785Exact fraction: 2048/17785Retained binary64: 0.1151532190047793false
11859/32Exact fraction: 59/32Retained binary64: 1.843751024/4505Exact fraction: 1024/4505Retained binary64: 0.227302996670366271888/4505Exact fraction: 1888/4505Retained binary64: 0.4190899001109878512/4505Exact fraction: 512/4505Retained binary64: 0.11365149833518313false
119119/64Exact fraction: 119/64Retained binary64: 1.8593754096/18257Exact fraction: 4096/18257Retained binary64: 0.224352303226159827616/18257Exact fraction: 7616/18257Retained binary64: 0.41715506381114092048/18257Exact fraction: 2048/18257Retained binary64: 0.11217615161307991false
12015/8Exact fraction: 15/8Retained binary64: 1.87564/289Exact fraction: 64/289Retained binary64: 0.22145328719723184120/289Exact fraction: 120/289Retained binary64: 0.4152249134948096732/289Exact fraction: 32/289Retained binary64: 0.11072664359861592false
121121/64Exact fraction: 121/64Retained binary64: 1.8906254096/18737Exact fraction: 4096/18737Retained binary64: 0.21860489939691527744/18737Exact fraction: 7744/18737Retained binary64: 0.41329988792229282048/18737Exact fraction: 2048/18737Retained binary64: 0.1093024496984576false
12261/32Exact fraction: 61/32Retained binary64: 1.906251024/4745Exact fraction: 1024/4745Retained binary64: 0.215806111696522671952/4745Exact fraction: 1952/4745Retained binary64: 0.4113804004214963512/4745Exact fraction: 512/4745Retained binary64: 0.10790305584826133false
123123/64Exact fraction: 123/64Retained binary64: 1.9218754096/19225Exact fraction: 4096/19225Retained binary64: 0.213055916775032527872/19225Exact fraction: 7872/19225Retained binary64: 0.40946684005201562048/19225Exact fraction: 2048/19225Retained binary64: 0.10652795838751626false
12431/16Exact fraction: 31/16Retained binary64: 1.9375256/1217Exact fraction: 256/1217Retained binary64: 0.21035332785538208496/1217Exact fraction: 496/1217Retained binary64: 0.4075595727198028128/1217Exact fraction: 128/1217Retained binary64: 0.10517666392769104false
125125/64Exact fraction: 125/64Retained binary64: 1.9531254096/19721Exact fraction: 4096/19721Retained binary64: 0.207697378429085758000/19721Exact fraction: 8000/19721Retained binary64: 0.40565894224430812048/19721Exact fraction: 2048/19721Retained binary64: 0.10384868921454288false
12663/32Exact fraction: 63/32Retained binary64: 1.968751024/4993Exact fraction: 1024/4993Retained binary64: 0.205087121970759072016/4993Exact fraction: 2016/4993Retained binary64: 0.4037652713799319512/4993Exact fraction: 512/4993Retained binary64: 0.10254356098537953false
127127/64Exact fraction: 127/64Retained binary64: 1.9843754096/20225Exact fraction: 4096/20225Retained binary64: 0.202521631644004948128/20225Exact fraction: 8128/20225Retained binary64: 0.40187886279357232048/20225Exact fraction: 2048/20225Retained binary64: 0.10126081582200247false
1282Exact fraction: 2Retained binary64: 2.01/5Exact fraction: 1/5Retained binary64: 0.22/5Exact fraction: 2/5Retained binary64: 0.41/10Exact fraction: 1/10Retained binary64: 0.1false
129129/64Exact fraction: 129/64Retained binary64: 2.0156254096/20737Exact fraction: 4096/20737Retained binary64: 0.197521338670010128256/20737Exact fraction: 8256/20737Retained binary64: 0.398128948256739172048/20737Exact fraction: 2048/20737Retained binary64: 0.09876066933500506false
13065/32Exact fraction: 65/32Retained binary64: 2.031251024/5249Exact fraction: 1024/5249Retained binary64: 0.195084778052962482080/5249Exact fraction: 2080/5249Retained binary64: 0.39626595542008003512/5249Exact fraction: 512/5249Retained binary64: 0.09754238902648124false
131131/64Exact fraction: 131/64Retained binary64: 2.0468754096/21257Exact fraction: 4096/21257Retained binary64: 0.19268946699910628384/21257Exact fraction: 8384/21257Retained binary64: 0.394411252763795462048/21257Exact fraction: 2048/21257Retained binary64: 0.0963447334995531false
13233/16Exact fraction: 33/16Retained binary64: 2.0625256/1345Exact fraction: 256/1345Retained binary64: 0.19033457249070632528/1345Exact fraction: 528/1345Retained binary64: 0.3925650557620818128/1345Exact fraction: 128/1345Retained binary64: 0.09516728624535316false
133133/64Exact fraction: 133/64Retained binary64: 2.0781254096/21785Exact fraction: 4096/21785Retained binary64: 0.188019279320633488512/21785Exact fraction: 8512/21785Retained binary64: 0.390727564838191432048/21785Exact fraction: 2048/21785Retained binary64: 0.09400963966031674false
13467/32Exact fraction: 67/32Retained binary64: 2.093751024/5513Exact fraction: 1024/5513Retained binary64: 0.18574278976963542144/5513Exact fraction: 2144/5513Retained binary64: 0.38889896608017416512/5513Exact fraction: 512/5513Retained binary64: 0.0928713948848177false
135135/64Exact fraction: 135/64Retained binary64: 2.1093754096/22321Exact fraction: 4096/22321Retained binary64: 0.183504323283007038640/22321Exact fraction: 8640/22321Retained binary64: 0.3870794319250932048/22321Exact fraction: 2048/22321Retained binary64: 0.09175216164150352false
13617/8Exact fraction: 17/8Retained binary64: 2.12564/353Exact fraction: 64/353Retained binary64: 0.1813031161473088136/353Exact fraction: 136/353Retained binary64: 0.3852691218130311332/353Exact fraction: 32/353Retained binary64: 0.0906515580736544false
137137/64Exact fraction: 137/64Retained binary64: 2.1406254096/22865Exact fraction: 4096/22865Retained binary64: 0.17913842116772368768/22865Exact fraction: 8768/22865Retained binary64: 0.383468182812158332048/22865Exact fraction: 2048/22865Retained binary64: 0.0895692105838618false
13869/32Exact fraction: 69/32Retained binary64: 2.156251024/5785Exact fraction: 1024/5785Retained binary64: 0.1770095073465862208/5785Exact fraction: 2208/5785Retained binary64: 0.38167675021607606512/5785Exact fraction: 512/5785Retained binary64: 0.088504753673293false
139139/64Exact fraction: 139/64Retained binary64: 2.1718754096/23417Exact fraction: 4096/23417Retained binary64: 0.174915659563564928896/23417Exact fraction: 8896/23417Retained binary64: 0.37989494811461762048/23417Exact fraction: 2048/23417Retained binary64: 0.08745782978178246false
14035/16Exact fraction: 35/16Retained binary64: 2.1875256/1481Exact fraction: 256/1481Retained binary64: 0.17285617825793384560/1481Exact fraction: 560/1481Retained binary64: 0.37812288993923027128/1481Exact fraction: 128/1481Retained binary64: 0.08642808912896692false
141141/64Exact fraction: 141/64Retained binary64: 2.2031254096/23977Exact fraction: 4096/23977Retained binary64: 0.170830379113316929024/23977Exact fraction: 9024/23977Retained binary64: 0.376360678984026362048/23977Exact fraction: 2048/23977Retained binary64: 0.08541518955665846false
14271/32Exact fraction: 71/32Retained binary64: 2.218751024/6065Exact fraction: 1024/6065Retained binary64: 0.16883759274525972272/6065Exact fraction: 2272/6065Retained binary64: 0.3746084089035449512/6065Exact fraction: 512/6065Retained binary64: 0.08441879637262985false
143143/64Exact fraction: 143/64Retained binary64: 2.2343754096/24545Exact fraction: 4096/24545Retained binary64: 0.166877164391933179152/24545Exact fraction: 9152/24545Retained binary64: 0.37286616418822572048/24545Exact fraction: 2048/24545Retained binary64: 0.08343858219596659false
1449/4Exact fraction: 9/4Retained binary64: 2.2516/97Exact fraction: 16/97Retained binary64: 0.1649484536082474236/97Exact fraction: 36/97Retained binary64: 0.37113402061855678/97Exact fraction: 8/97Retained binary64: 0.08247422680412371false
145145/64Exact fraction: 145/64Retained binary64: 2.2656254096/25121Exact fraction: 4096/25121Retained binary64: 0.16305083396361619280/25121Exact fraction: 9280/25121Retained binary64: 0.369412045698817752048/25121Exact fraction: 2048/25121Retained binary64: 0.08152541698180805false
14673/32Exact fraction: 73/32Retained binary64: 2.281251024/6353Exact fraction: 1024/6353Retained binary64: 0.16118369274358572336/6353Exact fraction: 2336/6353Retained binary64: 0.3677002990713049512/6353Exact fraction: 512/6353Retained binary64: 0.08059184637179286false
147147/64Exact fraction: 147/64Retained binary64: 2.2968754096/25705Exact fraction: 4096/25705Retained binary64: 0.15934643065551459408/25705Exact fraction: 9408/25705Retained binary64: 0.365998832911884842048/25705Exact fraction: 2048/25705Retained binary64: 0.07967321532775724false
14837/16Exact fraction: 37/16Retained binary64: 2.3125256/1625Exact fraction: 256/1625Retained binary64: 0.15753846153846154592/1625Exact fraction: 592/1625Retained binary64: 0.36430769230769233128/1625Exact fraction: 128/1625Retained binary64: 0.07876923076923077false
149149/64Exact fraction: 149/64Retained binary64: 2.3281254096/26297Exact fraction: 4096/26297Retained binary64: 0.155759212077423279536/26297Exact fraction: 9536/26297Retained binary64: 0.362626915617751072048/26297Exact fraction: 2048/26297Retained binary64: 0.07787960603871164false
15075/32Exact fraction: 75/32Retained binary64: 2.343751024/6649Exact fraction: 1024/6649Retained binary64: 0.15400812152203342400/6649Exact fraction: 2400/6649Retained binary64: 0.36095653481726575512/6649Exact fraction: 512/6649Retained binary64: 0.0770040607610167false
151151/64Exact fraction: 151/64Retained binary64: 2.3593754096/26897Exact fraction: 4096/26897Retained binary64: 0.152284641409822659664/26897Exact fraction: 9664/26897Retained binary64: 0.35929657582630032048/26897Exact fraction: 2048/26897Retained binary64: 0.07614232070491132false
15219/8Exact fraction: 19/8Retained binary64: 2.37564/425Exact fraction: 64/425Retained binary64: 0.15058823529411763152/425Exact fraction: 152/425Retained binary64: 0.3576470588235294332/425Exact fraction: 32/425Retained binary64: 0.07529411764705882false
153153/64Exact fraction: 153/64Retained binary64: 2.3906254096/27505Exact fraction: 4096/27505Retained binary64: 0.14891837847664069792/27505Exact fraction: 9792/27505Retained binary64: 0.356007998545718962048/27505Exact fraction: 2048/27505Retained binary64: 0.0744591892383203false
15477/32Exact fraction: 77/32Retained binary64: 2.406251024/6953Exact fraction: 1024/6953Retained binary64: 0.147274557744858342464/6953Exact fraction: 2464/6953Retained binary64: 0.35437940457356537512/6953Exact fraction: 512/6953Retained binary64: 0.07363727887242917false
155155/64Exact fraction: 155/64Retained binary64: 2.4218754096/28121Exact fraction: 4096/28121Retained binary64: 0.1456562711141149920/28121Exact fraction: 9920/28121Retained binary64: 0.35276128160449492048/28121Exact fraction: 2048/28121Retained binary64: 0.072828135557057false
15639/16Exact fraction: 39/16Retained binary64: 2.4375256/1777Exact fraction: 256/1777Retained binary64: 0.14406302757456388624/1777Exact fraction: 624/1777Retained binary64: 0.35115362971299946128/1777Exact fraction: 128/1777Retained binary64: 0.07203151378728194false
157157/64Exact fraction: 157/64Retained binary64: 2.4531254096/28745Exact fraction: 4096/28745Retained binary64: 0.142494346842929210048/28745Exact fraction: 10048/28745Retained binary64: 0.34955644459906072048/28745Exact fraction: 2048/28745Retained binary64: 0.0712471734214646false
15879/32Exact fraction: 79/32Retained binary64: 2.468751024/7265Exact fraction: 1024/7265Retained binary64: 0.140949759119064022528/7265Exact fraction: 2528/7265Retained binary64: 0.34796971782518926512/7265Exact fraction: 512/7265Retained binary64: 0.07047487955953201false
159159/64Exact fraction: 159/64Retained binary64: 2.4843754096/29377Exact fraction: 4096/29377Retained binary64: 0.1394288048473295610176/29377Exact fraction: 10176/29377Retained binary64: 0.346393437042584352048/29377Exact fraction: 2048/29377Retained binary64: 0.06971440242366478false
1605/2Exact fraction: 5/2Retained binary64: 2.54/29Exact fraction: 4/29Retained binary64: 0.1379310344827586210/29Exact fraction: 10/29Retained binary64: 0.34482758620689662/29Exact fraction: 2/29Retained binary64: 0.06896551724137931false
161161/64Exact fraction: 161/64Retained binary64: 2.5156254096/30017Exact fraction: 4096/30017Retained binary64: 0.1364560082619848710304/30017Exact fraction: 10304/30017Retained binary64: 0.34327214578405572048/30017Exact fraction: 2048/30017Retained binary64: 0.06822800413099243false
16281/32Exact fraction: 81/32Retained binary64: 2.531251024/7585Exact fraction: 1024/7585Retained binary64: 0.135003295978905732592/7585Exact fraction: 2592/7585Retained binary64: 0.34172709294660514512/7585Exact fraction: 512/7585Retained binary64: 0.06750164798945286false
163163/64Exact fraction: 163/64Retained binary64: 2.5468754096/30665Exact fraction: 4096/30665Retained binary64: 0.1335724767650415810432/30665Exact fraction: 10432/30665Retained binary64: 0.340192401760965272048/30665Exact fraction: 2048/30665Retained binary64: 0.06678623838252079false
16441/16Exact fraction: 41/16Retained binary64: 2.5625256/1937Exact fraction: 256/1937Retained binary64: 0.13216313887454828656/1937Exact fraction: 656/1937Retained binary64: 0.33866804336602996128/1937Exact fraction: 128/1937Retained binary64: 0.06608156943727414false
165165/64Exact fraction: 165/64Retained binary64: 2.5781254096/31321Exact fraction: 4096/31321Retained binary64: 0.1307748794738354510560/31321Exact fraction: 10560/31321Retained binary64: 0.3371539861434822048/31321Exact fraction: 2048/31321Retained binary64: 0.06538743973691773false
16683/32Exact fraction: 83/32Retained binary64: 2.593751024/7913Exact fraction: 1024/7913Retained binary64: 0.129407304435738672656/7913Exact fraction: 2656/7913Retained binary64: 0.3356501958801971512/7913Exact fraction: 512/7913Retained binary64: 0.06470365221786933false
167167/64Exact fraction: 167/64Retained binary64: 2.6093754096/31985Exact fraction: 4096/31985Retained binary64: 0.1280600281381897910688/31985Exact fraction: 10688/31985Retained binary64: 0.3341566359230892048/31985Exact fraction: 2048/31985Retained binary64: 0.06403001406909489false
16821/8Exact fraction: 21/8Retained binary64: 2.62564/505Exact fraction: 64/505Retained binary64: 0.12673267326732673168/505Exact fraction: 168/505Retained binary64: 0.332673267326732732/505Exact fraction: 32/505Retained binary64: 0.06336633663366337false
169169/64Exact fraction: 169/64Retained binary64: 2.6406254096/32657Exact fraction: 4096/32657Retained binary64: 0.1254248706249808710816/32657Exact fraction: 10816/32657Retained binary64: 0.33120004899409012048/32657Exact fraction: 2048/32657Retained binary64: 0.06271243531249043false
17085/32Exact fraction: 85/32Retained binary64: 2.656251024/8249Exact fraction: 1024/8249Retained binary64: 0.124136258940477642720/8249Exact fraction: 2720/8249Retained binary64: 0.3297369378106437512/8249Exact fraction: 512/8249Retained binary64: 0.06206812947023882false
171171/64Exact fraction: 171/64Retained binary64: 2.6718754096/33337Exact fraction: 4096/33337Retained binary64: 0.1228664846866844710944/33337Exact fraction: 10944/33337Retained binary64: 0.32828388877223512048/33337Exact fraction: 2048/33337Retained binary64: 0.061433242343342236false
17243/16Exact fraction: 43/16Retained binary64: 2.6875256/2105Exact fraction: 256/2105Retained binary64: 0.12161520190023753688/2105Exact fraction: 688/2105Retained binary64: 0.3268408551068884128/2105Exact fraction: 128/2105Retained binary64: 0.060807600950118765false
173173/64Exact fraction: 173/64Retained binary64: 2.7031254096/34025Exact fraction: 4096/34025Retained binary64: 0.1203820720058780311072/34025Exact fraction: 11072/34025Retained binary64: 0.32540778839088912048/34025Exact fraction: 2048/34025Retained binary64: 0.06019103600293901false
17487/32Exact fraction: 87/32Retained binary64: 2.718751024/8593Exact fraction: 1024/8593Retained binary64: 0.119166763644827192784/8593Exact fraction: 2784/8593Retained binary64: 0.32398463865937394512/8593Exact fraction: 512/8593Retained binary64: 0.05958338182241359false
175175/64Exact fraction: 175/64Retained binary64: 2.7343754096/34721Exact fraction: 4096/34721Retained binary64: 0.1179689525071282411200/34721Exact fraction: 11200/34721Retained binary64: 0.32257135451167882048/34721Exact fraction: 2048/34721Retained binary64: 0.05898447625356412false
17611/4Exact fraction: 11/4Retained binary64: 2.7516/137Exact fraction: 16/137Retained binary64: 0.1167883211678832144/137Exact fraction: 44/137Retained binary64: 0.321167883211678838/137Exact fraction: 8/137Retained binary64: 0.058394160583941604false
177177/64Exact fraction: 177/64Retained binary64: 2.7656254096/35425Exact fraction: 4096/35425Retained binary64: 0.1156245589273112211328/35425Exact fraction: 11328/35425Retained binary64: 0.31977417078334512048/35425Exact fraction: 2048/35425Retained binary64: 0.05781227946365561false
17889/32Exact fraction: 89/32Retained binary64: 2.781251024/8945Exact fraction: 1024/8945Retained binary64: 0.114477361654555622848/8945Exact fraction: 2848/8945Retained binary64: 0.3183901621017328512/8945Exact fraction: 512/8945Retained binary64: 0.05723868082727781false
179179/64Exact fraction: 179/64Retained binary64: 2.7968754096/36137Exact fraction: 4096/36137Retained binary64: 0.1133464316351661711456/36137Exact fraction: 11456/36137Retained binary64: 0.31701580097960542048/36137Exact fraction: 2048/36137Retained binary64: 0.056673215817583084false
18045/16Exact fraction: 45/16Retained binary64: 2.8125256/2281Exact fraction: 256/2281Retained binary64: 0.11223147742218326720/2281Exact fraction: 720/2281Retained binary64: 0.3156510302498904128/2281Exact fraction: 128/2281Retained binary64: 0.05611573871109163false
181181/64Exact fraction: 181/64Retained binary64: 2.8281254096/36857Exact fraction: 4096/36857Retained binary64: 0.1111322136907507411584/36857Exact fraction: 11584/36857Retained binary64: 0.314295791844154462048/36857Exact fraction: 2048/36857Retained binary64: 0.05556610684537537false
18291/32Exact fraction: 91/32Retained binary64: 2.843751024/9305Exact fraction: 1024/9305Retained binary64: 0.110048361096184852912/9305Exact fraction: 2912/9305Retained binary64: 0.31295002686727563512/9305Exact fraction: 512/9305Retained binary64: 0.055024180548092425false
183183/64Exact fraction: 183/64Retained binary64: 2.8593754096/37585Exact fraction: 4096/37585Retained binary64: 0.1089796461354263611712/37585Exact fraction: 11712/37585Retained binary64: 0.311613675668484752048/37585Exact fraction: 2048/37585Retained binary64: 0.05448982306771318false
18423/8Exact fraction: 23/8Retained binary64: 2.87564/593Exact fraction: 64/593Retained binary64: 0.10792580101180438184/593Exact fraction: 184/593Retained binary64: 0.310286677908937632/593Exact fraction: 32/593Retained binary64: 0.05396290050590219false
185185/64Exact fraction: 185/64Retained binary64: 2.8906254096/38321Exact fraction: 4096/38321Retained binary64: 0.1068865635030401111840/38321Exact fraction: 11840/38321Retained binary64: 0.308968972625975342048/38321Exact fraction: 2048/38321Retained binary64: 0.05344328175152006false
18693/32Exact fraction: 93/32Retained binary64: 2.906251024/9673Exact fraction: 1024/9673Retained binary64: 0.105861676832420132976/9673Exact fraction: 2976/9673Retained binary64: 0.30766049829422104512/9673Exact fraction: 512/9673Retained binary64: 0.05293083841621007false
187187/64Exact fraction: 187/64Retained binary64: 2.9218754096/39065Exact fraction: 4096/39065Retained binary64: 0.1048508895430692511968/39065Exact fraction: 11968/39065Retained binary64: 0.306361192883655432048/39065Exact fraction: 2048/39065Retained binary64: 0.05242544477153462false
18847/16Exact fraction: 47/16Retained binary64: 2.9375256/2465Exact fraction: 256/2465Retained binary64: 0.10385395537525355752/2465Exact fraction: 752/2465Retained binary64: 0.3050709939148073128/2465Exact fraction: 128/2465Retained binary64: 0.051926977687626774false
189189/64Exact fraction: 189/64Retained binary64: 2.9531254096/39817Exact fraction: 4096/39817Retained binary64: 0.102870633146645912096/39817Exact fraction: 12096/39817Retained binary64: 0.30378983851118872048/39817Exact fraction: 2048/39817Retained binary64: 0.05143531657332295false
19095/32Exact fraction: 95/32Retained binary64: 2.968751024/10049Exact fraction: 1024/10049Retained binary64: 0.101900686635486113040/10049Exact fraction: 3040/10049Retained binary64: 0.3025176634490994512/10049Exact fraction: 512/10049Retained binary64: 0.050950343317743056false
191191/64Exact fraction: 191/64Retained binary64: 2.9843754096/40577Exact fraction: 4096/40577Retained binary64: 0.1009438844665697212224/40577Exact fraction: 12224/40577Retained binary64: 0.301254405204919032048/40577Exact fraction: 2048/40577Retained binary64: 0.05047194223328486false
1923Exact fraction: 3Retained binary64: 3.01/10Exact fraction: 1/10Retained binary64: 0.13/10Exact fraction: 3/10Retained binary64: 0.31/20Exact fraction: 1/20Retained binary64: 0.05false
193193/64Exact fraction: 193/64Retained binary64: 3.0156254096/41345Exact fraction: 4096/41345Retained binary64: 0.0990688112226387812352/41345Exact fraction: 12352/41345Retained binary64: 0.298754383843272048/41345Exact fraction: 2048/41345Retained binary64: 0.04953440561131939false
19497/32Exact fraction: 97/32Retained binary64: 3.031251024/10433Exact fraction: 1024/10433Retained binary64: 0.098150100642193043104/10433Exact fraction: 3104/10433Retained binary64: 0.29751749257164767512/10433Exact fraction: 512/10433Retained binary64: 0.04907505032109652false
195195/64Exact fraction: 195/64Retained binary64: 3.0468754096/42121Exact fraction: 4096/42121Retained binary64: 0.0972436551838750212480/42121Exact fraction: 12480/42121Retained binary64: 0.296289261888369242048/42121Exact fraction: 2048/42121Retained binary64: 0.04862182759193751false
19649/16Exact fraction: 49/16Retained binary64: 3.0625256/2657Exact fraction: 256/2657Retained binary64: 0.09634926608957471784/2657Exact fraction: 784/2657Retained binary64: 0.2950696273993225128/2657Exact fraction: 128/2657Retained binary64: 0.048174633044787356false
197197/64Exact fraction: 197/64Retained binary64: 3.0781254096/42905Exact fraction: 4096/42905Retained binary64: 0.0954667288194849112608/42905Exact fraction: 12608/42905Retained binary64: 0.293858524647476972048/42905Exact fraction: 2048/42905Retained binary64: 0.047733364409742456false
19899/32Exact fraction: 99/32Retained binary64: 3.093751024/10825Exact fraction: 1024/10825Retained binary64: 0.09459584295612013168/10825Exact fraction: 3168/10825Retained binary64: 0.29265588914549656512/10825Exact fraction: 512/10825Retained binary64: 0.04729792147806005false
199199/64Exact fraction: 199/64Retained binary64: 3.1093754096/43697Exact fraction: 4096/43697Retained binary64: 0.0937364121106712112736/43697Exact fraction: 12736/43697Retained binary64: 0.29146165640661832048/43697Exact fraction: 2048/43697Retained binary64: 0.04686820605533561false
20025/8Exact fraction: 25/8Retained binary64: 3.12564/689Exact fraction: 64/689Retained binary64: 0.09288824383164006200/689Exact fraction: 200/689Retained binary64: 0.2902757619738751732/689Exact fraction: 32/689Retained binary64: 0.04644412191582003false
201201/64Exact fraction: 201/64Retained binary64: 3.1406254096/44497Exact fraction: 4096/44497Retained binary64: 0.0920511495156976912864/44497Exact fraction: 12864/44497Retained binary64: 0.289098141447738042048/44497Exact fraction: 2048/44497Retained binary64: 0.046025574757848846false
202101/32Exact fraction: 101/32Retained binary64: 3.156251024/11225Exact fraction: 1024/11225Retained binary64: 0.09122494432071273232/11225Exact fraction: 3232/11225Retained binary64: 0.28792873051224943512/11225Exact fraction: 512/11225Retained binary64: 0.04561247216035635false
203203/64Exact fraction: 203/64Retained binary64: 3.1718754096/45305Exact fraction: 4096/45305Retained binary64: 0.0904094470808961512992/45305Exact fraction: 12992/45305Retained binary64: 0.286767464959717442048/45305Exact fraction: 2048/45305Retained binary64: 0.04520472354044808false
20451/16Exact fraction: 51/16Retained binary64: 3.1875256/2857Exact fraction: 256/2857Retained binary64: 0.0896044802240112816/2857Exact fraction: 816/2857Retained binary64: 0.2856142807140357128/2857Exact fraction: 128/2857Retained binary64: 0.0448022401120056false
205205/64Exact fraction: 205/64Retained binary64: 3.2031254096/46121Exact fraction: 4096/46121Retained binary64: 0.0888098696905964713120/46121Exact fraction: 13120/46121Retained binary64: 0.284469113852691862048/46121Exact fraction: 2048/46121Retained binary64: 0.044404934845298234false
206103/32Exact fraction: 103/32Retained binary64: 3.218751024/11633Exact fraction: 1024/11633Retained binary64: 0.088025444855153443296/11633Exact fraction: 3296/11633Retained binary64: 0.28333190062752517512/11633Exact fraction: 512/11633Retained binary64: 0.04401272242757672false
207207/64Exact fraction: 207/64Retained binary64: 3.2343754096/46945Exact fraction: 4096/46945Retained binary64: 0.0872510384492491313248/46945Exact fraction: 13248/46945Retained binary64: 0.28220257748429012048/46945Exact fraction: 2048/46945Retained binary64: 0.043625519224624563false
20813/4Exact fraction: 13/4Retained binary64: 3.2516/185Exact fraction: 16/185Retained binary64: 0.0864864864864864952/185Exact fraction: 52/185Retained binary64: 0.28108108108108118/185Exact fraction: 8/185Retained binary64: 0.043243243243243246false
209209/64Exact fraction: 209/64Retained binary64: 3.2656254096/47777Exact fraction: 4096/47777Retained binary64: 0.0857316281892961113376/47777Exact fraction: 13376/47777Retained binary64: 0.27996734830567012048/47777Exact fraction: 2048/47777Retained binary64: 0.042865814094648055false
210105/32Exact fraction: 105/32Retained binary64: 3.281251024/12049Exact fraction: 1024/12049Retained binary64: 0.084986305917503533360/12049Exact fraction: 3360/12049Retained binary64: 0.27886131629180844512/12049Exact fraction: 512/12049Retained binary64: 0.042493152958751765false
211211/64Exact fraction: 211/64Retained binary64: 3.2968754096/48617Exact fraction: 4096/48617Retained binary64: 0.0842503650986280613504/48617Exact fraction: 13504/48617Retained binary64: 0.277762922434539372048/48617Exact fraction: 2048/48617Retained binary64: 0.04212518254931403false
21253/16Exact fraction: 53/16Retained binary64: 3.3125256/3065Exact fraction: 256/3065Retained binary64: 0.0835236541598695848/3065Exact fraction: 848/3065Retained binary64: 0.2766721044045677128/3065Exact fraction: 128/3065Retained binary64: 0.04176182707993475false
213213/64Exact fraction: 213/64Retained binary64: 3.3281254096/49465Exact fraction: 4096/49465Retained binary64: 0.0828060244617406213632/49465Exact fraction: 13632/49465Retained binary64: 0.275588800161730542048/49465Exact fraction: 2048/49465Retained binary64: 0.04140301223087031false
214107/32Exact fraction: 107/32Retained binary64: 3.343751024/12473Exact fraction: 1024/12473Retained binary64: 0.082097330233303943424/12473Exact fraction: 3424/12473Retained binary64: 0.27451294796761005512/12473Exact fraction: 512/12473Retained binary64: 0.04104866511665197false
215215/64Exact fraction: 215/64Retained binary64: 3.3593754096/50321Exact fraction: 4096/50321Retained binary64: 0.081397428508972413760/50321Exact fraction: 13760/50321Retained binary64: 0.273444486397329162048/50321Exact fraction: 2048/50321Retained binary64: 0.0406987142544862false
21627/8Exact fraction: 27/8Retained binary64: 3.37564/793Exact fraction: 64/793Retained binary64: 0.0807061790668348216/793Exact fraction: 216/793Retained binary64: 0.272383354350567532/793Exact fraction: 32/793Retained binary64: 0.0403530895334174false
217217/64Exact fraction: 217/64Retained binary64: 3.3906254096/51185Exact fraction: 4096/51185Retained binary64: 0.0800234443684673313888/51185Exact fraction: 13888/51185Retained binary64: 0.27132949106183452048/51185Exact fraction: 2048/51185Retained binary64: 0.04001172218423366false
218109/32Exact fraction: 109/32Retained binary64: 3.406251024/12905Exact fraction: 1024/12905Retained binary64: 0.079349089500193723488/12905Exact fraction: 3488/12905Retained binary64: 0.2702828361100349512/12905Exact fraction: 512/12905Retained binary64: 0.03967454475009686false
219219/64Exact fraction: 219/64Retained binary64: 3.4218754096/52057Exact fraction: 4096/52057Retained binary64: 0.0786829821157577214016/52057Exact fraction: 14016/52057Retained binary64: 0.269243329427358462048/52057Exact fraction: 2048/52057Retained binary64: 0.03934149105787886false
22055/16Exact fraction: 55/16Retained binary64: 3.4375256/3281Exact fraction: 256/3281Retained binary64: 0.07802499238037183880/3281Exact fraction: 880/3281Retained binary64: 0.26821091130752817128/3281Exact fraction: 128/3281Retained binary64: 0.039012496190185916false
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All four exact AC/DC response controls (4 rows)

Physical fields are Re(phasor exp(−iωt)). Peak-amplitude cycle averages require ω>0. At DC, the cycle coefficient is null/unassigned and constant power uses coefficient1, not1/2. No magnitude/phase curve is claimed.

All four exact AC/DC response controls
RowZ = ωτ_mRe(σ/σ₀)Im(σ/σ₀)Nonzero-frequency peak-cycle power coefficient (null at DC)Constant DC point?
0DC: no nonzero-frequency cycle-average coefficient is assigned here. Constant DC power has coefficient1.0Exact fraction: 0Retained binary64: 0.01Exact fraction: 1Retained binary64: 1.00Exact fraction: 0Retained binary64: 0.0nulltrue
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drudeAC: source scales, exact constants and scope flags (4 rows)

Physical fields are Re(phasor exp(−iωt)). Peak-amplitude cycle averages require ω>0. At DC, the cycle coefficient is null/unassigned and constant power uses coefficient1, not1/2. No magnitude/phase curve is claimed.

drudeAC: source scales, exact constants and scope flags
Quantity/source conditionAdmitted value
phasor ConventionRe(phasor exp(-iomega t)); peakamplitude;cycleaverages onlyomega>0
constant D C Power Coefficient1Exact fraction: 1Retained binary64: 1.0
Z1phasephasorarg+pi/4, physicalcurrentlagpi/4
native Magnitude Or Phase Curve Claimedfalse

Focus a labelled graph or table region and use arrow keys, or scroll sideways. Fixed-size readable diagrams stay within their own regions. Open the complete data disclosures for exact values, retained binary64 numbers and precision qualifications.

Exact normalized Drude conductivity controls for e^(−iωt)
Z = ωτ_mRe(σ/σ₀)Im(σ/σ₀)Interpretation
010DC response coefficient
11/21/2Physical current lags for the declared convention
21/52/5Both components retained
41/174/17Reactive component exceeds dissipative part
16

Material response beyond one scalar

Temperature, carrier density, band filling, disorder and applied bias can change conductivity. A short-range linear temperature approximation to resistivity, when experimentally justified over a stated range, is not a universal law extrapolating to every temperature or phase. No empirical temperature coefficient is selected for our symbolic models.

In an anisotropic material, a tensor can relate current to the local driving field. The symmetric dissipative part controls the quadratic work J·E, while an antisymmetric contribution can deflect current without the same local quadratic dissipation. Magnetic and Hall responses require their own field and sign conventions, developed with the later magnetic-force material.

For inhomogeneous systems, gradients of carrier concentration and chemical potential can drive diffusion and thermoelectric transport. A field-only Ohmic law may need replacement by coupled transport relations. Contact potentials and a measured terminal voltage also require a consistent electrochemical description; they are not determined solely by multiplying one bulk σ by a geometrical factor.

These qualifications connect electrical conduction to thermodynamics and transport rather than separating it from them. A positive local scalar σ describes a useful passive regime, but a material label such as metal or semiconductor does not by itself specify that regime.

17

Return path: where local resistance ends

The local resistance integral assumes that cross-sections can be assigned an appropriate local constitutive response. If a mean free path becomes comparable with a constriction, if contacts dominate or if phase-coherent and ballistic transport matter, a bulk ηℓ/A description can fail. Transmission through states and contacts then becomes part of the transport problem.

A microscopic band description also changes how carrier velocity, mass and scattering are interpreted. An effective mass and the occupied states can replace a naive classical free-particle picture. Such refinements can preserve a Drude-like low-frequency form in some regimes without validating every classical interpretation used to motivate it.

Letting a phenomenological scattering time tend to infinity is not a complete theory of superconductivity. Under a constant applied field, a collisionless Drude carrier accelerates instead of approaching a finite Ohmic steady drift. Superconducting electromagnetic equilibrium and flux response require additional physics beyond a zero-dissipation parameter limit.

Use these boundaries to choose the next model rather than to discard conservation laws. Charge continuity and consistent energy accounting still apply, but the constitutive relation, contact conditions and relevant quantum or electromagnetic degrees of freedom can change.

18

Audit a conduction calculation

State the carrier charges, drift directions, density definitions and surface orientation before reporting a current sign. Keep net charge density, particle flux and conventional current in separate columns. A neutral material can carry current, and a stationary charge distribution need not have zero field.

Next specify the geometry, local material law, temperature or bias regime and terminal constraints. Distinguish a complete three-dimensional field from a cross-section average. Check the resistance weighting, boundary conditions and integrated current conservation rather than substituting a convenient average area.

Reconcile terminal power with ∫J·E d³r under the declared approximation. Track where energy is supplied and where it is dissipated or stored. During transients, retain charge accumulation and the validity of any instantaneous constitutive or lumped-circuit assumptions.

Finally, label every time and frequency normalization. Momentum relaxation, bulk charge relaxation and signal propagation are not interchangeable clocks. A positive numerical fit, a plausible graph or a familiar formula is insufficient if its source, boundary or scale assumptions have changed.

Worked examples

A neutral region with two current contributions

Compute net charge density and conventional current for the prescribed three-species carrier model.

  1. Mobile positive charge gives ρ = +n₀q₀ and J = +2J₀x̂.
  2. Mobile negative charge gives ρ = −3n₀q₀ and J = (3n₀)(−q₀)(−v₀x̂) = +3J₀x̂.
  3. The stationary background contributes +2n₀q₀ of charge density and no current.
  4. Add charge and current separately: net charge is zero and total J = +5J₀x̂.

The model is neutral and current carrying. Opposite charge signs with opposite drifts can produce the same conventional-current direction.

Tilt the normal, not the current definition

For J = 5J₀x̂, area 6L₀² and normal (3/5,4/5,0), compute the signed current.

  1. The unit normal has x component 3/5.
  2. The normal current density is J·n̂ = 5J₀(3/5) = 3J₀.
  3. Multiply by the stated physical area to obtain I = 18J₀L₀².
  4. Reversing the orientation reverses the sign; it does not alter the moving charges.

I = 18J₀L₀² for the stated orientation. This equal-area patch calculation is distinct from changing the shape and area of a cut spanning the same finite wire.

Integrate a slow taper and inspect its midpoint

For the selected reduced wire, find total resistance, current under a 6V₀ drop and the midpoint potential and average current density.

  1. Integrate 60R₀∫₀¹(1 + s)^−2 ds = 30R₀.
  2. Then I = 6V₀/(30R₀) = I₀/5.
  3. At s = 1/2, f = 3/2 and accumulated resistance is 60(1 − 2/3)R₀ = 20R₀.
  4. Thus V/V₀ = 6 − (1/5)(20) = 2, and averaged J/J₀ = 1/[5(3/2)²] = 4/45.

R = 30R₀, I = I₀/5, midpoint V = 2V₀ and average axial J = (4/45)J₀. Half the geometrical length does not correspond to half the resistance or half the voltage drop.

Check terminal and local power independently

Show that the reduced taper’s integrated local power agrees with terminal power.

  1. Terminal power is IΔV = (I₀/5)(6V₀) = 6P₀/5.
  2. Local reduced power density is p₀/[25(1 + s)⁴]. Multiply by A(s) = πL₀²(1 + s)² and dx = 60L₀ ds.
  3. The dimensionless integral is ∫₀¹(1 + s)^−2 ds = 1/2.
  4. Therefore P = (60/25)(1/2)πL₀³p₀ = (6/5)P₀, using πL₀³p₀ = V₀I₀.

Both routes give 6P₀/5. The agreement checks dimensions, area weighting and energy conservation within the same reduced approximation.

One frequency with an explicit time convention

At Z = ωτ_m = 1, find the normalized Drude phasor conductivity and interpret its phase using e^(−iωt).

  1. σ/σ₀ = 1/(1 − i) = (1 + i)/2.
  2. For a real positive field phasor E_a, J(t) = (σ₀E_a/2)[cos(ωt) + sin(ωt)].
  3. This equals (σ₀E_a/√2)cos(ωt − π/4), so the physical current lags the field.
  4. The cycle-average work density is σ₀E_a²/4 for this peak-amplitude sinusoid at ω > 0.

The normalized coefficient is (1 + i)/2. A positive phasor argument here means a π/4 current lag under the declared negative-time-exponential convention.

Try it yourself

  1. An electron-like carrier with charge −q₀ drifts in negative x. Which way is its conventional current contribution?

    Show the answer and reasoning

    Positive x.

    The current density contains the product of signed charge and drift velocity. Both are negative in the x component.

  2. Reverse only the normal on the area-6L₀² patch with original normal +x̂. What happens to the selected current 30J₀L₀²?

    Show the answer and reasoning

    It becomes −30J₀L₀².

    The dot product with the normal changes sign. No physical carrier motion changed.

  3. Does ∇·J = 0 prove that the steady charge density ρ_q vanishes?

    Show the answer and reasoning

    No.

    Continuity then states only that charge is not changing locally. A nonzero stationary charge distribution can help establish the steady field.

  4. Does computing R = ΔV/I at one operating point prove an element is Ohmic?

    Show the answer and reasoning

    No.

    A ratio exists at a nonzero current, but Ohmic linearity requires the appropriate response over the stated regime. Differential and ratio resistance can differ.

  5. Why does the tapered wire’s average current density fall by four between its endpoints while I stays fixed?

    Show the answer and reasoning

    Its complete cross-sectional area increases by four.

    Current conservation fixes I, while average axial J = I/A. The uncomputed transverse corrections of a full three-dimensional solution are separate from this average.

  6. Compare the taper’s mean-area substitution R_wrong = (180/7)R₀ with the exact reduced result 30R₀.

    Show the answer and reasoning

    It is low by a relative 1/7.

    [(180/7) − 30]/30 = −1/7. This does not measure the reduced model’s error relative to a full three-dimensional wire.

  7. At T = t/τ_m = ln 2 in the zero-initial-drift Drude step, what fraction of the steady current has developed?

    Show the answer and reasoning

    1/2.

    The rise is 1 − e^(−T). The negative carrier velocity has the opposite sign while the conventional current is positive.

  8. At S = t/τ_q = ln 4 in the separate bulk-relaxation model, what fractions remain in the volume and have reached the bounding region?

    Show the answer and reasoning

    Bulk 1/4; accumulated boundary fraction 3/4.

    Their sum remains one. S and T use different time constants, so equal normalized numbers do not assert equal physical times.

  9. May ε/σ be treated as an exact microscopic decay time if the assumed instantaneous Ohmic law fails on that scale?

    Show the answer and reasoning

    No.

    Retain the relevant momentum inertia, spatial transport and field dynamics. The first-order charge-relaxation formula carries its constitutive validity conditions.

  10. Should the power density of a constant DC Ohmic field E be divided by two because sinusoidal phasor power has a one-half factor?

    Show the answer and reasoning

    No. Its power density is σE².

    The sinusoidal peak-amplitude factor comes from averaging over cycles for ω > 0. A constant DC field has no such cosine-squared cycle average.

What to carry forward

  • Current is signed charge flux through an oriented surface. Carrier direction, current direction, surface normal and net charge density must be distinguished.
  • Charge continuity holds locally and integrally. Steady current does not imply zero field or zero charge, and transients retain accumulation.
  • J = σE is a local constitutive model; R is a response of a complete geometry and state. Their familiar relations have scale and boundary assumptions.
  • A slowly tapered wire weights inverse area in its reduced resistance integral. Cross-section averages are not a complete purely axial three-dimensional field.
  • Terminal power and integrated J·E must agree under the same approximation. Charge circulates while the source transfers energy.
  • The Drude momentum time controls mean drift response. It is different from bulk ε/σ relaxation and from electromagnetic signal propagation.
  • Complex conductivity needs an explicit time convention and amplitude convention. Reactive response, dissipation and DC power have different interpretations.
  • Nonlocal, quantum, contact, material and electromagnetic effects can require a better constitutive model without changing charge conservation or consistent energy accounting.

References

Sources for the concepts and teaching approaches in this article.

  1. OpenStax, Rice University4 sources
    1. University Physics Volume 2: Electrical CurrentSamuel J. Ling, Jeff Sanny and William MoebsTextbook chapteropenstax.orgRead source
    2. University Physics Volume 2: Model of Conduction in MetalsSamuel J. Ling, Jeff Sanny and William MoebsTextbook chapteropenstax.orgRead source
    3. University Physics Volume 2: Resistivity and ResistanceSamuel J. Ling, Jeff Sanny and William MoebsTextbook chapteropenstax.orgRead source
    4. University Physics Volume 2: Electrical Energy and PowerSamuel J. Ling, Jeff Sanny and William MoebsTextbook chapteropenstax.orgRead source
  2. Massachusetts Institute of Technology3 sources
    1. Electromagnetic Fields and Energy: Conduction Constitutive LawsHermann A. Haus and James R. MelcherTextbook chapterweb.mit.eduRead source
    2. Electromagnetic Fields and Energy: Steady Ohmic ConductionHermann A. Haus and James R. MelcherTextbook chapterweb.mit.eduRead source
    3. Electromagnetic Fields and Energy: Charge Relaxation in Uniform ConductorsHermann A. Haus and James R. MelcherTextbook chapterweb.mit.eduRead source
  3. University of Cambridge2 sources
    1. Lectures on Electromagnetism: §§7.6.1–7.6.3, Drude response and longitudinal dynamicsDavid TongCourse notes · PDFdamtp.cam.ac.ukRead source
    2. Applications of Quantum Mechanics: Electron Dynamics in Solids, §§3.2.1–3.2.5David TongCourse notes · PDFdamtp.cam.ac.ukRead source