Physics · Article
Magnetic dipoles and magnetic materials
Connect a loop’s moment to torque and translation, derive magnetization currents and a finite magnet’s fields, and distinguish internal-field response, quantum populations, magnetic order and energy history.
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Back to PhysicsAfter reading, you can
- Define a magnetic moment, magnetization and the SI fields B and H without mixing their units.
- Distinguish torque, net force and mechanical interaction potential for a maintained test moment.
- Derive the finite uniformly magnetized sphere and check bound-current and one-sided boundary conditions.
- Separate intrinsic susceptibility from shape-dependent applied-field response in a controlled ellipsoid model.
- Derive independent spin-half populations, saturation and the conditional weak-field Curie approximation.
- Distinguish magnetic order, domain compensation, hysteresis and microscopic response mechanisms.
- Use point-dipole distributions, source/coenergy constraints and mean-field limits with their assumptions visible.
- Explain why material models, energy functions and dynamic losses need more information than one scalar permeability.
One moment, two kinds of motion
A small current loop can turn in a uniform magnetic field even when its centre has no magnetic force. A nonuniform field can also pull it. Which observation tells us about orientation, and which tells us about spatial variation? Keeping these questions separate is the first step from the force on a moving charge to the behaviour of a magnet.
Use B for the magnetic flux density in tesla. Every test-loop force or torque below uses the external field, excluding that loop’s own field. The loop is small enough that its disturbance of the prescribed sources can be neglected. A real magnetic body that changes its magnetization in response to the field needs a separate material model.
Four original models organize the lesson: a maintained test loop, a sphere with prescribed uniform magnetization, a reversible linear ellipsoid, and an equilibrium collection of independent spin-half moments. They have separate sources and constraints. In particular, the sphere’s prescribed magnetization is not a solution of the ellipsoid’s constitutive equation, and the independent spins do not model ferromagnetic domains.
Choose a first reading and a return path
For a first reading, follow the moment’s direction, compare torque with translation, and then use the finite magnetized sphere to meet magnetization M and the auxiliary field H. The response and statistical sections explain why an applied field, an internal field and a microscopic local field are different quantities.
Return later to the marked advanced sections for the point-dipole distribution, induced-moment energy, demagnetization limits, interacting-spin approximations and hysteretic loss. These sections refine the first models rather than changing their assumptions halfway through a calculation.
Keep a separate ledger for each model’s units. In the loop case m₀ = I₀A₀ is a magnetic-moment scale in A m², while M₀ in the sphere case is a magnetization scale in A/m. The number density n in the spin model has units m⁻³. Similar-looking symbols do not make these three quantities interchangeable.
Build a moment from a closed current
For one planar steady closed loop, the moment is m = IA n̂, where n̂ is the right-hand normal set by conventional current. The angle used in m·B is the angle between this normal and B, not between the wire’s plane and B. For coincident similar turns, sum their moments; the shortcut NIA assumes the specified current, area and orientation for every turn.
A localized stationary current distribution has m = ½∫r × J d³r under the usual localization and continuity conditions. For a complete steady source, shifting the coordinate origin does not change this moment. An isolated open segment carrying a permanent current is not a complete localized source: its return path and charge continuity cannot silently disappear.
Our test loop has three turns, current I₀ > 0, area A₀ per turn and normal (−1, 2, 2)/3. Thus m = m₀(−1, 2, 2), with m₀ = I₀A₀. The vector’s magnitude is 3m₀, so the unit normal and the turn count are consistent. We prescribe this moment; no susceptibility has been used to infer it.
Separate torque from translation
Adding the magnetic forces around a closed loop in a uniform external field gives zero net force, but opposite forces at different positions can still have a nonzero torque. In the small-loop description, τ = m × B_ext. Its units are A m² T = N m. The torque is zero for parallel or antiparallel vectors, although those orientations need not have the same stability.
For the maintained moment in the uniform field B_ext = B₀(1, 3, 2), the original case gives a nonzero cross product and zero translational force. This is not a statement that every extended magnet in every field has zero force. It depends on uniformity and on summing a complete source rather than one convenient wire segment.
Torque alone does not prove that a moment instantly aligns with the field. When m is proportional to angular momentum, the torque can produce precession. Damping or exchange with other degrees of freedom is needed for relaxation toward a lower-energy orientation. A thermal equilibrium calculation and an undamped dynamical trajectory answer different questions.
Hold the source and the moment fixed
At fixed moment, the orientation-dependent mechanical interaction potential is U_mech = −m·B_ext. It describes the mechanical coupling to prescribed external sources. It is not the entire electromagnetic energy of the maintained loop and its source circuit, and it does not include the loop’s self-field energy.
A spatial displacement with the moment’s components held fixed gives F = ∇(m·B_ext). For constant m, the identity is ∇(m·B) = (m·∇)B + m × (∇ × B). Replacing the gradient by (m·∇)B therefore needs the appropriate zero-curl condition in the test region. Do not discard the extra term merely because a field is called magnetic.
A field-dependent induced moment cannot be inserted into −m·B and differentiated as though it were independently maintained. Its internal response energy and the work of the field sources must be included. The advanced energy discussion derives a simple reversible example and distinguishes it from a general force law inside magnetic matter.
Translate in a compatible gradient
Set X = x/L₀, Y = y/L₀ and Z = z/L₀, with L₀ > 0. A local external field is B_ext/B₀ = (1 + 4X, 3 + Y, 2 − 5Z). Its diagonal gradient has trace 4 + 1 − 5 = 0, and its curl vanishes. It is therefore compatible with source-free magnetostatics in the stated local region. The distant apparatus maintaining that field is outside this model.
Keep the loop small compared with L₀ and hold its orientation during translation. Use the box |X| ≤ 1/2, |Y| ≤ 1/2 and |Z| ≤ 1/4. Move from A = (−1/4, 0, 1/5) to B = (1/4, −1/2, −1/5). These points and the straight connecting path lie within the box; the field’s y component stays positive there.
The force scale is m₀B₀/L₀, the torque and mechanical-energy scales are m₀B₀, and the fixed-m force is constant for this linear field. A force can do positive translational work while a controlled external agent extracts the same mechanical energy in an ideal quasistatic transfer. Maintaining the current and controlling orientation are part of the stated apparatus, not free operations.
A maintained moment: orientation and translation
A maintained test moment can turn without translating. The separate translation curve retains the fixed-current mechanical coupling and source-aware work balance.
Project the moment, field and torque
- m/m₀: maintained three-turn moment
- B_ext/B₀: prescribed uniform external field
- τ/(m₀B₀): retained torque, not a translation vector
Uniform F/(m₀B₀/L₀):
(0, 0, 0)Mechanical U/(m₀B₀): -9Translate a maintained moment, with the source ledger visible
- Original U_mech/(m₀B₀): fixed-m coupling only
- Both original translation endpoints
F/(m₀B₀/L₀):
(-4, 2, -10)ΔU/(m₀B₀): -1Force work A→B/(m₀B₀): 1External quasistatic work/(m₀B₀): -1- Small steady closed three-turn loop; fixed m/current and held orientation, fixed external sources. m₀=I₀A₀ (A m²), energy m₀B₀ (J), force m₀B₀/L₀ (N). Local gradient box only; potential excludes source/self energy.
- The first plot uses Q(v)=(vₓ+vᵧ/2,v_z+vᵧ/4). Vectors m/m₀, B/B₀ and τ/(m₀B₀) have distinct physical dimensions and scales; this schematic projection does not compare their physical lengths. Uniform-field F=0 is a separate source control.
- The energy plot reads the 21 retained U(s) samples. The displayed force, ΔU and work ledger are the original exact controls, not a new work curve. The held orientation/current apparatus and field sources matter to total energy.
- Tables preserve every original typed value: reduced rational strings, explicit approximate 70-significant-digit decimals and symbolic expressions. Plot coordinates only convert these stored values for display; no scientific solver, new case, re-rounding or measured dataset is introduced.
Moment, material fields and magnetic-current units in modern SI
All 21 maintained-moment translation records (21 rows)
Small steady closed three-turn loop; fixed m/current and held orientation, fixed external sources. m₀=I₀A₀ (A m²), energy m₀B₀ (J), force m₀B₀/L₀ (N). Local gradient box only; potential excludes source/self energy.
| Row | Translation parameter s | Position / L₀ · x | Position / L₀ · y | Position / L₀ · z | External B / B₀ · x | External B / B₀ · y | External B / B₀ · z | Torque / (m₀B₀) · x | Torque / (m₀B₀) · y | Torque / (m₀B₀) · z | Force / (m₀B₀/L₀) · x | Force / (m₀B₀/L₀) · y | Force / (m₀B₀/L₀) · z | Mechanical U / (m₀B₀) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0Exact rational | -1/4Exact rational | 0Exact rational | 1/5Exact rational | 0Exact rational | 3Exact rational | 1Exact rational | -4Exact rational | 1Exact rational | -3Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -8Exact rational |
1 | 1/20Exact rational | -9/40Exact rational | -1/40Exact rational | 9/50Exact rational | 1/10Exact rational | 119/40Exact rational | 11/10Exact rational | -15/4Exact rational | 13/10Exact rational | -127/40Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -161/20Exact rational |
2 | 1/10Exact rational | -1/5Exact rational | -1/20Exact rational | 4/25Exact rational | 1/5Exact rational | 59/20Exact rational | 6/5Exact rational | -7/2Exact rational | 8/5Exact rational | -67/20Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -81/10Exact rational |
3 | 3/20Exact rational | -7/40Exact rational | -3/40Exact rational | 7/50Exact rational | 3/10Exact rational | 117/40Exact rational | 13/10Exact rational | -13/4Exact rational | 19/10Exact rational | -141/40Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -163/20Exact rational |
4 | 1/5Exact rational | -3/20Exact rational | -1/10Exact rational | 3/25Exact rational | 2/5Exact rational | 29/10Exact rational | 7/5Exact rational | -3Exact rational | 11/5Exact rational | -37/10Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -41/5Exact rational |
5 | 1/4Exact rational | -1/8Exact rational | -1/8Exact rational | 1/10Exact rational | 1/2Exact rational | 23/8Exact rational | 3/2Exact rational | -11/4Exact rational | 5/2Exact rational | -31/8Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -33/4Exact rational |
6 | 3/10Exact rational | -1/10Exact rational | -3/20Exact rational | 2/25Exact rational | 3/5Exact rational | 57/20Exact rational | 8/5Exact rational | -5/2Exact rational | 14/5Exact rational | -81/20Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -83/10Exact rational |
7 | 7/20Exact rational | -3/40Exact rational | -7/40Exact rational | 3/50Exact rational | 7/10Exact rational | 113/40Exact rational | 17/10Exact rational | -9/4Exact rational | 31/10Exact rational | -169/40Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -167/20Exact rational |
8 | 2/5Exact rational | -1/20Exact rational | -1/5Exact rational | 1/25Exact rational | 4/5Exact rational | 14/5Exact rational | 9/5Exact rational | -2Exact rational | 17/5Exact rational | -22/5Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -42/5Exact rational |
9 | 9/20Exact rational | -1/40Exact rational | -9/40Exact rational | 1/50Exact rational | 9/10Exact rational | 111/40Exact rational | 19/10Exact rational | -7/4Exact rational | 37/10Exact rational | -183/40Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -169/20Exact rational |
10 | 1/2Exact rational | 0Exact rational | -1/4Exact rational | 0Exact rational | 1Exact rational | 11/4Exact rational | 2Exact rational | -3/2Exact rational | 4Exact rational | -19/4Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -17/2Exact rational |
11 | 11/20Exact rational | 1/40Exact rational | -11/40Exact rational | -1/50Exact rational | 11/10Exact rational | 109/40Exact rational | 21/10Exact rational | -5/4Exact rational | 43/10Exact rational | -197/40Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -171/20Exact rational |
12 | 3/5Exact rational | 1/20Exact rational | -3/10Exact rational | -1/25Exact rational | 6/5Exact rational | 27/10Exact rational | 11/5Exact rational | -1Exact rational | 23/5Exact rational | -51/10Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -43/5Exact rational |
13 | 13/20Exact rational | 3/40Exact rational | -13/40Exact rational | -3/50Exact rational | 13/10Exact rational | 107/40Exact rational | 23/10Exact rational | -3/4Exact rational | 49/10Exact rational | -211/40Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -173/20Exact rational |
14 | 7/10Exact rational | 1/10Exact rational | -7/20Exact rational | -2/25Exact rational | 7/5Exact rational | 53/20Exact rational | 12/5Exact rational | -1/2Exact rational | 26/5Exact rational | -109/20Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -87/10Exact rational |
15 | 3/4Exact rational | 1/8Exact rational | -3/8Exact rational | -1/10Exact rational | 3/2Exact rational | 21/8Exact rational | 5/2Exact rational | -1/4Exact rational | 11/2Exact rational | -45/8Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -35/4Exact rational |
16 | 4/5Exact rational | 3/20Exact rational | -2/5Exact rational | -3/25Exact rational | 8/5Exact rational | 13/5Exact rational | 13/5Exact rational | 0Exact rational | 29/5Exact rational | -29/5Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -44/5Exact rational |
17 | 17/20Exact rational | 7/40Exact rational | -17/40Exact rational | -7/50Exact rational | 17/10Exact rational | 103/40Exact rational | 27/10Exact rational | 1/4Exact rational | 61/10Exact rational | -239/40Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -177/20Exact rational |
18 | 9/10Exact rational | 1/5Exact rational | -9/20Exact rational | -4/25Exact rational | 9/5Exact rational | 51/20Exact rational | 14/5Exact rational | 1/2Exact rational | 32/5Exact rational | -123/20Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -89/10Exact rational |
19 | 19/20Exact rational | 9/40Exact rational | -19/40Exact rational | -9/50Exact rational | 19/10Exact rational | 101/40Exact rational | 29/10Exact rational | 3/4Exact rational | 67/10Exact rational | -253/40Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -179/20Exact rational |
20 | 1Exact rational | 1/4Exact rational | -1/2Exact rational | -1/5Exact rational | 2Exact rational | 5/2Exact rational | 3Exact rational | 1Exact rational | 7Exact rational | -13/2Exact rational | -4Exact rational | 2Exact rational | -10Exact rational | -9Exact rational |
Complete loop scales, controls and source constraints (86 rows)
Small steady closed three-turn loop; fixed m/current and held orientation, fixed external sources. m₀=I₀A₀ (A m²), energy m₀B₀ (J), force m₀B₀/L₀ (N). Local gradient box only; potential excludes source/self energy.
| Quantity or source condition | Retained admitted value |
|---|---|
definitions · Assumptions and regime | Small steady closed three-turn planar test loop; a_loop << L0 in the gradient case. Moment/current externally maintained, orientation held during translation; fixed external sources; dipole mechanical potential excludes self/source energy. No induced-moment substitution. |
definitions · turns | 3 |
definitions · current Over I0 | 1 |
definitions · area Per Turn Over A0 | 1 |
definitions · Loop unit normal · 1 | -1/3 |
definitions · Loop unit normal · 2 | 2/3 |
definitions · Loop unit normal · 3 | 2/3 |
definitions · uniform Field Over B0 · 1 | 1 |
definitions · uniform Field Over B0 · 2 | 3 |
definitions · uniform Field Over B0 · 3 | 2 |
definitions · gradient Field Over B0 · 1 | 1+4X |
definitions · gradient Field Over B0 · 2 | 3+Y |
definitions · gradient Field Over B0 · 3 | 2-5Z |
definitions · dimensionless Gradient Diagonal · 1 | 4 |
definitions · dimensionless Gradient Diagonal · 2 | 1 |
definitions · dimensionless Gradient Diagonal · 3 | -5 |
definitions · local Box · X=x/a · 1 | -1/2 |
definitions · local Box · X=x/a · 2 | 1/2 |
definitions · local Box · Y · 1 | -1/2 |
definitions · local Box · Y · 2 | 1/2 |
definitions · local Box · Z=z/a · 1 | -1/4 |
definitions · local Box · Z=z/a · 2 | 1/4 |
definitions · A · 1 | -1/4 |
definitions · A · 2 | 0 |
definitions · A · 3 | 1/5 |
definitions · B · 1 | 1/4 |
definitions · B · 2 | -1/2 |
definitions · B · 3 | -1/5 |
definitions · translation | r(s)=A+s(B-A),0<=s<=1 |
definitions · Original sampling rule | s=i/20,i=0..20 |
definitions · definitions · torque | m cross B_ext |
definitions · definitions · mechanical U | -m dot B_ext at fixed m |
definitions · definitions · force | gradient(m dot B_ext) at fixed m |
definitions · definitions · identity Check | The prescribed local external field is divergence-free and curl-free. |
definitions · Moment / m₀ · 1 | -1 |
definitions · Moment / m₀ · 2 | 2 |
definitions · Moment / m₀ · 3 | 2 |
uniform · Moment / m₀ · 1 | -1Exact rational |
uniform · Moment / m₀ · 2 | 2Exact rational |
uniform · Moment / m₀ · 3 | 2Exact rational |
uniform · Uniform external B / B₀ · 1 | 1Exact rational |
uniform · Uniform external B / B₀ · 2 | 3Exact rational |
uniform · Uniform external B / B₀ · 3 | 2Exact rational |
uniform · Force / (m₀B₀/L₀) · 1 | 0Exact rational |
uniform · Force / (m₀B₀/L₀) · 2 | 0Exact rational |
uniform · Force / (m₀B₀/L₀) · 3 | 0Exact rational |
uniform · Torque / (m₀B₀) · 1 | -2Exact rational |
uniform · Torque / (m₀B₀) · 2 | 4Exact rational |
uniform · Torque / (m₀B₀) · 3 | -5Exact rational |
uniform · Mechanical U / (m₀B₀) | -9Exact rational |
gradient · Force / (m₀B₀/L₀) · 1 | -4Exact rational |
gradient · Force / (m₀B₀/L₀) · 2 | 2Exact rational |
gradient · Force / (m₀B₀/L₀) · 3 | -10Exact rational |
gradient · Divergence B / (B₀/L₀) | 0Exact rational |
gradient · Curl B / (B₀/L₀) · 1 | 0Exact rational |
gradient · Curl B / (B₀/L₀) · 2 | 0Exact rational |
gradient · Curl B / (B₀/L₀) · 3 | 0Exact rational |
gradient · A→B displacement / L₀ · 1 | 1/2Exact rational |
gradient · A→B displacement / L₀ · 2 | -1/2Exact rational |
gradient · A→B displacement / L₀ · 3 | -2/5Exact rational |
gradient · Δ mechanical U / (m₀B₀) | -1Exact rational |
gradient · Force work A→B / (m₀B₀) | 1Exact rational |
gradient · Quasistatic external work A→B / (m₀B₀) | -1Exact rational |
mu0 | symbolic SI vacuum permeability, H/m; no exact numerical value asserted |
I0 | positive imposed loop-current scale, A |
A0 | positive turn-area scale, m^2 |
m0 | I0 A0, A m^2 = J/T |
B0 | positive external vacuum field scale, T |
L0 | positive translation length scale, m |
torque Scale | m0 B0, N m |
force Scale | m0 B0/L0, N |
energy Scale | m0 B0, J |
a | positive sphere radius, m; independent of L0 |
M0 | positive prescribed magnetization scale, A/m |
H0 | positive applied H scale, A/m |
mu S | positive magnitude of either localized spin-half moment projection, J/T; different from mu0 |
n | positive number density of localized noninteracting moments, m^-3 |
T | positive fixed absolute temperature, K |
k B | symbolic Boltzmann constant, J/K |
exact | Use reduced rational strings when the analytic output is rational; preserve symbolic pi coefficients. No binary float serialized as an exact rational. |
transcendental | Use100decimal-digit working precision; serialize70significant-digit decimal strings with explicit approximate status. Independent reviewer must recompute; no candidate tolerance self-approval. |
undefined | Use explicit source-excluded/one-sided/limit records; never hide undefined values as ordinary zero. |
dictionaries | Every record keeps units/normalization, branch/source conditions and control identity. |
0 | No actual measured material constants or measured hysteresis curve. |
1 | No native presentation, scientific verdict, current-source integration or publication. |
2 | Public prerequisites148–152 remain unfulfilled; all current drafts stay local. |
Focus a named plot or table region to scroll horizontally with the keyboard, or swipe within that region. Readable fixed-size diagrams remain locally contained. Full disclosures retain exact values, all records and source/domain conditions.
Read the dipole field on its domain
A localized magnetic source has the leading vacuum far field B_dip = μ₀[3(m·r̂)r̂ − m]/(4πr³). The ordinary expression applies away from its point origin, and the far-field approximation for a finite source requires distance large compared with that source’s size. Higher multipoles can matter when this separation is absent or the dipole moment cancels.
Along the positive or negative dipole axis, the exterior B points along m. In the equatorial plane it points opposite to m. These vector statements are not the same as a positive field magnitude. The r⁻³ variation also differs from the r⁻² field of a monopole; a magnet’s north and south labels do not assert isolated magnetic charges.
The finite sphere derived below has an exactly dipolar exterior because of its particular uniform magnetization and spherical geometry. That special result is not permission to replace an arbitrary magnet’s interior by the point formula. In particular, a finite magnet can have a regular nonzero field at its centre while the ordinary point expression is undefined there.
Average moments without inventing a transport current
Magnetization M is magnetic dipole moment per coarse-grained volume. The averaging volume must contain many microscopic constituents while remaining small enough to resolve the macroscopic variation of interest. Its units are A/m, since a moment has units A m² and volume has units m³.
A smooth magnetization can be represented by the bound volume current J_b = ∇ × M. At a material–vacuum surface with outward normal n̂, the bound sheet current is K_b = M × n̂. At an interface directed from medium 1 to medium 2, use (M₁ − M₂) × n̂. This representation reproduces the averaged magnetic source; it is not a claim that the microscopic currents literally form one visible surface wire.
Bound magnetization current is distinct from free charge transport supplied by terminals. In a stationary macroscopic magnetic problem, ∇ × B = μ₀(J_f + J_b), with the sheet terms treated at interfaces. In time-dependent material problems polarization current and displacement current also enter. Those dynamic laws and eddy-current effects must be restored when the magnetostatic approximation is inadequate.
| Quantity | SI units | Meaning in this lesson |
|---|---|---|
| m, m₀ | A m² = J/T | Localized magnetic moment; m₀ is the loop scale |
| M, M₀ | A/m | Moment density; M₀ is the sphere scale |
| B | T | Magnetic flux density used in the Lorentz force |
| H | A/m | Auxiliary macroscopic magnetic field |
| J_f, J_b | A/m² | Volume current density |
| K_f, K_b | A/m | Sheet current density |
| χ_m = M/H_int | 1 | Scalar susceptibility only in the stated linear model |
| H·dB | J/m³ | Reversible energy increment under stated conditions |
Define H with both field equations
Define H = B/μ₀ − M, so B = μ₀(H + M). In stationary magnetostatics, ∇ × H = J_f. However, ∇·B = 0 also gives ∇·H = −∇·M. Knowing the free current alone does not eliminate the influence of magnetization, geometry or boundary data on H.
For a unit normal n̂ from medium 1 to medium 2, n̂·(B₂ − B₁) = 0 and n̂ × (H₂ − H₁) = K_f. With no free sheet, tangential H is continuous. Tangential B can still jump because M jumps. Normal H need not be continuous, since continuity of normal B instead constrains the sum H + M.
In a current-free simply connected region one may write H = −∇Φ_H. The scalar potential still needs its boundary conditions, and a region linked by free current may not admit a single-valued global potential. Describing ∇·H through mathematical magnetic-pole densities is a representation of M, not evidence for physical magnetic monopoles.
Solve the prescribed magnetized sphere
Take a sphere of radius a with prescribed uniform M = M₀ ẑ inside and vacuum outside. There are no free currents. A regular interior scalar potential with this symmetry is Φ_in = C r cosθ, while a decaying exterior one is Φ_out = D cosθ/r². Tangential H continuity permits a continuous scalar potential at r = a, giving D = Ca³.
Normal B continuity gives 2D/a³ = −C + M₀. Substituting D = Ca³ yields C = M₀/3. Thus H_in = −M₀ ẑ/3 and B_in = 2μ₀M₀ ẑ/3. The exterior is the dipole field of m = (4πa³/3)M₀ ẑ. A negative interior H and positive interior B are compatible because B includes μ₀M.
This is a solved prescribed-source boundary problem. It does not state that all real permanent magnets maintain a perfectly uniform M, nor that a linear sphere with unknown M can skip its constitutive equation. Its finite interior, dipolar exterior and explicit sheet current let us check several field concepts without assigning a numerical value to μ₀.
Keep the interface sides visible
In the smooth interior, ∇ × M = 0, yet the curved surface carries K_b = M₀ ẑ × n̂. At the equator on the positive x axis, that sheet points along +y. On the north and south poles the sheet vanishes, but the normal jump in H remains part of the boundary problem.
At the equator, the z component of B changes from 2μ₀M₀/3 just inside to −μ₀M₀/3 just outside. It is tangential there; its jump represents the bound sheet current. H_z is −M₀/3 on both equatorial sides. At the north pole B_z is continuous at 2μ₀M₀/3, while H_z changes from −M₀/3 to 2M₀/3.
The table records limits rather than one unexplained ordinary field value on an ideal current sheet. A plotted line must not interpolate through the equatorial jump as if the sheet were a smooth material layer. The centre is an ordinary interior point of this finite sphere; it is not the singular origin of a point-dipole approximation.
A finite magnet, its boundary current and both field limits
A prescribed finite sphere has a regular interior, bound surface current and separately retained interface limits. Its point-dipole limit has a different source domain.
Finite-source geometry and signed meridian components
- Ten retained meridian observations M0…M9; positions only
- Finite surface: current cross (+y, into page), dot (−y, out); K=0 at poles
- Bₓ/(μ₀M₀) and Hₓ/M₀ coincide at these controls
- B_z/(μ₀M₀), separately retained
- H_z/M₀, separately retained
Keep the inside and outside branches separate
- Axis B_z/(μ₀M₀): blue solid, hollow circle
- Equator B_z/(μ₀M₀): orange dashed, hollow square
- Axis H_z/M₀: violet dotted, hollow diamond
- Equator H_z/M₀: green dash-dot, small hollow circle
- Uniform prescribed M=M₀ẑ inside radius a, vacuum outside, no free current. X=x/a,Z=z/a. B/(μ₀M₀), H/M₀ and K_b/M₀ have separate normalizations. Surface field records are one-sided limits; finite center regular, point-dipole origin excluded.
- View the meridian from −y: +x right,+z up,+y points into the page. A cross denotes K_b toward +y, a dot denotes −y, and the poles have K_b=0. Green normal markers are the six retained surface controls; M0…M9 identify the ten retained observation points.
- The component panel plots separate retained controls; it does not connect them into a trajectory or field line. Bₓ/(μ₀M₀) and Hₓ/M₀ coincide at these ten specified controls and share one blue circle. B_z and H_z remain separately keyed.
- The signed profile plot uses separate source branches. Hollow endpoint glyphs denote one-sided limits; equal limits can occupy the same location. The different shapes/dashes distinguish physical quantities, not two ordinary values assigned at the sheet. Complete surface tables retain every inside/outside vector.
- The finite Cartesian center is regular. The separate ideal point-dipole convention excludes its ordinary origin and has a distributional contact term; no delta spike or exterior singularity replaces the finite interior.
- Tables preserve every original typed value: reduced rational strings, explicit approximate 70-significant-digit decimals and symbolic expressions. Plot coordinates only convert these stored values for display; no scientific solver, new case, re-rounding or measured dataset is introduced.
Moment, material fields and magnetic-current units in modern SI
Finite prescribed sphere: signed axial components and one-sided surface limits
All 41 axis negative exterior records, with their own surface limits (41 rows)
Uniform prescribed M=M₀ẑ inside radius a, vacuum outside, no free current. X=x/a,Z=z/a. B/(μ₀M₀), H/M₀ and K_b/M₀ have separate normalizations. Surface field records are one-sided limits; finite center regular, point-dipole origin excluded.
| Row | Signed coordinate / a | X=x/a | Z=z/a | Source side/region | B / (μ₀M₀) · x | B / (μ₀M₀) · y | B / (μ₀M₀) · z | H / M₀ · x | H / M₀ · y | H / M₀ · z |
|---|---|---|---|---|---|---|---|---|---|---|
0 | -3Exact rational | 0Exact rational | -3Exact rational | exterior | 0Exact rational | 0Exact rational | 2/81Exact rational | 0Exact rational | 0Exact rational | 2/81Exact rational |
1 | -59/20Exact rational | 0Exact rational | -59/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/616137Exact rational | 0Exact rational | 0Exact rational | 16000/616137Exact rational |
2 | -29/10Exact rational | 0Exact rational | -29/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/73167Exact rational | 0Exact rational | 0Exact rational | 2000/73167Exact rational |
3 | -57/20Exact rational | 0Exact rational | -57/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/555579Exact rational | 0Exact rational | 0Exact rational | 16000/555579Exact rational |
4 | -14/5Exact rational | 0Exact rational | -14/5Exact rational | exterior | 0Exact rational | 0Exact rational | 125/4116Exact rational | 0Exact rational | 0Exact rational | 125/4116Exact rational |
5 | -11/4Exact rational | 0Exact rational | -11/4Exact rational | exterior | 0Exact rational | 0Exact rational | 128/3993Exact rational | 0Exact rational | 0Exact rational | 128/3993Exact rational |
6 | -27/10Exact rational | 0Exact rational | -27/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/59049Exact rational | 0Exact rational | 0Exact rational | 2000/59049Exact rational |
7 | -53/20Exact rational | 0Exact rational | -53/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/446631Exact rational | 0Exact rational | 0Exact rational | 16000/446631Exact rational |
8 | -13/5Exact rational | 0Exact rational | -13/5Exact rational | exterior | 0Exact rational | 0Exact rational | 250/6591Exact rational | 0Exact rational | 0Exact rational | 250/6591Exact rational |
9 | -51/20Exact rational | 0Exact rational | -51/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/397953Exact rational | 0Exact rational | 0Exact rational | 16000/397953Exact rational |
10 | -5/2Exact rational | 0Exact rational | -5/2Exact rational | exterior | 0Exact rational | 0Exact rational | 16/375Exact rational | 0Exact rational | 0Exact rational | 16/375Exact rational |
11 | -49/20Exact rational | 0Exact rational | -49/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/352947Exact rational | 0Exact rational | 0Exact rational | 16000/352947Exact rational |
12 | -12/5Exact rational | 0Exact rational | -12/5Exact rational | exterior | 0Exact rational | 0Exact rational | 125/2592Exact rational | 0Exact rational | 0Exact rational | 125/2592Exact rational |
13 | -47/20Exact rational | 0Exact rational | -47/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/311469Exact rational | 0Exact rational | 0Exact rational | 16000/311469Exact rational |
14 | -23/10Exact rational | 0Exact rational | -23/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/36501Exact rational | 0Exact rational | 0Exact rational | 2000/36501Exact rational |
15 | -9/4Exact rational | 0Exact rational | -9/4Exact rational | exterior | 0Exact rational | 0Exact rational | 128/2187Exact rational | 0Exact rational | 0Exact rational | 128/2187Exact rational |
16 | -11/5Exact rational | 0Exact rational | -11/5Exact rational | exterior | 0Exact rational | 0Exact rational | 250/3993Exact rational | 0Exact rational | 0Exact rational | 250/3993Exact rational |
17 | -43/20Exact rational | 0Exact rational | -43/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/238521Exact rational | 0Exact rational | 0Exact rational | 16000/238521Exact rational |
18 | -21/10Exact rational | 0Exact rational | -21/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/27783Exact rational | 0Exact rational | 0Exact rational | 2000/27783Exact rational |
19 | -41/20Exact rational | 0Exact rational | -41/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/206763Exact rational | 0Exact rational | 0Exact rational | 16000/206763Exact rational |
20 | -2Exact rational | 0Exact rational | -2Exact rational | exterior | 0Exact rational | 0Exact rational | 1/12Exact rational | 0Exact rational | 0Exact rational | 1/12Exact rational |
21 | -39/20Exact rational | 0Exact rational | -39/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/177957Exact rational | 0Exact rational | 0Exact rational | 16000/177957Exact rational |
22 | -19/10Exact rational | 0Exact rational | -19/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/20577Exact rational | 0Exact rational | 0Exact rational | 2000/20577Exact rational |
23 | -37/20Exact rational | 0Exact rational | -37/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/151959Exact rational | 0Exact rational | 0Exact rational | 16000/151959Exact rational |
24 | -9/5Exact rational | 0Exact rational | -9/5Exact rational | exterior | 0Exact rational | 0Exact rational | 250/2187Exact rational | 0Exact rational | 0Exact rational | 250/2187Exact rational |
25 | -7/4Exact rational | 0Exact rational | -7/4Exact rational | exterior | 0Exact rational | 0Exact rational | 128/1029Exact rational | 0Exact rational | 0Exact rational | 128/1029Exact rational |
26 | -17/10Exact rational | 0Exact rational | -17/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/14739Exact rational | 0Exact rational | 0Exact rational | 2000/14739Exact rational |
27 | -33/20Exact rational | 0Exact rational | -33/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/107811Exact rational | 0Exact rational | 0Exact rational | 16000/107811Exact rational |
28 | -8/5Exact rational | 0Exact rational | -8/5Exact rational | exterior | 0Exact rational | 0Exact rational | 125/768Exact rational | 0Exact rational | 0Exact rational | 125/768Exact rational |
29 | -31/20Exact rational | 0Exact rational | -31/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/89373Exact rational | 0Exact rational | 0Exact rational | 16000/89373Exact rational |
30 | -3/2Exact rational | 0Exact rational | -3/2Exact rational | exterior | 0Exact rational | 0Exact rational | 16/81Exact rational | 0Exact rational | 0Exact rational | 16/81Exact rational |
31 | -29/20Exact rational | 0Exact rational | -29/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/73167Exact rational | 0Exact rational | 0Exact rational | 16000/73167Exact rational |
32 | -7/5Exact rational | 0Exact rational | -7/5Exact rational | exterior | 0Exact rational | 0Exact rational | 250/1029Exact rational | 0Exact rational | 0Exact rational | 250/1029Exact rational |
33 | -27/20Exact rational | 0Exact rational | -27/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/59049Exact rational | 0Exact rational | 0Exact rational | 16000/59049Exact rational |
34 | -13/10Exact rational | 0Exact rational | -13/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/6591Exact rational | 0Exact rational | 0Exact rational | 2000/6591Exact rational |
35 | -5/4Exact rational | 0Exact rational | -5/4Exact rational | exterior | 0Exact rational | 0Exact rational | 128/375Exact rational | 0Exact rational | 0Exact rational | 128/375Exact rational |
36 | -6/5Exact rational | 0Exact rational | -6/5Exact rational | exterior | 0Exact rational | 0Exact rational | 125/324Exact rational | 0Exact rational | 0Exact rational | 125/324Exact rational |
37 | -23/20Exact rational | 0Exact rational | -23/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/36501Exact rational | 0Exact rational | 0Exact rational | 16000/36501Exact rational |
38 | -11/10Exact rational | 0Exact rational | -11/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/3993Exact rational | 0Exact rational | 0Exact rational | 2000/3993Exact rational |
39 | -21/20Exact rational | 0Exact rational | -21/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/27783Exact rational | 0Exact rational | 0Exact rational | 16000/27783Exact rational |
40One-sided source-branch limit only; not a unique ordinary interface field. | -1Exact rational | 0Exact rational | -1Exact rational | exterior-limit | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | 2/3Exact rational |
All 41 axis finite interior records, with their own surface limits (41 rows)
Uniform prescribed M=M₀ẑ inside radius a, vacuum outside, no free current. X=x/a,Z=z/a. B/(μ₀M₀), H/M₀ and K_b/M₀ have separate normalizations. Surface field records are one-sided limits; finite center regular, point-dipole origin excluded.
| Row | Signed coordinate / a | X=x/a | Z=z/a | Source side/region | B / (μ₀M₀) · x | B / (μ₀M₀) · y | B / (μ₀M₀) · z | H / M₀ · x | H / M₀ · y | H / M₀ · z |
|---|---|---|---|---|---|---|---|---|---|---|
0One-sided source-branch limit only; not a unique ordinary interface field. | -1Exact rational | 0Exact rational | -1Exact rational | interior-limit | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
1 | -19/20Exact rational | 0Exact rational | -19/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
2 | -9/10Exact rational | 0Exact rational | -9/10Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
3 | -17/20Exact rational | 0Exact rational | -17/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
4 | -4/5Exact rational | 0Exact rational | -4/5Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
5 | -3/4Exact rational | 0Exact rational | -3/4Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
6 | -7/10Exact rational | 0Exact rational | -7/10Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
7 | -13/20Exact rational | 0Exact rational | -13/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
8 | -3/5Exact rational | 0Exact rational | -3/5Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
9 | -11/20Exact rational | 0Exact rational | -11/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
10 | -1/2Exact rational | 0Exact rational | -1/2Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
11 | -9/20Exact rational | 0Exact rational | -9/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
12 | -2/5Exact rational | 0Exact rational | -2/5Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
13 | -7/20Exact rational | 0Exact rational | -7/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
14 | -3/10Exact rational | 0Exact rational | -3/10Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
15 | -1/4Exact rational | 0Exact rational | -1/4Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
16 | -1/5Exact rational | 0Exact rational | -1/5Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
17 | -3/20Exact rational | 0Exact rational | -3/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
18 | -1/10Exact rational | 0Exact rational | -1/10Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
19 | -1/20Exact rational | 0Exact rational | -1/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
20 | 0Exact rational | 0Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
21 | 1/20Exact rational | 0Exact rational | 1/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
22 | 1/10Exact rational | 0Exact rational | 1/10Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
23 | 3/20Exact rational | 0Exact rational | 3/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
24 | 1/5Exact rational | 0Exact rational | 1/5Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
25 | 1/4Exact rational | 0Exact rational | 1/4Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
26 | 3/10Exact rational | 0Exact rational | 3/10Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
27 | 7/20Exact rational | 0Exact rational | 7/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
28 | 2/5Exact rational | 0Exact rational | 2/5Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
29 | 9/20Exact rational | 0Exact rational | 9/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
30 | 1/2Exact rational | 0Exact rational | 1/2Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
31 | 11/20Exact rational | 0Exact rational | 11/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
32 | 3/5Exact rational | 0Exact rational | 3/5Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
33 | 13/20Exact rational | 0Exact rational | 13/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
34 | 7/10Exact rational | 0Exact rational | 7/10Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
35 | 3/4Exact rational | 0Exact rational | 3/4Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
36 | 4/5Exact rational | 0Exact rational | 4/5Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
37 | 17/20Exact rational | 0Exact rational | 17/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
38 | 9/10Exact rational | 0Exact rational | 9/10Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
39 | 19/20Exact rational | 0Exact rational | 19/20Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
40One-sided source-branch limit only; not a unique ordinary interface field. | 1Exact rational | 0Exact rational | 1Exact rational | interior-limit | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
All 41 axis positive exterior records, with their own surface limits (41 rows)
Uniform prescribed M=M₀ẑ inside radius a, vacuum outside, no free current. X=x/a,Z=z/a. B/(μ₀M₀), H/M₀ and K_b/M₀ have separate normalizations. Surface field records are one-sided limits; finite center regular, point-dipole origin excluded.
| Row | Signed coordinate / a | X=x/a | Z=z/a | Source side/region | B / (μ₀M₀) · x | B / (μ₀M₀) · y | B / (μ₀M₀) · z | H / M₀ · x | H / M₀ · y | H / M₀ · z |
|---|---|---|---|---|---|---|---|---|---|---|
0One-sided source-branch limit only; not a unique ordinary interface field. | 1Exact rational | 0Exact rational | 1Exact rational | exterior-limit | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | 2/3Exact rational |
1 | 21/20Exact rational | 0Exact rational | 21/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/27783Exact rational | 0Exact rational | 0Exact rational | 16000/27783Exact rational |
2 | 11/10Exact rational | 0Exact rational | 11/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/3993Exact rational | 0Exact rational | 0Exact rational | 2000/3993Exact rational |
3 | 23/20Exact rational | 0Exact rational | 23/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/36501Exact rational | 0Exact rational | 0Exact rational | 16000/36501Exact rational |
4 | 6/5Exact rational | 0Exact rational | 6/5Exact rational | exterior | 0Exact rational | 0Exact rational | 125/324Exact rational | 0Exact rational | 0Exact rational | 125/324Exact rational |
5 | 5/4Exact rational | 0Exact rational | 5/4Exact rational | exterior | 0Exact rational | 0Exact rational | 128/375Exact rational | 0Exact rational | 0Exact rational | 128/375Exact rational |
6 | 13/10Exact rational | 0Exact rational | 13/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/6591Exact rational | 0Exact rational | 0Exact rational | 2000/6591Exact rational |
7 | 27/20Exact rational | 0Exact rational | 27/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/59049Exact rational | 0Exact rational | 0Exact rational | 16000/59049Exact rational |
8 | 7/5Exact rational | 0Exact rational | 7/5Exact rational | exterior | 0Exact rational | 0Exact rational | 250/1029Exact rational | 0Exact rational | 0Exact rational | 250/1029Exact rational |
9 | 29/20Exact rational | 0Exact rational | 29/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/73167Exact rational | 0Exact rational | 0Exact rational | 16000/73167Exact rational |
10 | 3/2Exact rational | 0Exact rational | 3/2Exact rational | exterior | 0Exact rational | 0Exact rational | 16/81Exact rational | 0Exact rational | 0Exact rational | 16/81Exact rational |
11 | 31/20Exact rational | 0Exact rational | 31/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/89373Exact rational | 0Exact rational | 0Exact rational | 16000/89373Exact rational |
12 | 8/5Exact rational | 0Exact rational | 8/5Exact rational | exterior | 0Exact rational | 0Exact rational | 125/768Exact rational | 0Exact rational | 0Exact rational | 125/768Exact rational |
13 | 33/20Exact rational | 0Exact rational | 33/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/107811Exact rational | 0Exact rational | 0Exact rational | 16000/107811Exact rational |
14 | 17/10Exact rational | 0Exact rational | 17/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/14739Exact rational | 0Exact rational | 0Exact rational | 2000/14739Exact rational |
15 | 7/4Exact rational | 0Exact rational | 7/4Exact rational | exterior | 0Exact rational | 0Exact rational | 128/1029Exact rational | 0Exact rational | 0Exact rational | 128/1029Exact rational |
16 | 9/5Exact rational | 0Exact rational | 9/5Exact rational | exterior | 0Exact rational | 0Exact rational | 250/2187Exact rational | 0Exact rational | 0Exact rational | 250/2187Exact rational |
17 | 37/20Exact rational | 0Exact rational | 37/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/151959Exact rational | 0Exact rational | 0Exact rational | 16000/151959Exact rational |
18 | 19/10Exact rational | 0Exact rational | 19/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/20577Exact rational | 0Exact rational | 0Exact rational | 2000/20577Exact rational |
19 | 39/20Exact rational | 0Exact rational | 39/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/177957Exact rational | 0Exact rational | 0Exact rational | 16000/177957Exact rational |
20 | 2Exact rational | 0Exact rational | 2Exact rational | exterior | 0Exact rational | 0Exact rational | 1/12Exact rational | 0Exact rational | 0Exact rational | 1/12Exact rational |
21 | 41/20Exact rational | 0Exact rational | 41/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/206763Exact rational | 0Exact rational | 0Exact rational | 16000/206763Exact rational |
22 | 21/10Exact rational | 0Exact rational | 21/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/27783Exact rational | 0Exact rational | 0Exact rational | 2000/27783Exact rational |
23 | 43/20Exact rational | 0Exact rational | 43/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/238521Exact rational | 0Exact rational | 0Exact rational | 16000/238521Exact rational |
24 | 11/5Exact rational | 0Exact rational | 11/5Exact rational | exterior | 0Exact rational | 0Exact rational | 250/3993Exact rational | 0Exact rational | 0Exact rational | 250/3993Exact rational |
25 | 9/4Exact rational | 0Exact rational | 9/4Exact rational | exterior | 0Exact rational | 0Exact rational | 128/2187Exact rational | 0Exact rational | 0Exact rational | 128/2187Exact rational |
26 | 23/10Exact rational | 0Exact rational | 23/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/36501Exact rational | 0Exact rational | 0Exact rational | 2000/36501Exact rational |
27 | 47/20Exact rational | 0Exact rational | 47/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/311469Exact rational | 0Exact rational | 0Exact rational | 16000/311469Exact rational |
28 | 12/5Exact rational | 0Exact rational | 12/5Exact rational | exterior | 0Exact rational | 0Exact rational | 125/2592Exact rational | 0Exact rational | 0Exact rational | 125/2592Exact rational |
29 | 49/20Exact rational | 0Exact rational | 49/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/352947Exact rational | 0Exact rational | 0Exact rational | 16000/352947Exact rational |
30 | 5/2Exact rational | 0Exact rational | 5/2Exact rational | exterior | 0Exact rational | 0Exact rational | 16/375Exact rational | 0Exact rational | 0Exact rational | 16/375Exact rational |
31 | 51/20Exact rational | 0Exact rational | 51/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/397953Exact rational | 0Exact rational | 0Exact rational | 16000/397953Exact rational |
32 | 13/5Exact rational | 0Exact rational | 13/5Exact rational | exterior | 0Exact rational | 0Exact rational | 250/6591Exact rational | 0Exact rational | 0Exact rational | 250/6591Exact rational |
33 | 53/20Exact rational | 0Exact rational | 53/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/446631Exact rational | 0Exact rational | 0Exact rational | 16000/446631Exact rational |
34 | 27/10Exact rational | 0Exact rational | 27/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/59049Exact rational | 0Exact rational | 0Exact rational | 2000/59049Exact rational |
35 | 11/4Exact rational | 0Exact rational | 11/4Exact rational | exterior | 0Exact rational | 0Exact rational | 128/3993Exact rational | 0Exact rational | 0Exact rational | 128/3993Exact rational |
36 | 14/5Exact rational | 0Exact rational | 14/5Exact rational | exterior | 0Exact rational | 0Exact rational | 125/4116Exact rational | 0Exact rational | 0Exact rational | 125/4116Exact rational |
37 | 57/20Exact rational | 0Exact rational | 57/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/555579Exact rational | 0Exact rational | 0Exact rational | 16000/555579Exact rational |
38 | 29/10Exact rational | 0Exact rational | 29/10Exact rational | exterior | 0Exact rational | 0Exact rational | 2000/73167Exact rational | 0Exact rational | 0Exact rational | 2000/73167Exact rational |
39 | 59/20Exact rational | 0Exact rational | 59/20Exact rational | exterior | 0Exact rational | 0Exact rational | 16000/616137Exact rational | 0Exact rational | 0Exact rational | 16000/616137Exact rational |
40 | 3Exact rational | 0Exact rational | 3Exact rational | exterior | 0Exact rational | 0Exact rational | 2/81Exact rational | 0Exact rational | 0Exact rational | 2/81Exact rational |
All 41 equator negative exterior records, with their own surface limits (41 rows)
Uniform prescribed M=M₀ẑ inside radius a, vacuum outside, no free current. X=x/a,Z=z/a. B/(μ₀M₀), H/M₀ and K_b/M₀ have separate normalizations. Surface field records are one-sided limits; finite center regular, point-dipole origin excluded.
| Row | Signed coordinate / a | X=x/a | Z=z/a | Source side/region | B / (μ₀M₀) · x | B / (μ₀M₀) · y | B / (μ₀M₀) · z | H / M₀ · x | H / M₀ · y | H / M₀ · z |
|---|---|---|---|---|---|---|---|---|---|---|
0 | -3Exact rational | -3Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1/81Exact rational | 0Exact rational | 0Exact rational | -1/81Exact rational |
1 | -59/20Exact rational | -59/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/616137Exact rational | 0Exact rational | 0Exact rational | -8000/616137Exact rational |
2 | -29/10Exact rational | -29/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/73167Exact rational | 0Exact rational | 0Exact rational | -1000/73167Exact rational |
3 | -57/20Exact rational | -57/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/555579Exact rational | 0Exact rational | 0Exact rational | -8000/555579Exact rational |
4 | -14/5Exact rational | -14/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/8232Exact rational | 0Exact rational | 0Exact rational | -125/8232Exact rational |
5 | -11/4Exact rational | -11/4Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -64/3993Exact rational | 0Exact rational | 0Exact rational | -64/3993Exact rational |
6 | -27/10Exact rational | -27/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/59049Exact rational | 0Exact rational | 0Exact rational | -1000/59049Exact rational |
7 | -53/20Exact rational | -53/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/446631Exact rational | 0Exact rational | 0Exact rational | -8000/446631Exact rational |
8 | -13/5Exact rational | -13/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/6591Exact rational | 0Exact rational | 0Exact rational | -125/6591Exact rational |
9 | -51/20Exact rational | -51/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/397953Exact rational | 0Exact rational | 0Exact rational | -8000/397953Exact rational |
10 | -5/2Exact rational | -5/2Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8/375Exact rational | 0Exact rational | 0Exact rational | -8/375Exact rational |
11 | -49/20Exact rational | -49/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/352947Exact rational | 0Exact rational | 0Exact rational | -8000/352947Exact rational |
12 | -12/5Exact rational | -12/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/5184Exact rational | 0Exact rational | 0Exact rational | -125/5184Exact rational |
13 | -47/20Exact rational | -47/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/311469Exact rational | 0Exact rational | 0Exact rational | -8000/311469Exact rational |
14 | -23/10Exact rational | -23/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/36501Exact rational | 0Exact rational | 0Exact rational | -1000/36501Exact rational |
15 | -9/4Exact rational | -9/4Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -64/2187Exact rational | 0Exact rational | 0Exact rational | -64/2187Exact rational |
16 | -11/5Exact rational | -11/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/3993Exact rational | 0Exact rational | 0Exact rational | -125/3993Exact rational |
17 | -43/20Exact rational | -43/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/238521Exact rational | 0Exact rational | 0Exact rational | -8000/238521Exact rational |
18 | -21/10Exact rational | -21/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/27783Exact rational | 0Exact rational | 0Exact rational | -1000/27783Exact rational |
19 | -41/20Exact rational | -41/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/206763Exact rational | 0Exact rational | 0Exact rational | -8000/206763Exact rational |
20 | -2Exact rational | -2Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1/24Exact rational | 0Exact rational | 0Exact rational | -1/24Exact rational |
21 | -39/20Exact rational | -39/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/177957Exact rational | 0Exact rational | 0Exact rational | -8000/177957Exact rational |
22 | -19/10Exact rational | -19/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/20577Exact rational | 0Exact rational | 0Exact rational | -1000/20577Exact rational |
23 | -37/20Exact rational | -37/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/151959Exact rational | 0Exact rational | 0Exact rational | -8000/151959Exact rational |
24 | -9/5Exact rational | -9/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/2187Exact rational | 0Exact rational | 0Exact rational | -125/2187Exact rational |
25 | -7/4Exact rational | -7/4Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -64/1029Exact rational | 0Exact rational | 0Exact rational | -64/1029Exact rational |
26 | -17/10Exact rational | -17/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/14739Exact rational | 0Exact rational | 0Exact rational | -1000/14739Exact rational |
27 | -33/20Exact rational | -33/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/107811Exact rational | 0Exact rational | 0Exact rational | -8000/107811Exact rational |
28 | -8/5Exact rational | -8/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/1536Exact rational | 0Exact rational | 0Exact rational | -125/1536Exact rational |
29 | -31/20Exact rational | -31/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/89373Exact rational | 0Exact rational | 0Exact rational | -8000/89373Exact rational |
30 | -3/2Exact rational | -3/2Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8/81Exact rational | 0Exact rational | 0Exact rational | -8/81Exact rational |
31 | -29/20Exact rational | -29/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/73167Exact rational | 0Exact rational | 0Exact rational | -8000/73167Exact rational |
32 | -7/5Exact rational | -7/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/1029Exact rational | 0Exact rational | 0Exact rational | -125/1029Exact rational |
33 | -27/20Exact rational | -27/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/59049Exact rational | 0Exact rational | 0Exact rational | -8000/59049Exact rational |
34 | -13/10Exact rational | -13/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/6591Exact rational | 0Exact rational | 0Exact rational | -1000/6591Exact rational |
35 | -5/4Exact rational | -5/4Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -64/375Exact rational | 0Exact rational | 0Exact rational | -64/375Exact rational |
36 | -6/5Exact rational | -6/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/648Exact rational | 0Exact rational | 0Exact rational | -125/648Exact rational |
37 | -23/20Exact rational | -23/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/36501Exact rational | 0Exact rational | 0Exact rational | -8000/36501Exact rational |
38 | -11/10Exact rational | -11/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/3993Exact rational | 0Exact rational | 0Exact rational | -1000/3993Exact rational |
39 | -21/20Exact rational | -21/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/27783Exact rational | 0Exact rational | 0Exact rational | -8000/27783Exact rational |
40One-sided source-branch limit only; not a unique ordinary interface field. | -1Exact rational | -1Exact rational | 0Exact rational | exterior-limit | 0Exact rational | 0Exact rational | -1/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
All 41 equator finite interior records, with their own surface limits (41 rows)
Uniform prescribed M=M₀ẑ inside radius a, vacuum outside, no free current. X=x/a,Z=z/a. B/(μ₀M₀), H/M₀ and K_b/M₀ have separate normalizations. Surface field records are one-sided limits; finite center regular, point-dipole origin excluded.
| Row | Signed coordinate / a | X=x/a | Z=z/a | Source side/region | B / (μ₀M₀) · x | B / (μ₀M₀) · y | B / (μ₀M₀) · z | H / M₀ · x | H / M₀ · y | H / M₀ · z |
|---|---|---|---|---|---|---|---|---|---|---|
0One-sided source-branch limit only; not a unique ordinary interface field. | -1Exact rational | -1Exact rational | 0Exact rational | interior-limit | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
1 | -19/20Exact rational | -19/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
2 | -9/10Exact rational | -9/10Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
3 | -17/20Exact rational | -17/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
4 | -4/5Exact rational | -4/5Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
5 | -3/4Exact rational | -3/4Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
6 | -7/10Exact rational | -7/10Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
7 | -13/20Exact rational | -13/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
8 | -3/5Exact rational | -3/5Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
9 | -11/20Exact rational | -11/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
10 | -1/2Exact rational | -1/2Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
11 | -9/20Exact rational | -9/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
12 | -2/5Exact rational | -2/5Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
13 | -7/20Exact rational | -7/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
14 | -3/10Exact rational | -3/10Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
15 | -1/4Exact rational | -1/4Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
16 | -1/5Exact rational | -1/5Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
17 | -3/20Exact rational | -3/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
18 | -1/10Exact rational | -1/10Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
19 | -1/20Exact rational | -1/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
20 | 0Exact rational | 0Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
21 | 1/20Exact rational | 1/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
22 | 1/10Exact rational | 1/10Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
23 | 3/20Exact rational | 3/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
24 | 1/5Exact rational | 1/5Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
25 | 1/4Exact rational | 1/4Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
26 | 3/10Exact rational | 3/10Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
27 | 7/20Exact rational | 7/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
28 | 2/5Exact rational | 2/5Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
29 | 9/20Exact rational | 9/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
30 | 1/2Exact rational | 1/2Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
31 | 11/20Exact rational | 11/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
32 | 3/5Exact rational | 3/5Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
33 | 13/20Exact rational | 13/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
34 | 7/10Exact rational | 7/10Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
35 | 3/4Exact rational | 3/4Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
36 | 4/5Exact rational | 4/5Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
37 | 17/20Exact rational | 17/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
38 | 9/10Exact rational | 9/10Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
39 | 19/20Exact rational | 19/20Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
40One-sided source-branch limit only; not a unique ordinary interface field. | 1Exact rational | 1Exact rational | 0Exact rational | interior-limit | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
All 41 equator positive exterior records, with their own surface limits (41 rows)
Uniform prescribed M=M₀ẑ inside radius a, vacuum outside, no free current. X=x/a,Z=z/a. B/(μ₀M₀), H/M₀ and K_b/M₀ have separate normalizations. Surface field records are one-sided limits; finite center regular, point-dipole origin excluded.
| Row | Signed coordinate / a | X=x/a | Z=z/a | Source side/region | B / (μ₀M₀) · x | B / (μ₀M₀) · y | B / (μ₀M₀) · z | H / M₀ · x | H / M₀ · y | H / M₀ · z |
|---|---|---|---|---|---|---|---|---|---|---|
0One-sided source-branch limit only; not a unique ordinary interface field. | 1Exact rational | 1Exact rational | 0Exact rational | exterior-limit | 0Exact rational | 0Exact rational | -1/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
1 | 21/20Exact rational | 21/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/27783Exact rational | 0Exact rational | 0Exact rational | -8000/27783Exact rational |
2 | 11/10Exact rational | 11/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/3993Exact rational | 0Exact rational | 0Exact rational | -1000/3993Exact rational |
3 | 23/20Exact rational | 23/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/36501Exact rational | 0Exact rational | 0Exact rational | -8000/36501Exact rational |
4 | 6/5Exact rational | 6/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/648Exact rational | 0Exact rational | 0Exact rational | -125/648Exact rational |
5 | 5/4Exact rational | 5/4Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -64/375Exact rational | 0Exact rational | 0Exact rational | -64/375Exact rational |
6 | 13/10Exact rational | 13/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/6591Exact rational | 0Exact rational | 0Exact rational | -1000/6591Exact rational |
7 | 27/20Exact rational | 27/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/59049Exact rational | 0Exact rational | 0Exact rational | -8000/59049Exact rational |
8 | 7/5Exact rational | 7/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/1029Exact rational | 0Exact rational | 0Exact rational | -125/1029Exact rational |
9 | 29/20Exact rational | 29/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/73167Exact rational | 0Exact rational | 0Exact rational | -8000/73167Exact rational |
10 | 3/2Exact rational | 3/2Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8/81Exact rational | 0Exact rational | 0Exact rational | -8/81Exact rational |
11 | 31/20Exact rational | 31/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/89373Exact rational | 0Exact rational | 0Exact rational | -8000/89373Exact rational |
12 | 8/5Exact rational | 8/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/1536Exact rational | 0Exact rational | 0Exact rational | -125/1536Exact rational |
13 | 33/20Exact rational | 33/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/107811Exact rational | 0Exact rational | 0Exact rational | -8000/107811Exact rational |
14 | 17/10Exact rational | 17/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/14739Exact rational | 0Exact rational | 0Exact rational | -1000/14739Exact rational |
15 | 7/4Exact rational | 7/4Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -64/1029Exact rational | 0Exact rational | 0Exact rational | -64/1029Exact rational |
16 | 9/5Exact rational | 9/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/2187Exact rational | 0Exact rational | 0Exact rational | -125/2187Exact rational |
17 | 37/20Exact rational | 37/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/151959Exact rational | 0Exact rational | 0Exact rational | -8000/151959Exact rational |
18 | 19/10Exact rational | 19/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/20577Exact rational | 0Exact rational | 0Exact rational | -1000/20577Exact rational |
19 | 39/20Exact rational | 39/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/177957Exact rational | 0Exact rational | 0Exact rational | -8000/177957Exact rational |
20 | 2Exact rational | 2Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1/24Exact rational | 0Exact rational | 0Exact rational | -1/24Exact rational |
21 | 41/20Exact rational | 41/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/206763Exact rational | 0Exact rational | 0Exact rational | -8000/206763Exact rational |
22 | 21/10Exact rational | 21/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/27783Exact rational | 0Exact rational | 0Exact rational | -1000/27783Exact rational |
23 | 43/20Exact rational | 43/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/238521Exact rational | 0Exact rational | 0Exact rational | -8000/238521Exact rational |
24 | 11/5Exact rational | 11/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/3993Exact rational | 0Exact rational | 0Exact rational | -125/3993Exact rational |
25 | 9/4Exact rational | 9/4Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -64/2187Exact rational | 0Exact rational | 0Exact rational | -64/2187Exact rational |
26 | 23/10Exact rational | 23/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/36501Exact rational | 0Exact rational | 0Exact rational | -1000/36501Exact rational |
27 | 47/20Exact rational | 47/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/311469Exact rational | 0Exact rational | 0Exact rational | -8000/311469Exact rational |
28 | 12/5Exact rational | 12/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/5184Exact rational | 0Exact rational | 0Exact rational | -125/5184Exact rational |
29 | 49/20Exact rational | 49/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/352947Exact rational | 0Exact rational | 0Exact rational | -8000/352947Exact rational |
30 | 5/2Exact rational | 5/2Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8/375Exact rational | 0Exact rational | 0Exact rational | -8/375Exact rational |
31 | 51/20Exact rational | 51/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/397953Exact rational | 0Exact rational | 0Exact rational | -8000/397953Exact rational |
32 | 13/5Exact rational | 13/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/6591Exact rational | 0Exact rational | 0Exact rational | -125/6591Exact rational |
33 | 53/20Exact rational | 53/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/446631Exact rational | 0Exact rational | 0Exact rational | -8000/446631Exact rational |
34 | 27/10Exact rational | 27/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/59049Exact rational | 0Exact rational | 0Exact rational | -1000/59049Exact rational |
35 | 11/4Exact rational | 11/4Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -64/3993Exact rational | 0Exact rational | 0Exact rational | -64/3993Exact rational |
36 | 14/5Exact rational | 14/5Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -125/8232Exact rational | 0Exact rational | 0Exact rational | -125/8232Exact rational |
37 | 57/20Exact rational | 57/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/555579Exact rational | 0Exact rational | 0Exact rational | -8000/555579Exact rational |
38 | 29/10Exact rational | 29/10Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1000/73167Exact rational | 0Exact rational | 0Exact rational | -1000/73167Exact rational |
39 | 59/20Exact rational | 59/20Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -8000/616137Exact rational | 0Exact rational | 0Exact rational | -8000/616137Exact rational |
40 | 3Exact rational | 3Exact rational | 0Exact rational | exterior | 0Exact rational | 0Exact rational | -1/81Exact rational | 0Exact rational | 0Exact rational | -1/81Exact rational |
All 10 finite meridian field controls (10 rows)
Uniform prescribed M=M₀ẑ inside radius a, vacuum outside, no free current. X=x/a,Z=z/a. B/(μ₀M₀), H/M₀ and K_b/M₀ have separate normalizations. Surface field records are one-sided limits; finite center regular, point-dipole origin excluded.
| Row | X=x/a | Z=z/a | R² | Finite-source region | B / (μ₀M₀) · x | B / (μ₀M₀) · y | B / (μ₀M₀) · z | H / M₀ · x | H / M₀ · y | H / M₀ · z |
|---|---|---|---|---|---|---|---|---|---|---|
0Regular finite-source center; no radial unit vector or point-source singular value is assigned. | 0Exact rational | 0Exact rational | 0Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
1 | 1/2Exact rational | 0Exact rational | 1/4Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
2 | 0Exact rational | 1/2Exact rational | 1/4Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
3 | 1/3Exact rational | -1/3Exact rational | 2/9Exact rational | interior | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational |
4 | 3/2Exact rational | 0Exact rational | 9/4Exact rational | exterior | 0Exact rational | 0Exact rational | -8/81Exact rational | 0Exact rational | 0Exact rational | -8/81Exact rational |
5 | 0Exact rational | 2Exact rational | 4Exact rational | exterior | 0Exact rational | 0Exact rational | 1/12Exact rational | 0Exact rational | 0Exact rational | 1/12Exact rational |
6 | 0Exact rational | -2Exact rational | 4Exact rational | exterior | 0Exact rational | 0Exact rational | 1/12Exact rational | 0Exact rational | 0Exact rational | 1/12Exact rational |
7 | 2Exact rational | 1Exact rational | 5Exact rational | exterior | 0.03577708763999663514254677869970041976704989375378441158833435592656833Approximate: 70 significant digits; calculation provenance | 0Exact rational | -0.01192569587999887838084892623323347325568329791792813719611145197552278Approximate: 70 significant digits; calculation provenance | 0.03577708763999663514254677869970041976704989375378441158833435592656833Approximate: 70 significant digits; calculation provenance | 0Exact rational | -0.01192569587999887838084892623323347325568329791792813719611145197552278Approximate: 70 significant digits; calculation provenance |
8 | -2Exact rational | 1Exact rational | 5Exact rational | exterior | -0.03577708763999663514254677869970041976704989375378441158833435592656833Approximate: 70 significant digits; calculation provenance | 0Exact rational | -0.01192569587999887838084892623323347325568329791792813719611145197552278Approximate: 70 significant digits; calculation provenance | -0.03577708763999663514254677869970041976704989375378441158833435592656833Approximate: 70 significant digits; calculation provenance | 0Exact rational | -0.01192569587999887838084892623323347325568329791792813719611145197552278Approximate: 70 significant digits; calculation provenance |
9 | 1Exact rational | -2Exact rational | 5Exact rational | exterior | -0.03577708763999663514254677869970041976704989375378441158833435592656833Approximate: 70 significant digits; calculation provenance | 0Exact rational | 0.04173993557999607433297124181631715639489154271274848018639008191432972Approximate: 70 significant digits; calculation provenance | -0.03577708763999663514254677869970041976704989375378441158833435592656833Approximate: 70 significant digits; calculation provenance | 0Exact rational | 0.04173993557999607433297124181631715639489154271274848018639008191432972Approximate: 70 significant digits; calculation provenance |
All 6 one-sided surface controls and bound-current relations (6 rows)
Uniform prescribed M=M₀ẑ inside radius a, vacuum outside, no free current. X=x/a,Z=z/a. B/(μ₀M₀), H/M₀ and K_b/M₀ have separate normalizations. Surface field records are one-sided limits; finite center regular, point-dipole origin excluded.
| Row | X=x/a | Z=z/a | Outward unit normal · x | Outward unit normal · y | Outward unit normal · z | Bound K / M₀ · x | Bound K / M₀ · y | Bound K / M₀ · z | Interior B limit / (μ₀M₀) · x | Interior B limit / (μ₀M₀) · y | Interior B limit / (μ₀M₀) · z | Exterior B limit / (μ₀M₀) · x | Exterior B limit / (μ₀M₀) · y | Exterior B limit / (μ₀M₀) · z | Interior H limit / M₀ · x | Interior H limit / M₀ · y | Interior H limit / M₀ · z | Exterior H limit / M₀ · x | Exterior H limit / M₀ · y | Exterior H limit / M₀ · z | Normal B jump / (μ₀M₀) | n × ΔH / M₀ · x | n × ΔH / M₀ · y | n × ΔH / M₀ · z | Normal H jump / M₀ | n × ΔB / (μ₀M₀) · x | n × ΔB / (μ₀M₀) · y | n × ΔB / (μ₀M₀) · z |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0Both inside/outside limits retained; n points from material to vacuum. Current and normal/tangent relations have distinct SI scales. | 1Exact rational | 0Exact rational | 1Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 1Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 1Exact rational | 0Exact rational |
1Both inside/outside limits retained; n points from material to vacuum. Current and normal/tangent relations have distinct SI scales. | -1Exact rational | 0Exact rational | -1Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | -1Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | -1Exact rational | 0Exact rational |
2Both inside/outside limits retained; n points from material to vacuum. Current and normal/tangent relations have distinct SI scales. | 0Exact rational | 1Exact rational | 0Exact rational | 0Exact rational | 1Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 1Exact rational | 0Exact rational | 0Exact rational | 0Exact rational |
3Both inside/outside limits retained; n points from material to vacuum. Current and normal/tangent relations have distinct SI scales. | 0Exact rational | -1Exact rational | 0Exact rational | 0Exact rational | -1Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational | 0Exact rational | 0Exact rational | 2/3Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | -1Exact rational | 0Exact rational | 0Exact rational | 0Exact rational |
4Both inside/outside limits retained; n points from material to vacuum. Current and normal/tangent relations have distinct SI scales. | 3/5Exact rational | 4/5Exact rational | 3/5Exact rational | 0Exact rational | 4/5Exact rational | 0Exact rational | 3/5Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 2/3Exact rational | 12/25Exact rational | 0Exact rational | 23/75Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational | 12/25Exact rational | 0Exact rational | 23/75Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 4/5Exact rational | 0Exact rational | 3/5Exact rational | 0Exact rational |
5Both inside/outside limits retained; n points from material to vacuum. Current and normal/tangent relations have distinct SI scales. | -3/5Exact rational | 4/5Exact rational | -3/5Exact rational | 0Exact rational | 4/5Exact rational | 0Exact rational | -3/5Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 2/3Exact rational | -12/25Exact rational | 0Exact rational | 23/75Exact rational | 0Exact rational | 0Exact rational | -1/3Exact rational | -12/25Exact rational | 0Exact rational | 23/75Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 0Exact rational | 4/5Exact rational | 0Exact rational | -3/5Exact rational | 0Exact rational |
Complete finite-source definitions, branch identities and point-limit exclusions (85 rows)
Uniform prescribed M=M₀ẑ inside radius a, vacuum outside, no free current. X=x/a,Z=z/a. B/(μ₀M₀), H/M₀ and K_b/M₀ have separate normalizations. Surface field records are one-sided limits; finite center regular, point-dipole origin excluded.
| Quantity or source condition | Retained admitted value |
|---|---|
definitions · Assumptions and regime | Uniform prescribed stationary M=M0 zhat inside radius a, vacuum outside, no free currents; coarse-grained source, not an inferred constitutive response. Surface values are one-sided limits. |
definitions · M Inside Over M0 · 1 | 0 |
definitions · M Inside Over M0 · 2 | 0 |
definitions · M Inside Over M0 · 3 | 1 |
definitions · M Outside Over M0 · 1 | 0 |
definitions · M Outside Over M0 · 2 | 0 |
definitions · M Outside Over M0 · 3 | 0 |
definitions · coordinates | X=x/a,Y=y/a,Z=z/a,R=sqrt(X^2+Y^2+Z^2) |
definitions · moment | (4pi/3)a^3 M0 zhat |
definitions · volume Bound Current | curl M=0 in each open region |
definitions · surface Bound Current Over M0 | zhat cross outward n |
definitions · interior · B / (μ₀M₀) · 1 | 0 |
definitions · interior · B / (μ₀M₀) · 2 | 0 |
definitions · interior · B / (μ₀M₀) · 3 | 2/3 |
definitions · interior · H / M₀ · 1 | 0 |
definitions · interior · H / M₀ · 2 | 0 |
definitions · interior · H / M₀ · 3 | -1/3 |
definitions · exterior | B/(mu0 M0)=[3(zhat dot Rhat)Rhat-zhat]/(3R^3),H/M0=B/(mu0 M0),R>1 |
definitions · axis Branches · 1 · domain · 1 | -3 |
definitions · axis Branches · 1 · domain · 2 | -1 |
definitions · axis Branches · 1 · grid | -3+i/20,i=0..40 |
definitions · axis Branches · 1 · surface | exterior limit at -1 |
definitions · axis Branches · 2 · domain · 1 | -1 |
definitions · axis Branches · 2 · domain · 2 | 1 |
definitions · axis Branches · 2 · grid | -1+i/20,i=0..40 |
definitions · axis Branches · 2 · surface | interior limits at +/-1 |
definitions · axis Branches · 3 · domain · 1 | 1 |
definitions · axis Branches · 3 · domain · 2 | 3 |
definitions · axis Branches · 3 · grid | 1+i/20,i=0..40 |
definitions · axis Branches · 3 · surface | exterior limit at +1 |
definitions · equatorial Branches | The same signed coordinate grids at Z=0, with distinct boundary sides. |
definitions · meridian Control Points · 1 · 1 | 0 |
definitions · meridian Control Points · 1 · 2 | 0 |
definitions · meridian Control Points · 2 · 1 | 1/2 |
definitions · meridian Control Points · 2 · 2 | 0 |
definitions · meridian Control Points · 3 · 1 | 0 |
definitions · meridian Control Points · 3 · 2 | 1/2 |
definitions · meridian Control Points · 4 · 1 | 1/3 |
definitions · meridian Control Points · 4 · 2 | -1/3 |
definitions · meridian Control Points · 5 · 1 | 3/2 |
definitions · meridian Control Points · 5 · 2 | 0 |
definitions · meridian Control Points · 6 · 1 | 0 |
definitions · meridian Control Points · 6 · 2 | 2 |
definitions · meridian Control Points · 7 · 1 | 0 |
definitions · meridian Control Points · 7 · 2 | -2 |
definitions · meridian Control Points · 8 · 1 | 2 |
definitions · meridian Control Points · 8 · 2 | 1 |
definitions · meridian Control Points · 9 · 1 | -2 |
definitions · meridian Control Points · 9 · 2 | 1 |
definitions · meridian Control Points · 10 · 1 | 1 |
definitions · meridian Control Points · 10 · 2 | -2 |
definitions · surface Controls · 1 · 1 | 1 |
definitions · surface Controls · 1 · 2 | 0 |
definitions · surface Controls · 2 · 1 | -1 |
definitions · surface Controls · 2 · 2 | 0 |
definitions · surface Controls · 3 · 1 | 0 |
definitions · surface Controls · 3 · 2 | 1 |
definitions · surface Controls · 4 · 1 | 0 |
definitions · surface Controls · 4 · 2 | -1 |
definitions · surface Controls · 5 · 1 | 3/5 |
definitions · surface Controls · 5 · 2 | 4/5 |
definitions · surface Controls · 6 · 1 | -3/5 |
definitions · surface Controls · 6 · 2 | 4/5 |
definitions · Interface/domain rules | Do not bridge a tangential B jump. Normal B and tangential H are continuous without free sheets; total sheet current equals bound current. Never substitute the exterior point-dipole formula for the finite interior or a source value at R=0. |
definitions · Distributional point-limit convention | For an Ampèrian point-dipole limit, the exterior singular field is distributional and has contact term +(2mu0/3)m delta^3(r). No ordinary central sample or numeric delta spike. |
profiles · 1 · orientation | axis |
profiles · 1 · interior Finite At Centre | true |
profiles · 1 · cross Interface Interpolation Allowed | false |
profiles · 1 · branches · 1 · id | negativeExterior |
profiles · 1 · branches · 2 · id | interior |
profiles · 1 · branches · 3 · id | positiveExterior |
profiles · 2 · orientation | equator |
profiles · 2 · interior Finite At Centre | true |
profiles · 2 · cross Interface Interpolation Allowed | false |
profiles · 2 · branches · 1 · id | negativeExterior |
profiles · 2 · branches · 2 · id | interior |
profiles · 2 · branches · 3 · id | positiveExterior |
Volume bound current / (M₀/a) · x | 0Exact rational |
Volume bound current / (M₀/a) · y | 0Exact rational |
Volume bound current / (M₀/a) · z | 0Exact rational |
Total moment / (M₀a³) | (4pi/3) zhatSymbolic expression |
point Limit · Ordinary point-dipole origin valuePoint source is excluded; finite sphere center is a separate regular solution. | Not an ordinary numerical value |
point Limit · Point-source origin exclusion | source-excluded-for-point-dipole-only;finite-sphere-interior-is-separate |
point Limit · Point-contact coefficient / (μ₀m) | 2/3Exact rational |
point Limit · Distribution interpretation | three-dimensional Dirac distribution; not an ordinary numerical sample |
Focus a named plot or table region to scroll horizontally with the keyboard, or swipe within that region. Readable fixed-size diagrams remain locally contained. Full disclosures retain exact values, all records and source/domain conditions.
| Observation | Region or side | B_z/(μ₀M₀) | H_z/M₀ | Interpretation |
|---|---|---|---|---|
| Centre | Interior | 2/3 | −1/3 | Regular finite field |
| z = 2a | Exterior axis | 1/12 | 1/12 | Vacuum, field along +z |
| x = 2a, z = 0 | Exterior equator | −1/24 | −1/24 | Vacuum, field along −z |
| North pole | Interior limit | 2/3 | −1/3 | Normal B matches exterior |
| North pole | Exterior limit | 2/3 | 2/3 | Normal H jump is allowed |
| Positive-x equator | Interior limit | 2/3 | −1/3 | Tangential B on inside |
| Positive-x equator | Exterior limit | −1/3 | −1/3 | Bound sheet accounts for B jump |
Separate intrinsic and apparent response
A scalar constitutive approximation M = χ_m H_int assumes local, linear, isotropic and reversible response in the stated regime. It gives μ = μ₀(1 + χ_m), not a universal constant permeability for every magnetic material. The H in this equation is the internal macroscopic field, not automatically the externally applied field.
For a uniformly magnetized ellipsoid in a uniform applied field along a principal axis, H_int = H_app − NM. The principal demagnetizing factors in SI sum to 1, and a sphere has N = 1/3 on every axis. Solving the constitutive equation gives H_int/H_app = 1/(1 + Nχ_m), M/H_app = χ_m/(1 + Nχ_m), and B/(μ₀H_app) = (1 + χ_m)/(1 + Nχ_m).
Our two illustrative susceptibilities are 7/3 and −1/4. Both give positive scalar permeability and positive denominators for the selected shapes. The controls N = 1/8, 1/3 and 5/8 represent prolate, spherical and oblate axisymmetric ellipsoids along their symmetry axis. The curve endpoints N = 0 and 1 are aspect-ratio limits, not additional finite ellipsoids. No named material is assigned these values.
A measurement using H_app gives an apparent susceptibility χ_app = M/H_app. Recovering χ_m requires the geometry and an applicable uniform-field model: χ_m = χ_app/(1 − Nχ_app). An arbitrary cylinder or irregular specimen usually has a nonuniform demagnetizing field, so one cannot insert a convenient single N without checking its approximation. NIST’s cited cgs ellipsoid factors sum to 4π; they must not be imported unchanged into this SI convention.
Internal field, apparent response and a specified shape
Internal and applied fields give different susceptibilities. These two illustrative linear ellipsoid families keep finite shape controls separate from limiting aspect ratios.
Intrinsic response and the shape seen by an applied field
- χ=7/3: H_int/H₀ (dotted)
- χ=7/3: M/H₀=χ_app (solid)
- χ=7/3: B/(μ₀H₀) (dashed)
- χ=-1/4: H_int/H₀ (dotted)
- χ=-1/4: M/H₀=χ_app (solid)
- χ=-1/4: B/(μ₀H₀) (dashed)
- Uniform applied H₀ẑ and uniformly magnetized axisymmetric ellipsoid; SI Nₓ+Nᵧ+N_z=1. χ=7/3 or−1/4, χ>−1 and positive denominator. H/H₀,M/H₀ and B/(μ₀H₀). No arbitrary-body uniform-field or measured-material claim.
- Blue denotes χ=7/3; orange denotes χ=−1/4. Quantity dashes are shared between families: H/H₀ dotted, M/H₀ solid, B/(μ₀H₀) dashed. M/H₀ is the applied-field susceptibility, distinct from intrinsic χ.
- The original controls N=1/8,1/3,5/8 are finite prolate, spherical and oblate examples. Hollow N=0,1 endpoints represent ideal limiting aspect ratios. An arbitrary finite rod or sheet does not inherit a uniform-field ellipsoid result.
- The negative susceptibility is an illustrative positive-permeability response, not a claimed measured diamagnetic dataset. Tensor, nonlinear, ordered, hysteretic and dynamic material behavior remains in the full manuscript.
- Tables preserve every original typed value: reduced rational strings, explicit approximate 70-significant-digit decimals and symbolic expressions. Plot coordinates only convert these stored values for display; no scientific solver, new case, re-rounding or measured dataset is introduced.
Illustrative finite ellipsoids: intrinsic versus applied-field response
All 82 original shape rows for intrinsic χ=7/3 (82 rows)
Uniform applied H₀ẑ and uniformly magnetized axisymmetric ellipsoid; SI Nₓ+Nᵧ+N_z=1. χ=7/3 or−1/4, χ>−1 and positive denominator. H/H₀,M/H₀ and B/(μ₀H₀). No arbitrary-body uniform-field or measured-material claim.
| Row | SI selected-axis N | SI Nₓ | SI Nᵧ | SI N_z | Intrinsic χ | Relative permeability 1+χ | 1+Nχ | Internal H / H₀ | M/H₀ = apparent χ | B / (μ₀H₀) | Applied-field susceptibility | Recovered intrinsic χ | Original finite control? | Finite shape or endpoint limit | Axisymmetric shape class |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0N=0 or1 is a limiting aspect ratio, not an ordinary finite ellipsoid. | 0Exact rational | 1/2Exact rational | 1/2Exact rational | 0Exact rational | 7/3Exact rational | 10/3Exact rational | 1Exact rational | 1Exact rational | 7/3Exact rational | 10/3Exact rational | 7/3Exact rational | 7/3Exact rational | false | limiting-aspect-ratio | prolate |
1 | 1/80Exact rational | 79/160Exact rational | 79/160Exact rational | 1/80Exact rational | 7/3Exact rational | 10/3Exact rational | 247/240Exact rational | 240/247Exact rational | 560/247Exact rational | 800/247Exact rational | 560/247Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
2 | 1/40Exact rational | 39/80Exact rational | 39/80Exact rational | 1/40Exact rational | 7/3Exact rational | 10/3Exact rational | 127/120Exact rational | 120/127Exact rational | 280/127Exact rational | 400/127Exact rational | 280/127Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
3 | 3/80Exact rational | 77/160Exact rational | 77/160Exact rational | 3/80Exact rational | 7/3Exact rational | 10/3Exact rational | 87/80Exact rational | 80/87Exact rational | 560/261Exact rational | 800/261Exact rational | 560/261Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
4 | 1/20Exact rational | 19/40Exact rational | 19/40Exact rational | 1/20Exact rational | 7/3Exact rational | 10/3Exact rational | 67/60Exact rational | 60/67Exact rational | 140/67Exact rational | 200/67Exact rational | 140/67Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
5 | 1/16Exact rational | 15/32Exact rational | 15/32Exact rational | 1/16Exact rational | 7/3Exact rational | 10/3Exact rational | 55/48Exact rational | 48/55Exact rational | 112/55Exact rational | 32/11Exact rational | 112/55Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
6 | 3/40Exact rational | 37/80Exact rational | 37/80Exact rational | 3/40Exact rational | 7/3Exact rational | 10/3Exact rational | 47/40Exact rational | 40/47Exact rational | 280/141Exact rational | 400/141Exact rational | 280/141Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
7 | 7/80Exact rational | 73/160Exact rational | 73/160Exact rational | 7/80Exact rational | 7/3Exact rational | 10/3Exact rational | 289/240Exact rational | 240/289Exact rational | 560/289Exact rational | 800/289Exact rational | 560/289Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
8 | 1/10Exact rational | 9/20Exact rational | 9/20Exact rational | 1/10Exact rational | 7/3Exact rational | 10/3Exact rational | 37/30Exact rational | 30/37Exact rational | 70/37Exact rational | 100/37Exact rational | 70/37Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
9 | 9/80Exact rational | 71/160Exact rational | 71/160Exact rational | 9/80Exact rational | 7/3Exact rational | 10/3Exact rational | 101/80Exact rational | 80/101Exact rational | 560/303Exact rational | 800/303Exact rational | 560/303Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
10 | 1/8Exact rational | 7/16Exact rational | 7/16Exact rational | 1/8Exact rational | 7/3Exact rational | 10/3Exact rational | 31/24Exact rational | 24/31Exact rational | 56/31Exact rational | 80/31Exact rational | 56/31Exact rational | 7/3Exact rational | true | finite-axisymmetric-ellipsoid | prolate |
11 | 11/80Exact rational | 69/160Exact rational | 69/160Exact rational | 11/80Exact rational | 7/3Exact rational | 10/3Exact rational | 317/240Exact rational | 240/317Exact rational | 560/317Exact rational | 800/317Exact rational | 560/317Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
12 | 3/20Exact rational | 17/40Exact rational | 17/40Exact rational | 3/20Exact rational | 7/3Exact rational | 10/3Exact rational | 27/20Exact rational | 20/27Exact rational | 140/81Exact rational | 200/81Exact rational | 140/81Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
13 | 13/80Exact rational | 67/160Exact rational | 67/160Exact rational | 13/80Exact rational | 7/3Exact rational | 10/3Exact rational | 331/240Exact rational | 240/331Exact rational | 560/331Exact rational | 800/331Exact rational | 560/331Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
14 | 7/40Exact rational | 33/80Exact rational | 33/80Exact rational | 7/40Exact rational | 7/3Exact rational | 10/3Exact rational | 169/120Exact rational | 120/169Exact rational | 280/169Exact rational | 400/169Exact rational | 280/169Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
15 | 3/16Exact rational | 13/32Exact rational | 13/32Exact rational | 3/16Exact rational | 7/3Exact rational | 10/3Exact rational | 23/16Exact rational | 16/23Exact rational | 112/69Exact rational | 160/69Exact rational | 112/69Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
16 | 1/5Exact rational | 2/5Exact rational | 2/5Exact rational | 1/5Exact rational | 7/3Exact rational | 10/3Exact rational | 22/15Exact rational | 15/22Exact rational | 35/22Exact rational | 25/11Exact rational | 35/22Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
17 | 17/80Exact rational | 63/160Exact rational | 63/160Exact rational | 17/80Exact rational | 7/3Exact rational | 10/3Exact rational | 359/240Exact rational | 240/359Exact rational | 560/359Exact rational | 800/359Exact rational | 560/359Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
18 | 9/40Exact rational | 31/80Exact rational | 31/80Exact rational | 9/40Exact rational | 7/3Exact rational | 10/3Exact rational | 61/40Exact rational | 40/61Exact rational | 280/183Exact rational | 400/183Exact rational | 280/183Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
19 | 19/80Exact rational | 61/160Exact rational | 61/160Exact rational | 19/80Exact rational | 7/3Exact rational | 10/3Exact rational | 373/240Exact rational | 240/373Exact rational | 560/373Exact rational | 800/373Exact rational | 560/373Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
20 | 1/4Exact rational | 3/8Exact rational | 3/8Exact rational | 1/4Exact rational | 7/3Exact rational | 10/3Exact rational | 19/12Exact rational | 12/19Exact rational | 28/19Exact rational | 40/19Exact rational | 28/19Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
21 | 21/80Exact rational | 59/160Exact rational | 59/160Exact rational | 21/80Exact rational | 7/3Exact rational | 10/3Exact rational | 129/80Exact rational | 80/129Exact rational | 560/387Exact rational | 800/387Exact rational | 560/387Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
22 | 11/40Exact rational | 29/80Exact rational | 29/80Exact rational | 11/40Exact rational | 7/3Exact rational | 10/3Exact rational | 197/120Exact rational | 120/197Exact rational | 280/197Exact rational | 400/197Exact rational | 280/197Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
23 | 23/80Exact rational | 57/160Exact rational | 57/160Exact rational | 23/80Exact rational | 7/3Exact rational | 10/3Exact rational | 401/240Exact rational | 240/401Exact rational | 560/401Exact rational | 800/401Exact rational | 560/401Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
24 | 3/10Exact rational | 7/20Exact rational | 7/20Exact rational | 3/10Exact rational | 7/3Exact rational | 10/3Exact rational | 17/10Exact rational | 10/17Exact rational | 70/51Exact rational | 100/51Exact rational | 70/51Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
25 | 5/16Exact rational | 11/32Exact rational | 11/32Exact rational | 5/16Exact rational | 7/3Exact rational | 10/3Exact rational | 83/48Exact rational | 48/83Exact rational | 112/83Exact rational | 160/83Exact rational | 112/83Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
26 | 13/40Exact rational | 27/80Exact rational | 27/80Exact rational | 13/40Exact rational | 7/3Exact rational | 10/3Exact rational | 211/120Exact rational | 120/211Exact rational | 280/211Exact rational | 400/211Exact rational | 280/211Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
27 | 1/3Exact rational | 1/3Exact rational | 1/3Exact rational | 1/3Exact rational | 7/3Exact rational | 10/3Exact rational | 16/9Exact rational | 9/16Exact rational | 21/16Exact rational | 15/8Exact rational | 21/16Exact rational | 7/3Exact rational | true | finite-axisymmetric-ellipsoid | sphere |
28 | 27/80Exact rational | 53/160Exact rational | 53/160Exact rational | 27/80Exact rational | 7/3Exact rational | 10/3Exact rational | 143/80Exact rational | 80/143Exact rational | 560/429Exact rational | 800/429Exact rational | 560/429Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
29 | 7/20Exact rational | 13/40Exact rational | 13/40Exact rational | 7/20Exact rational | 7/3Exact rational | 10/3Exact rational | 109/60Exact rational | 60/109Exact rational | 140/109Exact rational | 200/109Exact rational | 140/109Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
30 | 29/80Exact rational | 51/160Exact rational | 51/160Exact rational | 29/80Exact rational | 7/3Exact rational | 10/3Exact rational | 443/240Exact rational | 240/443Exact rational | 560/443Exact rational | 800/443Exact rational | 560/443Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
31 | 3/8Exact rational | 5/16Exact rational | 5/16Exact rational | 3/8Exact rational | 7/3Exact rational | 10/3Exact rational | 15/8Exact rational | 8/15Exact rational | 56/45Exact rational | 16/9Exact rational | 56/45Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
32 | 31/80Exact rational | 49/160Exact rational | 49/160Exact rational | 31/80Exact rational | 7/3Exact rational | 10/3Exact rational | 457/240Exact rational | 240/457Exact rational | 560/457Exact rational | 800/457Exact rational | 560/457Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
33 | 2/5Exact rational | 3/10Exact rational | 3/10Exact rational | 2/5Exact rational | 7/3Exact rational | 10/3Exact rational | 29/15Exact rational | 15/29Exact rational | 35/29Exact rational | 50/29Exact rational | 35/29Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
34 | 33/80Exact rational | 47/160Exact rational | 47/160Exact rational | 33/80Exact rational | 7/3Exact rational | 10/3Exact rational | 157/80Exact rational | 80/157Exact rational | 560/471Exact rational | 800/471Exact rational | 560/471Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
35 | 17/40Exact rational | 23/80Exact rational | 23/80Exact rational | 17/40Exact rational | 7/3Exact rational | 10/3Exact rational | 239/120Exact rational | 120/239Exact rational | 280/239Exact rational | 400/239Exact rational | 280/239Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
36 | 7/16Exact rational | 9/32Exact rational | 9/32Exact rational | 7/16Exact rational | 7/3Exact rational | 10/3Exact rational | 97/48Exact rational | 48/97Exact rational | 112/97Exact rational | 160/97Exact rational | 112/97Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
37 | 9/20Exact rational | 11/40Exact rational | 11/40Exact rational | 9/20Exact rational | 7/3Exact rational | 10/3Exact rational | 41/20Exact rational | 20/41Exact rational | 140/123Exact rational | 200/123Exact rational | 140/123Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
38 | 37/80Exact rational | 43/160Exact rational | 43/160Exact rational | 37/80Exact rational | 7/3Exact rational | 10/3Exact rational | 499/240Exact rational | 240/499Exact rational | 560/499Exact rational | 800/499Exact rational | 560/499Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
39 | 19/40Exact rational | 21/80Exact rational | 21/80Exact rational | 19/40Exact rational | 7/3Exact rational | 10/3Exact rational | 253/120Exact rational | 120/253Exact rational | 280/253Exact rational | 400/253Exact rational | 280/253Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
40 | 39/80Exact rational | 41/160Exact rational | 41/160Exact rational | 39/80Exact rational | 7/3Exact rational | 10/3Exact rational | 171/80Exact rational | 80/171Exact rational | 560/513Exact rational | 800/513Exact rational | 560/513Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
41 | 1/2Exact rational | 1/4Exact rational | 1/4Exact rational | 1/2Exact rational | 7/3Exact rational | 10/3Exact rational | 13/6Exact rational | 6/13Exact rational | 14/13Exact rational | 20/13Exact rational | 14/13Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
42 | 41/80Exact rational | 39/160Exact rational | 39/160Exact rational | 41/80Exact rational | 7/3Exact rational | 10/3Exact rational | 527/240Exact rational | 240/527Exact rational | 560/527Exact rational | 800/527Exact rational | 560/527Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
43 | 21/40Exact rational | 19/80Exact rational | 19/80Exact rational | 21/40Exact rational | 7/3Exact rational | 10/3Exact rational | 89/40Exact rational | 40/89Exact rational | 280/267Exact rational | 400/267Exact rational | 280/267Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
44 | 43/80Exact rational | 37/160Exact rational | 37/160Exact rational | 43/80Exact rational | 7/3Exact rational | 10/3Exact rational | 541/240Exact rational | 240/541Exact rational | 560/541Exact rational | 800/541Exact rational | 560/541Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
45 | 11/20Exact rational | 9/40Exact rational | 9/40Exact rational | 11/20Exact rational | 7/3Exact rational | 10/3Exact rational | 137/60Exact rational | 60/137Exact rational | 140/137Exact rational | 200/137Exact rational | 140/137Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
46 | 9/16Exact rational | 7/32Exact rational | 7/32Exact rational | 9/16Exact rational | 7/3Exact rational | 10/3Exact rational | 37/16Exact rational | 16/37Exact rational | 112/111Exact rational | 160/111Exact rational | 112/111Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
47 | 23/40Exact rational | 17/80Exact rational | 17/80Exact rational | 23/40Exact rational | 7/3Exact rational | 10/3Exact rational | 281/120Exact rational | 120/281Exact rational | 280/281Exact rational | 400/281Exact rational | 280/281Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
48 | 47/80Exact rational | 33/160Exact rational | 33/160Exact rational | 47/80Exact rational | 7/3Exact rational | 10/3Exact rational | 569/240Exact rational | 240/569Exact rational | 560/569Exact rational | 800/569Exact rational | 560/569Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
49 | 3/5Exact rational | 1/5Exact rational | 1/5Exact rational | 3/5Exact rational | 7/3Exact rational | 10/3Exact rational | 12/5Exact rational | 5/12Exact rational | 35/36Exact rational | 25/18Exact rational | 35/36Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
50 | 49/80Exact rational | 31/160Exact rational | 31/160Exact rational | 49/80Exact rational | 7/3Exact rational | 10/3Exact rational | 583/240Exact rational | 240/583Exact rational | 560/583Exact rational | 800/583Exact rational | 560/583Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
51 | 5/8Exact rational | 3/16Exact rational | 3/16Exact rational | 5/8Exact rational | 7/3Exact rational | 10/3Exact rational | 59/24Exact rational | 24/59Exact rational | 56/59Exact rational | 80/59Exact rational | 56/59Exact rational | 7/3Exact rational | true | finite-axisymmetric-ellipsoid | oblate |
52 | 51/80Exact rational | 29/160Exact rational | 29/160Exact rational | 51/80Exact rational | 7/3Exact rational | 10/3Exact rational | 199/80Exact rational | 80/199Exact rational | 560/597Exact rational | 800/597Exact rational | 560/597Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
53 | 13/20Exact rational | 7/40Exact rational | 7/40Exact rational | 13/20Exact rational | 7/3Exact rational | 10/3Exact rational | 151/60Exact rational | 60/151Exact rational | 140/151Exact rational | 200/151Exact rational | 140/151Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
54 | 53/80Exact rational | 27/160Exact rational | 27/160Exact rational | 53/80Exact rational | 7/3Exact rational | 10/3Exact rational | 611/240Exact rational | 240/611Exact rational | 560/611Exact rational | 800/611Exact rational | 560/611Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
55 | 27/40Exact rational | 13/80Exact rational | 13/80Exact rational | 27/40Exact rational | 7/3Exact rational | 10/3Exact rational | 103/40Exact rational | 40/103Exact rational | 280/309Exact rational | 400/309Exact rational | 280/309Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
56 | 11/16Exact rational | 5/32Exact rational | 5/32Exact rational | 11/16Exact rational | 7/3Exact rational | 10/3Exact rational | 125/48Exact rational | 48/125Exact rational | 112/125Exact rational | 32/25Exact rational | 112/125Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
57 | 7/10Exact rational | 3/20Exact rational | 3/20Exact rational | 7/10Exact rational | 7/3Exact rational | 10/3Exact rational | 79/30Exact rational | 30/79Exact rational | 70/79Exact rational | 100/79Exact rational | 70/79Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
58 | 57/80Exact rational | 23/160Exact rational | 23/160Exact rational | 57/80Exact rational | 7/3Exact rational | 10/3Exact rational | 213/80Exact rational | 80/213Exact rational | 560/639Exact rational | 800/639Exact rational | 560/639Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
59 | 29/40Exact rational | 11/80Exact rational | 11/80Exact rational | 29/40Exact rational | 7/3Exact rational | 10/3Exact rational | 323/120Exact rational | 120/323Exact rational | 280/323Exact rational | 400/323Exact rational | 280/323Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
60 | 59/80Exact rational | 21/160Exact rational | 21/160Exact rational | 59/80Exact rational | 7/3Exact rational | 10/3Exact rational | 653/240Exact rational | 240/653Exact rational | 560/653Exact rational | 800/653Exact rational | 560/653Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
61 | 3/4Exact rational | 1/8Exact rational | 1/8Exact rational | 3/4Exact rational | 7/3Exact rational | 10/3Exact rational | 11/4Exact rational | 4/11Exact rational | 28/33Exact rational | 40/33Exact rational | 28/33Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
62 | 61/80Exact rational | 19/160Exact rational | 19/160Exact rational | 61/80Exact rational | 7/3Exact rational | 10/3Exact rational | 667/240Exact rational | 240/667Exact rational | 560/667Exact rational | 800/667Exact rational | 560/667Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
63 | 31/40Exact rational | 9/80Exact rational | 9/80Exact rational | 31/40Exact rational | 7/3Exact rational | 10/3Exact rational | 337/120Exact rational | 120/337Exact rational | 280/337Exact rational | 400/337Exact rational | 280/337Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
64 | 63/80Exact rational | 17/160Exact rational | 17/160Exact rational | 63/80Exact rational | 7/3Exact rational | 10/3Exact rational | 227/80Exact rational | 80/227Exact rational | 560/681Exact rational | 800/681Exact rational | 560/681Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
65 | 4/5Exact rational | 1/10Exact rational | 1/10Exact rational | 4/5Exact rational | 7/3Exact rational | 10/3Exact rational | 43/15Exact rational | 15/43Exact rational | 35/43Exact rational | 50/43Exact rational | 35/43Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
66 | 13/16Exact rational | 3/32Exact rational | 3/32Exact rational | 13/16Exact rational | 7/3Exact rational | 10/3Exact rational | 139/48Exact rational | 48/139Exact rational | 112/139Exact rational | 160/139Exact rational | 112/139Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
67 | 33/40Exact rational | 7/80Exact rational | 7/80Exact rational | 33/40Exact rational | 7/3Exact rational | 10/3Exact rational | 117/40Exact rational | 40/117Exact rational | 280/351Exact rational | 400/351Exact rational | 280/351Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
68 | 67/80Exact rational | 13/160Exact rational | 13/160Exact rational | 67/80Exact rational | 7/3Exact rational | 10/3Exact rational | 709/240Exact rational | 240/709Exact rational | 560/709Exact rational | 800/709Exact rational | 560/709Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
69 | 17/20Exact rational | 3/40Exact rational | 3/40Exact rational | 17/20Exact rational | 7/3Exact rational | 10/3Exact rational | 179/60Exact rational | 60/179Exact rational | 140/179Exact rational | 200/179Exact rational | 140/179Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
70 | 69/80Exact rational | 11/160Exact rational | 11/160Exact rational | 69/80Exact rational | 7/3Exact rational | 10/3Exact rational | 241/80Exact rational | 80/241Exact rational | 560/723Exact rational | 800/723Exact rational | 560/723Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
71 | 7/8Exact rational | 1/16Exact rational | 1/16Exact rational | 7/8Exact rational | 7/3Exact rational | 10/3Exact rational | 73/24Exact rational | 24/73Exact rational | 56/73Exact rational | 80/73Exact rational | 56/73Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
72 | 71/80Exact rational | 9/160Exact rational | 9/160Exact rational | 71/80Exact rational | 7/3Exact rational | 10/3Exact rational | 737/240Exact rational | 240/737Exact rational | 560/737Exact rational | 800/737Exact rational | 560/737Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
73 | 9/10Exact rational | 1/20Exact rational | 1/20Exact rational | 9/10Exact rational | 7/3Exact rational | 10/3Exact rational | 31/10Exact rational | 10/31Exact rational | 70/93Exact rational | 100/93Exact rational | 70/93Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
74 | 73/80Exact rational | 7/160Exact rational | 7/160Exact rational | 73/80Exact rational | 7/3Exact rational | 10/3Exact rational | 751/240Exact rational | 240/751Exact rational | 560/751Exact rational | 800/751Exact rational | 560/751Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
75 | 37/40Exact rational | 3/80Exact rational | 3/80Exact rational | 37/40Exact rational | 7/3Exact rational | 10/3Exact rational | 379/120Exact rational | 120/379Exact rational | 280/379Exact rational | 400/379Exact rational | 280/379Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
76 | 15/16Exact rational | 1/32Exact rational | 1/32Exact rational | 15/16Exact rational | 7/3Exact rational | 10/3Exact rational | 51/16Exact rational | 16/51Exact rational | 112/153Exact rational | 160/153Exact rational | 112/153Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
77 | 19/20Exact rational | 1/40Exact rational | 1/40Exact rational | 19/20Exact rational | 7/3Exact rational | 10/3Exact rational | 193/60Exact rational | 60/193Exact rational | 140/193Exact rational | 200/193Exact rational | 140/193Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
78 | 77/80Exact rational | 3/160Exact rational | 3/160Exact rational | 77/80Exact rational | 7/3Exact rational | 10/3Exact rational | 779/240Exact rational | 240/779Exact rational | 560/779Exact rational | 800/779Exact rational | 560/779Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
79 | 39/40Exact rational | 1/80Exact rational | 1/80Exact rational | 39/40Exact rational | 7/3Exact rational | 10/3Exact rational | 131/40Exact rational | 40/131Exact rational | 280/393Exact rational | 400/393Exact rational | 280/393Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
80 | 79/80Exact rational | 1/160Exact rational | 1/160Exact rational | 79/80Exact rational | 7/3Exact rational | 10/3Exact rational | 793/240Exact rational | 240/793Exact rational | 560/793Exact rational | 800/793Exact rational | 560/793Exact rational | 7/3Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
81N=0 or1 is a limiting aspect ratio, not an ordinary finite ellipsoid. | 1Exact rational | 0Exact rational | 0Exact rational | 1Exact rational | 7/3Exact rational | 10/3Exact rational | 10/3Exact rational | 3/10Exact rational | 7/10Exact rational | 1Exact rational | 7/10Exact rational | 7/3Exact rational | false | limiting-aspect-ratio | oblate |
All 82 original shape rows for intrinsic χ=-1/4 (82 rows)
Uniform applied H₀ẑ and uniformly magnetized axisymmetric ellipsoid; SI Nₓ+Nᵧ+N_z=1. χ=7/3 or−1/4, χ>−1 and positive denominator. H/H₀,M/H₀ and B/(μ₀H₀). No arbitrary-body uniform-field or measured-material claim.
| Row | SI selected-axis N | SI Nₓ | SI Nᵧ | SI N_z | Intrinsic χ | Relative permeability 1+χ | 1+Nχ | Internal H / H₀ | M/H₀ = apparent χ | B / (μ₀H₀) | Applied-field susceptibility | Recovered intrinsic χ | Original finite control? | Finite shape or endpoint limit | Axisymmetric shape class |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0N=0 or1 is a limiting aspect ratio, not an ordinary finite ellipsoid. | 0Exact rational | 1/2Exact rational | 1/2Exact rational | 0Exact rational | -1/4Exact rational | 3/4Exact rational | 1Exact rational | 1Exact rational | -1/4Exact rational | 3/4Exact rational | -1/4Exact rational | -1/4Exact rational | false | limiting-aspect-ratio | prolate |
1 | 1/80Exact rational | 79/160Exact rational | 79/160Exact rational | 1/80Exact rational | -1/4Exact rational | 3/4Exact rational | 319/320Exact rational | 320/319Exact rational | -80/319Exact rational | 240/319Exact rational | -80/319Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
2 | 1/40Exact rational | 39/80Exact rational | 39/80Exact rational | 1/40Exact rational | -1/4Exact rational | 3/4Exact rational | 159/160Exact rational | 160/159Exact rational | -40/159Exact rational | 40/53Exact rational | -40/159Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
3 | 3/80Exact rational | 77/160Exact rational | 77/160Exact rational | 3/80Exact rational | -1/4Exact rational | 3/4Exact rational | 317/320Exact rational | 320/317Exact rational | -80/317Exact rational | 240/317Exact rational | -80/317Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
4 | 1/20Exact rational | 19/40Exact rational | 19/40Exact rational | 1/20Exact rational | -1/4Exact rational | 3/4Exact rational | 79/80Exact rational | 80/79Exact rational | -20/79Exact rational | 60/79Exact rational | -20/79Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
5 | 1/16Exact rational | 15/32Exact rational | 15/32Exact rational | 1/16Exact rational | -1/4Exact rational | 3/4Exact rational | 63/64Exact rational | 64/63Exact rational | -16/63Exact rational | 16/21Exact rational | -16/63Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
6 | 3/40Exact rational | 37/80Exact rational | 37/80Exact rational | 3/40Exact rational | -1/4Exact rational | 3/4Exact rational | 157/160Exact rational | 160/157Exact rational | -40/157Exact rational | 120/157Exact rational | -40/157Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
7 | 7/80Exact rational | 73/160Exact rational | 73/160Exact rational | 7/80Exact rational | -1/4Exact rational | 3/4Exact rational | 313/320Exact rational | 320/313Exact rational | -80/313Exact rational | 240/313Exact rational | -80/313Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
8 | 1/10Exact rational | 9/20Exact rational | 9/20Exact rational | 1/10Exact rational | -1/4Exact rational | 3/4Exact rational | 39/40Exact rational | 40/39Exact rational | -10/39Exact rational | 10/13Exact rational | -10/39Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
9 | 9/80Exact rational | 71/160Exact rational | 71/160Exact rational | 9/80Exact rational | -1/4Exact rational | 3/4Exact rational | 311/320Exact rational | 320/311Exact rational | -80/311Exact rational | 240/311Exact rational | -80/311Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
10 | 1/8Exact rational | 7/16Exact rational | 7/16Exact rational | 1/8Exact rational | -1/4Exact rational | 3/4Exact rational | 31/32Exact rational | 32/31Exact rational | -8/31Exact rational | 24/31Exact rational | -8/31Exact rational | -1/4Exact rational | true | finite-axisymmetric-ellipsoid | prolate |
11 | 11/80Exact rational | 69/160Exact rational | 69/160Exact rational | 11/80Exact rational | -1/4Exact rational | 3/4Exact rational | 309/320Exact rational | 320/309Exact rational | -80/309Exact rational | 80/103Exact rational | -80/309Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
12 | 3/20Exact rational | 17/40Exact rational | 17/40Exact rational | 3/20Exact rational | -1/4Exact rational | 3/4Exact rational | 77/80Exact rational | 80/77Exact rational | -20/77Exact rational | 60/77Exact rational | -20/77Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
13 | 13/80Exact rational | 67/160Exact rational | 67/160Exact rational | 13/80Exact rational | -1/4Exact rational | 3/4Exact rational | 307/320Exact rational | 320/307Exact rational | -80/307Exact rational | 240/307Exact rational | -80/307Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
14 | 7/40Exact rational | 33/80Exact rational | 33/80Exact rational | 7/40Exact rational | -1/4Exact rational | 3/4Exact rational | 153/160Exact rational | 160/153Exact rational | -40/153Exact rational | 40/51Exact rational | -40/153Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
15 | 3/16Exact rational | 13/32Exact rational | 13/32Exact rational | 3/16Exact rational | -1/4Exact rational | 3/4Exact rational | 61/64Exact rational | 64/61Exact rational | -16/61Exact rational | 48/61Exact rational | -16/61Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
16 | 1/5Exact rational | 2/5Exact rational | 2/5Exact rational | 1/5Exact rational | -1/4Exact rational | 3/4Exact rational | 19/20Exact rational | 20/19Exact rational | -5/19Exact rational | 15/19Exact rational | -5/19Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
17 | 17/80Exact rational | 63/160Exact rational | 63/160Exact rational | 17/80Exact rational | -1/4Exact rational | 3/4Exact rational | 303/320Exact rational | 320/303Exact rational | -80/303Exact rational | 80/101Exact rational | -80/303Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
18 | 9/40Exact rational | 31/80Exact rational | 31/80Exact rational | 9/40Exact rational | -1/4Exact rational | 3/4Exact rational | 151/160Exact rational | 160/151Exact rational | -40/151Exact rational | 120/151Exact rational | -40/151Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
19 | 19/80Exact rational | 61/160Exact rational | 61/160Exact rational | 19/80Exact rational | -1/4Exact rational | 3/4Exact rational | 301/320Exact rational | 320/301Exact rational | -80/301Exact rational | 240/301Exact rational | -80/301Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
20 | 1/4Exact rational | 3/8Exact rational | 3/8Exact rational | 1/4Exact rational | -1/4Exact rational | 3/4Exact rational | 15/16Exact rational | 16/15Exact rational | -4/15Exact rational | 4/5Exact rational | -4/15Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
21 | 21/80Exact rational | 59/160Exact rational | 59/160Exact rational | 21/80Exact rational | -1/4Exact rational | 3/4Exact rational | 299/320Exact rational | 320/299Exact rational | -80/299Exact rational | 240/299Exact rational | -80/299Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
22 | 11/40Exact rational | 29/80Exact rational | 29/80Exact rational | 11/40Exact rational | -1/4Exact rational | 3/4Exact rational | 149/160Exact rational | 160/149Exact rational | -40/149Exact rational | 120/149Exact rational | -40/149Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
23 | 23/80Exact rational | 57/160Exact rational | 57/160Exact rational | 23/80Exact rational | -1/4Exact rational | 3/4Exact rational | 297/320Exact rational | 320/297Exact rational | -80/297Exact rational | 80/99Exact rational | -80/297Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
24 | 3/10Exact rational | 7/20Exact rational | 7/20Exact rational | 3/10Exact rational | -1/4Exact rational | 3/4Exact rational | 37/40Exact rational | 40/37Exact rational | -10/37Exact rational | 30/37Exact rational | -10/37Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
25 | 5/16Exact rational | 11/32Exact rational | 11/32Exact rational | 5/16Exact rational | -1/4Exact rational | 3/4Exact rational | 59/64Exact rational | 64/59Exact rational | -16/59Exact rational | 48/59Exact rational | -16/59Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
26 | 13/40Exact rational | 27/80Exact rational | 27/80Exact rational | 13/40Exact rational | -1/4Exact rational | 3/4Exact rational | 147/160Exact rational | 160/147Exact rational | -40/147Exact rational | 40/49Exact rational | -40/147Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | prolate |
27 | 1/3Exact rational | 1/3Exact rational | 1/3Exact rational | 1/3Exact rational | -1/4Exact rational | 3/4Exact rational | 11/12Exact rational | 12/11Exact rational | -3/11Exact rational | 9/11Exact rational | -3/11Exact rational | -1/4Exact rational | true | finite-axisymmetric-ellipsoid | sphere |
28 | 27/80Exact rational | 53/160Exact rational | 53/160Exact rational | 27/80Exact rational | -1/4Exact rational | 3/4Exact rational | 293/320Exact rational | 320/293Exact rational | -80/293Exact rational | 240/293Exact rational | -80/293Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
29 | 7/20Exact rational | 13/40Exact rational | 13/40Exact rational | 7/20Exact rational | -1/4Exact rational | 3/4Exact rational | 73/80Exact rational | 80/73Exact rational | -20/73Exact rational | 60/73Exact rational | -20/73Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
30 | 29/80Exact rational | 51/160Exact rational | 51/160Exact rational | 29/80Exact rational | -1/4Exact rational | 3/4Exact rational | 291/320Exact rational | 320/291Exact rational | -80/291Exact rational | 80/97Exact rational | -80/291Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
31 | 3/8Exact rational | 5/16Exact rational | 5/16Exact rational | 3/8Exact rational | -1/4Exact rational | 3/4Exact rational | 29/32Exact rational | 32/29Exact rational | -8/29Exact rational | 24/29Exact rational | -8/29Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
32 | 31/80Exact rational | 49/160Exact rational | 49/160Exact rational | 31/80Exact rational | -1/4Exact rational | 3/4Exact rational | 289/320Exact rational | 320/289Exact rational | -80/289Exact rational | 240/289Exact rational | -80/289Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
33 | 2/5Exact rational | 3/10Exact rational | 3/10Exact rational | 2/5Exact rational | -1/4Exact rational | 3/4Exact rational | 9/10Exact rational | 10/9Exact rational | -5/18Exact rational | 5/6Exact rational | -5/18Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
34 | 33/80Exact rational | 47/160Exact rational | 47/160Exact rational | 33/80Exact rational | -1/4Exact rational | 3/4Exact rational | 287/320Exact rational | 320/287Exact rational | -80/287Exact rational | 240/287Exact rational | -80/287Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
35 | 17/40Exact rational | 23/80Exact rational | 23/80Exact rational | 17/40Exact rational | -1/4Exact rational | 3/4Exact rational | 143/160Exact rational | 160/143Exact rational | -40/143Exact rational | 120/143Exact rational | -40/143Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
36 | 7/16Exact rational | 9/32Exact rational | 9/32Exact rational | 7/16Exact rational | -1/4Exact rational | 3/4Exact rational | 57/64Exact rational | 64/57Exact rational | -16/57Exact rational | 16/19Exact rational | -16/57Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
37 | 9/20Exact rational | 11/40Exact rational | 11/40Exact rational | 9/20Exact rational | -1/4Exact rational | 3/4Exact rational | 71/80Exact rational | 80/71Exact rational | -20/71Exact rational | 60/71Exact rational | -20/71Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
38 | 37/80Exact rational | 43/160Exact rational | 43/160Exact rational | 37/80Exact rational | -1/4Exact rational | 3/4Exact rational | 283/320Exact rational | 320/283Exact rational | -80/283Exact rational | 240/283Exact rational | -80/283Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
39 | 19/40Exact rational | 21/80Exact rational | 21/80Exact rational | 19/40Exact rational | -1/4Exact rational | 3/4Exact rational | 141/160Exact rational | 160/141Exact rational | -40/141Exact rational | 40/47Exact rational | -40/141Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
40 | 39/80Exact rational | 41/160Exact rational | 41/160Exact rational | 39/80Exact rational | -1/4Exact rational | 3/4Exact rational | 281/320Exact rational | 320/281Exact rational | -80/281Exact rational | 240/281Exact rational | -80/281Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
41 | 1/2Exact rational | 1/4Exact rational | 1/4Exact rational | 1/2Exact rational | -1/4Exact rational | 3/4Exact rational | 7/8Exact rational | 8/7Exact rational | -2/7Exact rational | 6/7Exact rational | -2/7Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
42 | 41/80Exact rational | 39/160Exact rational | 39/160Exact rational | 41/80Exact rational | -1/4Exact rational | 3/4Exact rational | 279/320Exact rational | 320/279Exact rational | -80/279Exact rational | 80/93Exact rational | -80/279Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
43 | 21/40Exact rational | 19/80Exact rational | 19/80Exact rational | 21/40Exact rational | -1/4Exact rational | 3/4Exact rational | 139/160Exact rational | 160/139Exact rational | -40/139Exact rational | 120/139Exact rational | -40/139Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
44 | 43/80Exact rational | 37/160Exact rational | 37/160Exact rational | 43/80Exact rational | -1/4Exact rational | 3/4Exact rational | 277/320Exact rational | 320/277Exact rational | -80/277Exact rational | 240/277Exact rational | -80/277Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
45 | 11/20Exact rational | 9/40Exact rational | 9/40Exact rational | 11/20Exact rational | -1/4Exact rational | 3/4Exact rational | 69/80Exact rational | 80/69Exact rational | -20/69Exact rational | 20/23Exact rational | -20/69Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
46 | 9/16Exact rational | 7/32Exact rational | 7/32Exact rational | 9/16Exact rational | -1/4Exact rational | 3/4Exact rational | 55/64Exact rational | 64/55Exact rational | -16/55Exact rational | 48/55Exact rational | -16/55Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
47 | 23/40Exact rational | 17/80Exact rational | 17/80Exact rational | 23/40Exact rational | -1/4Exact rational | 3/4Exact rational | 137/160Exact rational | 160/137Exact rational | -40/137Exact rational | 120/137Exact rational | -40/137Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
48 | 47/80Exact rational | 33/160Exact rational | 33/160Exact rational | 47/80Exact rational | -1/4Exact rational | 3/4Exact rational | 273/320Exact rational | 320/273Exact rational | -80/273Exact rational | 80/91Exact rational | -80/273Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
49 | 3/5Exact rational | 1/5Exact rational | 1/5Exact rational | 3/5Exact rational | -1/4Exact rational | 3/4Exact rational | 17/20Exact rational | 20/17Exact rational | -5/17Exact rational | 15/17Exact rational | -5/17Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
50 | 49/80Exact rational | 31/160Exact rational | 31/160Exact rational | 49/80Exact rational | -1/4Exact rational | 3/4Exact rational | 271/320Exact rational | 320/271Exact rational | -80/271Exact rational | 240/271Exact rational | -80/271Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
51 | 5/8Exact rational | 3/16Exact rational | 3/16Exact rational | 5/8Exact rational | -1/4Exact rational | 3/4Exact rational | 27/32Exact rational | 32/27Exact rational | -8/27Exact rational | 8/9Exact rational | -8/27Exact rational | -1/4Exact rational | true | finite-axisymmetric-ellipsoid | oblate |
52 | 51/80Exact rational | 29/160Exact rational | 29/160Exact rational | 51/80Exact rational | -1/4Exact rational | 3/4Exact rational | 269/320Exact rational | 320/269Exact rational | -80/269Exact rational | 240/269Exact rational | -80/269Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
53 | 13/20Exact rational | 7/40Exact rational | 7/40Exact rational | 13/20Exact rational | -1/4Exact rational | 3/4Exact rational | 67/80Exact rational | 80/67Exact rational | -20/67Exact rational | 60/67Exact rational | -20/67Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
54 | 53/80Exact rational | 27/160Exact rational | 27/160Exact rational | 53/80Exact rational | -1/4Exact rational | 3/4Exact rational | 267/320Exact rational | 320/267Exact rational | -80/267Exact rational | 80/89Exact rational | -80/267Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
55 | 27/40Exact rational | 13/80Exact rational | 13/80Exact rational | 27/40Exact rational | -1/4Exact rational | 3/4Exact rational | 133/160Exact rational | 160/133Exact rational | -40/133Exact rational | 120/133Exact rational | -40/133Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
56 | 11/16Exact rational | 5/32Exact rational | 5/32Exact rational | 11/16Exact rational | -1/4Exact rational | 3/4Exact rational | 53/64Exact rational | 64/53Exact rational | -16/53Exact rational | 48/53Exact rational | -16/53Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
57 | 7/10Exact rational | 3/20Exact rational | 3/20Exact rational | 7/10Exact rational | -1/4Exact rational | 3/4Exact rational | 33/40Exact rational | 40/33Exact rational | -10/33Exact rational | 10/11Exact rational | -10/33Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
58 | 57/80Exact rational | 23/160Exact rational | 23/160Exact rational | 57/80Exact rational | -1/4Exact rational | 3/4Exact rational | 263/320Exact rational | 320/263Exact rational | -80/263Exact rational | 240/263Exact rational | -80/263Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
59 | 29/40Exact rational | 11/80Exact rational | 11/80Exact rational | 29/40Exact rational | -1/4Exact rational | 3/4Exact rational | 131/160Exact rational | 160/131Exact rational | -40/131Exact rational | 120/131Exact rational | -40/131Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
60 | 59/80Exact rational | 21/160Exact rational | 21/160Exact rational | 59/80Exact rational | -1/4Exact rational | 3/4Exact rational | 261/320Exact rational | 320/261Exact rational | -80/261Exact rational | 80/87Exact rational | -80/261Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
61 | 3/4Exact rational | 1/8Exact rational | 1/8Exact rational | 3/4Exact rational | -1/4Exact rational | 3/4Exact rational | 13/16Exact rational | 16/13Exact rational | -4/13Exact rational | 12/13Exact rational | -4/13Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
62 | 61/80Exact rational | 19/160Exact rational | 19/160Exact rational | 61/80Exact rational | -1/4Exact rational | 3/4Exact rational | 259/320Exact rational | 320/259Exact rational | -80/259Exact rational | 240/259Exact rational | -80/259Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
63 | 31/40Exact rational | 9/80Exact rational | 9/80Exact rational | 31/40Exact rational | -1/4Exact rational | 3/4Exact rational | 129/160Exact rational | 160/129Exact rational | -40/129Exact rational | 40/43Exact rational | -40/129Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
64 | 63/80Exact rational | 17/160Exact rational | 17/160Exact rational | 63/80Exact rational | -1/4Exact rational | 3/4Exact rational | 257/320Exact rational | 320/257Exact rational | -80/257Exact rational | 240/257Exact rational | -80/257Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
65 | 4/5Exact rational | 1/10Exact rational | 1/10Exact rational | 4/5Exact rational | -1/4Exact rational | 3/4Exact rational | 4/5Exact rational | 5/4Exact rational | -5/16Exact rational | 15/16Exact rational | -5/16Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
66 | 13/16Exact rational | 3/32Exact rational | 3/32Exact rational | 13/16Exact rational | -1/4Exact rational | 3/4Exact rational | 51/64Exact rational | 64/51Exact rational | -16/51Exact rational | 16/17Exact rational | -16/51Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
67 | 33/40Exact rational | 7/80Exact rational | 7/80Exact rational | 33/40Exact rational | -1/4Exact rational | 3/4Exact rational | 127/160Exact rational | 160/127Exact rational | -40/127Exact rational | 120/127Exact rational | -40/127Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
68 | 67/80Exact rational | 13/160Exact rational | 13/160Exact rational | 67/80Exact rational | -1/4Exact rational | 3/4Exact rational | 253/320Exact rational | 320/253Exact rational | -80/253Exact rational | 240/253Exact rational | -80/253Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
69 | 17/20Exact rational | 3/40Exact rational | 3/40Exact rational | 17/20Exact rational | -1/4Exact rational | 3/4Exact rational | 63/80Exact rational | 80/63Exact rational | -20/63Exact rational | 20/21Exact rational | -20/63Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
70 | 69/80Exact rational | 11/160Exact rational | 11/160Exact rational | 69/80Exact rational | -1/4Exact rational | 3/4Exact rational | 251/320Exact rational | 320/251Exact rational | -80/251Exact rational | 240/251Exact rational | -80/251Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
71 | 7/8Exact rational | 1/16Exact rational | 1/16Exact rational | 7/8Exact rational | -1/4Exact rational | 3/4Exact rational | 25/32Exact rational | 32/25Exact rational | -8/25Exact rational | 24/25Exact rational | -8/25Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
72 | 71/80Exact rational | 9/160Exact rational | 9/160Exact rational | 71/80Exact rational | -1/4Exact rational | 3/4Exact rational | 249/320Exact rational | 320/249Exact rational | -80/249Exact rational | 80/83Exact rational | -80/249Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
73 | 9/10Exact rational | 1/20Exact rational | 1/20Exact rational | 9/10Exact rational | -1/4Exact rational | 3/4Exact rational | 31/40Exact rational | 40/31Exact rational | -10/31Exact rational | 30/31Exact rational | -10/31Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
74 | 73/80Exact rational | 7/160Exact rational | 7/160Exact rational | 73/80Exact rational | -1/4Exact rational | 3/4Exact rational | 247/320Exact rational | 320/247Exact rational | -80/247Exact rational | 240/247Exact rational | -80/247Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
75 | 37/40Exact rational | 3/80Exact rational | 3/80Exact rational | 37/40Exact rational | -1/4Exact rational | 3/4Exact rational | 123/160Exact rational | 160/123Exact rational | -40/123Exact rational | 40/41Exact rational | -40/123Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
76 | 15/16Exact rational | 1/32Exact rational | 1/32Exact rational | 15/16Exact rational | -1/4Exact rational | 3/4Exact rational | 49/64Exact rational | 64/49Exact rational | -16/49Exact rational | 48/49Exact rational | -16/49Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
77 | 19/20Exact rational | 1/40Exact rational | 1/40Exact rational | 19/20Exact rational | -1/4Exact rational | 3/4Exact rational | 61/80Exact rational | 80/61Exact rational | -20/61Exact rational | 60/61Exact rational | -20/61Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
78 | 77/80Exact rational | 3/160Exact rational | 3/160Exact rational | 77/80Exact rational | -1/4Exact rational | 3/4Exact rational | 243/320Exact rational | 320/243Exact rational | -80/243Exact rational | 80/81Exact rational | -80/243Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
79 | 39/40Exact rational | 1/80Exact rational | 1/80Exact rational | 39/40Exact rational | -1/4Exact rational | 3/4Exact rational | 121/160Exact rational | 160/121Exact rational | -40/121Exact rational | 120/121Exact rational | -40/121Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
80 | 79/80Exact rational | 1/160Exact rational | 1/160Exact rational | 79/80Exact rational | -1/4Exact rational | 3/4Exact rational | 241/320Exact rational | 320/241Exact rational | -80/241Exact rational | 240/241Exact rational | -80/241Exact rational | -1/4Exact rational | false | finite-axisymmetric-ellipsoid | oblate |
81N=0 or1 is a limiting aspect ratio, not an ordinary finite ellipsoid. | 1Exact rational | 0Exact rational | 0Exact rational | 1Exact rational | -1/4Exact rational | 3/4Exact rational | 3/4Exact rational | 4/3Exact rational | -1/3Exact rational | 1Exact rational | -1/3Exact rational | -1/4Exact rational | false | limiting-aspect-ratio | oblate |
Complete ellipsoid definitions and intrinsic controls (19 rows)
Uniform applied H₀ẑ and uniformly magnetized axisymmetric ellipsoid; SI Nₓ+Nᵧ+N_z=1. χ=7/3 or−1/4, χ>−1 and positive denominator. H/H₀,M/H₀ and B/(μ₀H₀). No arbitrary-body uniform-field or measured-material claim.
| Quantity or source condition | Retained admitted value |
|---|---|
definitions · Assumptions and regime | Uniform applied H along a principal axis of a uniformly magnetized ellipsoid; local isotropic linear reversible response. Illustrative positive-permeability constitutive models, not named measured materials or ferromagnetic hysteresis. |
definitions · applied H Over H0 · 1 | 0 |
definitions · applied H Over H0 · 2 | 0 |
definitions · applied H Over H0 · 3 | 1 |
definitions · Intrinsic χ · 1 | 7/3 |
definitions · Intrinsic χ · 2 | -1/4 |
definitions · principal N | Nz=N,Nx=Ny=(1-N)/2; SI sum=1 |
definitions · actual Finite Controls · 1 | 1/8 |
definitions · actual Finite Controls · 2 | 1/3 |
definitions · actual Finite Controls · 3 | 5/8 |
definitions · curve Grid | N=i/80,i=0..80; endpoints0 and1 marked limiting aspect ratios, not finite ellipsoids |
definitions · definitions · Internal H / H₀ | 1/(1+N chi) |
definitions · definitions · M/H₀ = apparent χ | chi/(1+N chi) |
definitions · definitions · B / (μ₀H₀) | (1+chi)/(1+N chi) |
definitions · definitions · Applied-field susceptibility | M/H_app |
definitions · definitions · inverse Chi | chi=chi_app/(1-N chi_app) on its valid branch |
definitions · Validity limits | All chosen chi>-1; require positive denominator. No arbitrary-body uniform-field claim. NIST cgs factors must be divided by4pi before SI interpretation. |
families · 1 · Intrinsic χ | 7/3Exact rational |
families · 2 · Intrinsic χ | -1/4Exact rational |
Focus a named plot or table region to scroll horizontally with the keyboard, or swipe within that region. Readable fixed-size diagrams remain locally contained. Full disclosures retain exact values, all records and source/domain conditions.
| χ_m | N along applied field | H_int/H₀ | M/H₀ = χ_app | B/(μ₀H₀) | Shape class |
|---|---|---|---|---|---|
| 7/3 | 1/8 | 24/31 | 56/31 | 80/31 | prolate |
| 7/3 | 1/3 | 9/16 | 21/16 | 15/8 | sphere |
| 7/3 | 5/8 | 24/59 | 56/59 | 80/59 | oblate |
| -1/4 | 1/8 | 32/31 | -8/31 | 24/31 | prolate |
| -1/4 | 1/3 | 12/11 | -3/11 | 9/11 | sphere |
| -1/4 | 5/8 | 32/27 | -8/27 | 8/9 | oblate |
Know when scalar permeability stops being enough
A positive scalar χ_m does not by itself identify ferromagnetic order. A reversible positive-response model can have no remanence or hysteresis. Conversely, a magnetically ordered specimen can have a small net moment because its domains or sublattices compensate. A constitutive slope and an ordering mechanism are different pieces of information.
An anisotropic crystal can require M_i = Σ_j χ_ij H_j even in a linear regime. Strong fields can produce saturation or other nonlinear response. Secant M/H, the differential slope dM/dH and a susceptibility measured against applied field need not agree. State field direction, temperature, history and frequency before comparing quoted response values.
At changing fields, finite conductivity can produce eddy currents, and relaxation can make the response delayed and frequency dependent. A complex permeability then describes a dynamical response rather than the static real constant used above. These limitations motivate induction and material-transport models; they do not erase the usefulness of a well-specified low-field magnetostatic approximation.
Count two quantum states
Consider localized independent spin-half moments whose projections along the chosen local field are +μ_s and −μ_s. Here μ_s is a positive moment magnitude; it is not the vacuum permeability μ₀. Let n be their number density, T > 0 the temperature, and u = μ_s B_loc/(k_BT). This is a two-state equilibrium model, not a classical spinning-sphere picture of electron spin.
The two energies are −μ_sB_loc and +μ_sB_loc. Their Boltzmann weights give Z₁ = 2cosh u, p_+ = exp(u)/(2cosh u), p_− = exp(−u)/(2cosh u), and M/(nμ_s) = p_+ − p_− = tanh u. Probabilities sum to one, the response reverses with the field, and its magnitude remains below the saturation value for finite u.
The single-spin free energy is f = −k_BT ln(2cosh u), so −∂f/∂B_loc is its mean projected moment. Differentiating again gives ∂M/∂B_loc = nμ_s² sech²u/(k_BT). This derivative has units A/(m T), whereas a susceptibility defined by M/H is dimensionless. Confusing those definitions introduces both unit and μ₀ errors.
The original table rounds selected numerical model values to four decimal places. The full model retains higher precision for reproducible calculations, not as a claim to measured material accuracy. At u = 0, the two states remain populated even though their mean magnetization cancels.
Independent quantum states, not an ordered-material curve
Independent two-state moments have thermal populations, saturation and fluctuations. These curves do not describe exchange order, domains or a general measured material.
Count the two states and their mean moment
- p₊: positive-projection population
- p₋: negative-projection population
- M/(nμ_s): population difference
Exact worked u:
ln(3)p₊: 9/10p₋: 1/10M/(nμ_s): 4/5Variance: 9/25Fluctuation, saturation and a limited straight line
- Normalized variance; response factor sech²u
- Stored weak-field straight line u; valid only |u|≪1
- Actual M/(nμ_s); bounded saturation
- Localized noninteracting two-state moments ±μ_s at fixed T>0 in prescribed local B. u=μ_sB_loc/(k_BT); M/(nμ_s), probabilities and variance dimensionless. Free energy per spin/(k_BT). dM/dB has A/(m T), not dimensionless M/H; extra local-field assumptions are required.
- Positive/negative projections have energies−μ_sB_loc and+μ_sB_loc. At u=0 each probability is exactly1/2 while the mean is0 and normalized variance1. The symbolic u=ln3 worked case is retained exactly in its data disclosure rather than re-evaluated for plotting.
- The weak-field straight line equals the stored u values and is valid only for |u|≪1. It is intentionally shown outside the saturation range, rather than clipped or promoted to the exact response.
- Variance multiplies nμ_s²/(k_BT) in ∂M/∂B_loc, which has A/(m T) units. Dimensionless M/H requires the stated extra local/internal-field relation; no unconditional μ₀ conversion or general Curie/Pauli equivalence is implied.
- All moments are independent at fixed positive T. No exchange-driven spontaneous order, domains, hysteresis, demagnetization solver or zero-temperature B/T value is supplied.
- Tables preserve every original typed value: reduced rational strings, explicit approximate 70-significant-digit decimals and symbolic expressions. Plot coordinates only convert these stored values for display; no scientific solver, new case, re-rounding or measured dataset is introduced.
Independent spin-half model: selected equilibrium populations and response
All 241 independent spin-half equilibrium and weak-field records (241 rows)
Localized noninteracting two-state moments ±μ_s at fixed T>0 in prescribed local B. u=μ_sB_loc/(k_BT); M/(nμ_s), probabilities and variance dimensionless. Free energy per spin/(k_BT). dM/dB has A/(m T), not dimensionless M/H; extra local-field assumptions are required.
| Row | u=μ_s B_loc/(k_BT) | Positive-projection probability p₊ | Negative-projection probability p₋ | M / (n μ_s) | Normalized moment variance | Free energy per spin / (k_BT) | Weak-field linear M / (n μ_s) | State energy / (k_BT) · 1 | State energy / (k_BT) · 2 | Original finite control? | Weak-field approximation regime-limited? |
|---|---|---|---|---|---|---|---|---|---|---|---|
0 | -3Exact rational | 0.002472623156634774334059907372255762451093072649858754088060592434667649Approximate: 70 significant digits; calculation provenance | 0.9975273768433652256659400926277442375489069273501412459119394075653324Approximate: 70 significant digits; calculation provenance | -0.9950547536867304513318801852554884750978138547002824918238788151306647Approximate: 70 significant digits; calculation provenance | 0.009866037165440191273156169683523487327321609118799727204249490046423324Approximate: 70 significant digits; calculation provenance | -3.002475685137730449530859682043522968776733924661974105498096348803967Approximate: 70 significant digits; calculation provenance | -3Exact rational | 3Exact rational | -3Exact rational | false | true |
1 | -119/40Exact rational | 0.002599067762323347165063588694139463043460681486686029365342954690886876Approximate: 70 significant digits; calculation provenance | 0.9974009322376766528349364113058605369565393185133139706346570453091131Approximate: 70 significant digits; calculation provenance | -0.9948018644753533056698728226117210739130786370266279412693140906182262Approximate: 70 significant digits; calculation provenance | 0.01036925043636079469613672963456150908132490889540545861834989267236642Approximate: 70 significant digits; calculation provenance | -2.977602451202738716159791544151885411176318200598736263610052074097091Approximate: 70 significant digits; calculation provenance | -119/40Exact rational | 119/40Exact rational | -119/40Exact rational | false | true |
2 | -59/20Exact rational | 0.002731960763011060103502845792818668056722569027062419410275638313988061Approximate: 70 significant digits; calculation provenance | 0.9972680392369889398964971542071813319432774309729375805897243616860119Approximate: 70 significant digits; calculation provenance | -0.9945360784739778797929943084143626638865548619458751611794487233720239Approximate: 70 significant digits; calculation provenance | 0.01089798861360171251918494673345255242580407102798406569025223779295274Approximate: 70 significant digits; calculation provenance | -2.952735699378536118575704789814597983757527754003024838293187314583929Approximate: 70 significant digits; calculation provenance | -59/20Exact rational | 59/20Exact rational | -59/20Exact rational | false | true |
3 | -117/40Exact rational | 0.002871629155700398773125206248555764679244421121101659074514538061869767Approximate: 70 significant digits; calculation provenance | 0.9971283708442996012268747937514442353207555788788983409254854619381302Approximate: 70 significant digits; calculation provenance | -0.9942567416885992024537495875028884706415111577577966818509709238762605Approximate: 70 significant digits; calculation provenance | 0.01145353160677012075210260217862755025646883862866094393805835459926523Approximate: 70 significant digits; calculation provenance | -2.927875760193138135709121651006142711642355275414223984028350968609581Approximate: 70 significant digits; calculation provenance | -117/40Exact rational | 117/40Exact rational | -117/40Exact rational | false | true |
4 | -29/10Exact rational | 0.003018416324708423933200602495310494682854987466409130795553131767600816Approximate: 70 significant digits; calculation provenance | 0.9969815836752915760667993975046895053171450125335908692044468682323992Approximate: 70 significant digits; calculation provenance | -0.9939631673505831521335987950093790106342900250671817384088937364647984Approximate: 70 significant digits; calculation provenance | 0.01203722195039663049398212265025761043980025152584939647448307835804373Approximate: 70 significant digits; calculation provenance | -2.903022980930831619273345223197070524139264109464995710695004586745200Approximate: 70 significant digits; calculation provenance | -29/10Exact rational | 29/10Exact rational | -29/10Exact rational | false | true |
5 | -23/8Exact rational | 0.003172682842485189501963168586501728567181065348639959013004131707463052Approximate: 70 significant digits; calculation provenance | 0.9968273171575148104980368314134982714328189346513600409869958682925369Approximate: 70 significant digits; calculation provenance | -0.9936546343150296209960736628269965428656378693027200819739917365850739Approximate: 70 significant digits; calculation provenance | 0.01265046770426475840073175614827287378220214854479127092432028735160467Approximate: 70 significant digits; calculation provenance | -2.878177726471409926664346345528242255156173268151804689970618026722808Approximate: 70 significant digits; calculation provenance | -23/8Exact rational | 23/8Exact rational | -23/8Exact rational | false | true |
6 | -57/20Exact rational | 0.003334807307413344929713838000704687362694300148557676829327262682326375Approximate: 70 significant digits; calculation provenance | 0.9966651926925866550702861619992953126373056998514423231706727373176736Approximate: 70 significant digits; calculation provenance | -0.9933303853851733101405723239985906252746113997028846463413454746353472Approximate: 70 significant digits; calculation provenance | 0.01329474547054306994432329971962495479953930399714121586480621332260334Approximate: 70 significant digits; calculation provenance | -2.853340380170367420773267272297273702481679805583579137435795629181163Approximate: 70 significant digits; calculation provenance | -57/20Exact rational | 57/20Exact rational | -57/20Exact rational | false | true |
7 | -113/40Exact rational | 0.003505187220066338795280907804964733894452067342759305062095229360192615Approximate: 70 significant digits; calculation provenance | 0.9964948127799336612047190921950352661055479326572406949379047706398074Approximate: 70 significant digits; calculation provenance | -0.9929896255598673224094381843900705322110958653144813898758095412796148Approximate: 70 significant digits; calculation provenance | 0.01397160353047448962834434294173221318664538036772587676469519534146454Approximate: 70 significant digits; calculation provenance | -2.828511344781939187856797524113716463701628833044059255609038372713170Approximate: 70 significant digits; calculation provenance | -113/40Exact rational | 113/40Exact rational | -113/40Exact rational | false | true |
8 | -14/5Exact rational | 0.003684239899435987759693183711989038229525405371753872443305992277365704Approximate: 70 significant digits; calculation provenance | 0.9963157601005640122403068162880109617704745946282461275566940077226343Approximate: 70 significant digits; calculation provenance | -0.9926315202011280244806136325760219235409491892564922551133880154452686Approximate: 70 significant digits; calculation provenance | 0.01468266510319756664997007649520936827850354570826105693693409332118851Approximate: 70 significant digits; calculation provenance | -2.803691043426946538491740763648030278942176489730812671802569774194763Approximate: 70 significant digits; calculation provenance | -14/5Exact rational | 14/5Exact rational | -14/5Exact rational | false | true |
9 | -111/40Exact rational | 0.003872403440671031115485027028596470715964624897769503557278890660564056Approximate: 70 significant digits; calculation provenance | 0.9961275965593289688845149729714035292840353751022304964427211093394359Approximate: 70 significant digits; calculation provenance | -0.9922551931186579377690299459428070585680707502044609928854422186788719Approximate: 70 significant digits; calculation provenance | 0.01542963172905484110193869693588184693255623936218789082704486434591069Approximate: 70 significant digits; calculation provenance | -2.778879920607485177684989946277863003800013489713684954959085612835163Approximate: 70 significant digits; calculation provenance | -111/40Exact rational | 111/40Exact rational | -111/40Exact rational | false | true |
10 | -11/4Exact rational | 0.004070137715896127325164605428574834567247756263379406613233961378703273Approximate: 70 significant digits; calculation provenance | 0.9959298622841038726748353945714251654327522437366205933867660386212967Approximate: 70 significant digits; calculation provenance | -0.9918597245682077453496707891428503308655044874732411867735320772425935Approximate: 70 significant digits; calculation provenance | 0.01621428677947906872276645330446422220413547634109097514793157050260235Approximate: 70 significant digits; calculation provenance | -2.754078443270570720891516424678363665743836656757163253421493949418678Approximate: 70 significant digits; calculation provenance | -11/4Exact rational | 11/4Exact rational | -11/4Exact rational | false | true |
11 | -109/40Exact rational | 0.004277925419704973417886566947082113783655697447862897806790907858803211Approximate: 70 significant digits; calculation provenance | 0.9957220745802950265821134330529178862163443025521371021932090921411968Approximate: 70 significant digits; calculation provenance | -0.9914441491605900531642268661058357724326886051042742043864181842823936Approximate: 70 significant digits; calculation provenance | 0.01703849909523366177966750980623620604254016810132469584070768036734183Approximate: 70 significant digits; calculation provenance | -2.729287101922935327494560735418711758436793382962302988937662510208796Approximate: 70 significant digits; calculation provenance | -109/40Exact rational | 109/40Exact rational | -109/40Exact rational | false | true |
12 | -27/10Exact rational | 0.004496273160941180225656375374984618182434912457393560807611467131503573Approximate: 70 significant digits; calculation provenance | 0.9955037268390588197743436246250153818175650875426064391923885328684964Approximate: 70 significant digits; calculation provenance | -0.9910074536781176395486872492500307636351301750852128783847770657369929Approximate: 70 significant digits; calculation provenance | 0.01790422675441352093312322618721838019388078601390980649415127660422605Approximate: 70 significant digits; calculation provenance | -2.704506411799249425423352946213323472595839270586710745037151428061255Approximate: 70 significant digits; calculation provenance | -27/10Exact rational | 27/10Exact rational | -27/10Exact rational | false | true |
13 | -107/40Exact rational | 0.004725712602395480080323934976488241922491961275454811366019476151592069Approximate: 70 significant digits; calculation provenance | 0.9952742873976045199196760650235117580775080387245451886339805238484079Approximate: 70 significant digits; calculation provenance | -0.9905485747952090398393521300470235161550160774490903772679610476968159Approximate: 70 significant digits; calculation provenance | 0.01881352097118016247808369995389228311726053467292184155270135531608162Approximate: 70 significant digits; calculation provenance | -2.679736914086123578331872455234968834681888715646503936984968473222179Approximate: 70 significant digits; calculation provenance | -107/40Exact rational | 107/40Exact rational | -107/40Exact rational | false | true |
14 | -53/20Exact rational | 0.004966801650056960281408071062605449086954852909134669124956416822244517Approximate: 70 significant digits; calculation provenance | 0.9950331983499430397185919289373945509130451470908653308750435831777555Approximate: 70 significant digits; calculation provenance | -0.9900663966998860794371838578747891018260902941817306617500871663555110Approximate: 70 significant digits; calculation provenance | 0.01976853012570380695227481469911063601864296689355636060612855171344093Approximate: 70 significant digits; calculation provenance | -2.654979177204327229610104627794477411895141749525834275046289850648567Approximate: 70 significant digits; calculation provenance | -53/20Exact rational | 53/20Exact rational | -53/20Exact rational | false | true |
15 | -21/8Exact rational | 0.005220125693558397103191452519744471871652514498745877026301612922525574Approximate: 70 significant digits; calculation provenance | 0.9947798743064416028968085474802555281283474855012541229736983870774744Approximate: 70 significant digits; calculation provenance | -0.9895597486128832057936170949605110562566949710025082459473967741549489Approximate: 70 significant digits; calculation provenance | 0.02077150392520739426724643106090801540461937975714340910874636913789263Approximate: 70 significant digits; calculation provenance | -2.630233798151743031782728942771398758285003162072799827312006926724049Approximate: 70 significant digits; calculation provenance | -21/8Exact rational | 21/8Exact rational | -21/8Exact rational | false | true |
16 | -13/5Exact rational | 0.005486298899450405329518781620919341250348587593622942293760553479030875Approximate: 70 significant digits; calculation provenance | 0.9945137011005495946704812183790806587496514124063770577062394465209691Approximate: 70 significant digits; calculation provenance | -0.9890274022010991893409624367581613174993028248127541154124788930419383Approximate: 70 significant digits; calculation provenance | 0.02182479769534517840316277403338055957421933728773555330119580288611362Approximate: 70 significant digits; calculation provenance | -2.605501403909657374081844759492338301303046114327419802138325257519104Approximate: 70 significant digits; calculation provenance | -13/5Exact rational | 13/5Exact rational | -13/5Exact rational | false | true |
17 | -103/40Exact rational | 0.005765965558924905117441257867539929188560904787342372209553606709049077Approximate: 70 significant digits; calculation provenance | 0.9942340344410750948825587421324600708114390952126576277904463932909509Approximate: 70 significant digits; calculation provenance | -0.9884680688821501897651174842649201416228781904253152555808927865819018Approximate: 70 significant digits; calculation provenance | 0.02293087680039278769589293576863518658308199828896721672569453564310053Approximate: 70 significant digits; calculation provenance | -2.580782652915069140754634405824594435197212717132404173580432300102199Approximate: 70 significant digits; calculation provenance | -103/40Exact rational | 103/40Exact rational | -103/40Exact rational | false | true |
18 | -51/20Exact rational | 0.006059801491584114130450046838696289607882563578385341398438141571410681Approximate: 70 significant digits; calculation provenance | 0.9939401985084158858695499531613037103921174364216146586015618584285893Approximate: 70 significant digits; calculation provenance | -0.9878803970168317717390999063226074207842348728432293172031237168571786Approximate: 70 significant digits; calculation provenance | 0.02409232118986683630404589969884237987609217296360337828171928682096901Approximate: 70 significant digits; calculation provenance | -2.556078236601779195130616868369368184630344962862353709422606608491073Approximate: 70 significant digits; calculation provenance | -51/20Exact rational | 51/20Exact rational | -51/20Exact rational | false | true |
19 | -101/40Exact rational | 0.006368515506815546552812922475580091515572544031748327462097365071381622Approximate: 70 significant digits; calculation provenance | 0.9936314844931844534471870775244199084844274559682516725379026349286184Approximate: 70 significant digits; calculation provenance | -0.9872629689863689068943741550488398169688549119365033450758052698572368Approximate: 70 significant digits; calculation provenance | 0.02531183006821998590016539787997788414520092927543268381186044260271151Approximate: 70 significant digits; calculation provenance | -2.531388881013102052336979588252781932051164896704553908555161932307090Approximate: 70 significant digits; calculation provenance | -101/40Exact rational | 101/40Exact rational | -101/40Exact rational | false | true |
20 | -5/2Exact rational | 0.006692850924284855559361980381325180393744008907296530269212243059884084Approximate: 70 significant digits; calculation provenance | 0.9933071490757151444406380196186748196062559910927034697307877569401159Approximate: 70 significant digits; calculation provenance | -0.9866142981514302888812760392373496392125119821854069394615755138802318Approximate: 70 significant digits; calculation provenance | 0.02659222668316061965599413801657694702051866597994826696310613928330415Approximate: 70 significant digits; calculation provenance | -2.506715348489118068616416687732642075114892172964320326936005156413407Approximate: 70 significant digits; calculation provenance | -5/2Exact rational | 5/2Exact rational | -5/2Exact rational | false | true |
21 | -99/40Exact rational | 0.007033587154995161020830207523487590518009114958675275811960448561712974Approximate: 70 significant digits; calculation provenance | 0.9929664128450048389791697924765124094819908850413247241880395514382870Approximate: 70 significant digits; calculation provenance | -0.9859328256900096779583395849530248189639817700826494483760791028765741Approximate: 70 significant digits; calculation provenance | 0.02793646322691299239027468571970199102953956924724894613503541038794434Approximate: 70 significant digits; calculation provenance | -2.482058439431458551383859151335636693997314795269051676369912207931352Approximate: 70 significant digits; calculation provenance | -99/40Exact rational | 99/40Exact rational | -99/40Exact rational | false | true |
22 | -49/20Exact rational | 0.007391541344281973035367942228144277582005679327059314712172214256881033Approximate: 70 significant digits; calculation provenance | 0.9926084586557180269646320577718557224179943206729406852878277857431190Approximate: 70 significant digits; calculation provenance | -0.9852169173114360539292641155437114448359886413458813705756555714862379Approximate: 70 significant digits; calculation provenance | 0.02934762584335097311333692832355415661967272111385268674285365634998205Approximate: 70 significant digits; calculation provenance | -2.457418994148686709973886823389828700691971382230745445191656595492077Approximate: 70 significant digits; calculation provenance | -49/20Exact rational | 49/20Exact rational | -49/20Exact rational | false | true |
23 | -97/40Exact rational | 0.007767570078015000930718153150821357484018738743892133282617097897185118Approximate: 70 significant digits; calculation provenance | 0.9922324299219849990692818468491786425159812612561078667173829021028149Approximate: 70 significant digits; calculation provenance | -0.9844648598439699981385636936983572850319625225122157334347658042056298Approximate: 70 significant digits; calculation provenance | 0.03082893973239250785229190388344780911262209144870056590903646081336027Approximate: 70 significant digits; calculation provenance | -2.432797894785403457543945579071933055164915618662570648625045945593596Approximate: 70 significant digits; calculation provenance | -97/40Exact rational | 97/40Exact rational | -97/40Exact rational | false | true |
24 | -12/5Exact rational | 0.008162571153159895180135425033062508935303348511577403817452290851449158Approximate: 70 significant digits; calculation provenance | 0.9918374288468401048198645749669374910646966514884225961825477091485508Approximate: 70 significant digits; calculation provenance | -0.9836748576936802096397291499338749821293933029768451923650954182971017Approximate: 70 significant digits; calculation provenance | 0.03238377434131798847662658059430935272768412155309234222172150138303010Approximate: 70 significant digits; calculation provenance | -2.408196067338267772800755888240396519764878990641158907654638223011920Approximate: 70 significant digits; calculation provenance | -12/5Exact rational | 12/5Exact rational | -12/5Exact rational | false | true |
25 | -19/8Exact rational | 0.008577485413711984988230991899205671765020003950640162122671614160886069Approximate: 70 significant digits; calculation provenance | 0.9914225145862880150117690081007943282349799960493598378773283858391139Approximate: 70 significant digits; calculation provenance | -0.9828450291725760300235380162015886564699599920987196757546567716782279Approximate: 70 significant digits; calculation provenance | 0.03401564863075817250383932942880915939057652961384290936617588721635706Approximate: 70 significant digits; calculation provenance | -2.383614483762175557829804739019990758022824225207226103099087749411652Approximate: 70 significant digits; calculation provenance | -19/8Exact rational | 19/8Exact rational | -19/8Exact rational | false | true |
26 | -47/20Exact rational | 0.009013298652847824518019228165068154408773139053596212013670366647865082Approximate: 70 significant digits; calculation provenance | 0.9909867013471521754819807718349318455912268609464037879863296333521349Approximate: 70 significant digits; calculation provenance | -0.9819734026943043509639615436698636911824537218928075759726592667042698Approximate: 70 significant digits; calculation provenance | 0.03572823640096958443897487536317688565195252494626068086123104102527095Approximate: 70 significant digits; calculation provenance | -2.359054164169887486004273513617968604531616212752084297942504324530114Approximate: 70 significant digits; calculation provenance | -47/20Exact rational | 47/20Exact rational | -47/20Exact rational | false | true |
27 | -93/40Exact rational | 0.009471043581946110302204712221834367286282574614997084776990920226188461Approximate: 70 significant digits; calculation provenance | 0.9905289564180538896977952877781656327137174253850029152230090797738115Approximate: 70 significant digits; calculation provenance | -0.9810579128361077793955905755563312654274348507700058304460181595476231Approximate: 70 significant digits; calculation provenance | 0.03752537166165995077933435710982926451892240465651946234925713826332787Approximate: 70 significant digits; calculation provenance | -2.334516179128433890997617766135538998466585480699536860001770059584107Approximate: 70 significant digits; calculation provenance | -93/40Exact rational | 93/40Exact rational | -93/40Exact rational | false | true |
28 | -23/10Exact rational | 0.009951801866904321572949038449437181216616909165435646812116277995735790Approximate: 70 significant digits; calculation provenance | 0.9900481981330956784270509615505628187833830908345643531878837220042642Approximate: 70 significant digits; calculation provenance | -0.9800963962661913568541019231011256375667661816691287063757674440085284Approximate: 70 significant digits; calculation provenance | 0.03941105402602480493103177624000740328073769971724618794406231752561093Approximate: 70 significant digits; calculation provenance | -2.310001652055651831254680117182091813836876559564107587209782541401929Approximate: 70 significant digits; calculation provenance | -23/10Exact rational | 23/10Exact rational | -23/10Exact rational | false | true |
29 | -91/40Exact rational | 0.01045670623191806939118256616035351572703227946399865365770971845952493Approximate: 70 significant digits; calculation provenance | 0.9895432937680819306088174338396464842729677205360013463422902815404751Approximate: 70 significant digits; calculation provenance | -0.9790865875361638612176348676792929685459354410720026926845805630809501Approximate: 70 significant digits; calculation provenance | 0.04138945410678974080789674030756565077285084139568256165698378917621376Approximate: 70 significant digits; calculation provenance | -2.285511761720224447004342689608316839619116789362933632028241714537530Approximate: 70 significant digits; calculation provenance | -91/40Exact rational | 91/40Exact rational | -91/40Exact rational | false | true |
30 | -9/4Exact rational | 0.01098694263059318003863537849690841706762338441435871321258568924410270Approximate: 70 significant digits; calculation provenance | 0.9890130573694068199613646215030915829323766155856412867874143107558973Approximate: 70 significant digits; calculation provenance | -0.9780261147388136399227292430061831658647532311712825735748286215117946Approximate: 70 significant digits; calculation provenance | 0.04346491888890093700651012184112977910849967105229201430856291891240252Approximate: 70 significant digits; calculation provenance | -2.261047744848593816321644218716262933425798414904328961905696305324554Approximate: 70 significant digits; calculation provenance | -9/4Exact rational | 9/4Exact rational | -9/4Exact rational | false | true |
31 | -89/40Exact rational | 0.01154375248392228998546310801575071651115780734625949080743146844883387Approximate: 70 significant digits; calculation provenance | 0.9884562475160777100145368919842492834888421926537405091925685315511661Approximate: 70 significant digits; calculation provenance | -0.9769124950321554200290737839684985669776843853074810183851370631023323Approximate: 70 significant digits; calculation provenance | 0.04564197705004891178221514329123996758997351435807668397223220139938829Approximate: 70 significant digits; calculation provenance | -2.236610898842103741884889908335484168247435006266814082094457663177413Approximate: 70 significant digits; calculation provenance | -89/40Exact rational | 89/40Exact rational | -89/40Exact rational | false | true |
32 | -11/5Exact rational | 0.01212843498427423979284666598494267360934748297100562534245916304133280Approximate: 70 significant digits; calculation provenance | 0.9878715650157257602071533340150573263906525170289943746575408369586672Approximate: 70 significant digits; calculation provenance | -0.9757431300314515204143066680301146527813050340579887493150816739173344Approximate: 70 significant digits; calculation provenance | 0.04792534419642589005438183177647843978959740278429874376297693893828274Approximate: 70 significant digits; calculation provenance | -2.212202584607696098784946568941603330001682446155611976678025089663730Approximate: 70 significant digits; calculation provenance | -11/5Exact rational | 11/5Exact rational | -11/5Exact rational | false | true |
33 | -87/40Exact rational | 0.01274234946411160232406955350640691619062434084922332798636391795973025Approximate: 70 significant digits; calculation provenance | 0.9872576505358883976759304464935930838093756591507766720136360820402697Approximate: 70 significant digits; calculation provenance | -0.9745153010717767953518608929871861676187513183015533440272721640805395Approximate: 70 significant digits; calculation provenance | 0.05031992797698422834057985356994404594722972522867628930466470540732502Approximate: 70 significant digits; calculation provenance | -2.187824229505431179550563771528029172833784612918315531964549179186845Approximate: 70 significant digits; calculation provenance | -87/40Exact rational | 87/40Exact rational | -87/40Exact rational | false | true |
34 | -43/20Exact rational | 0.01338691782766477752036858831405329845216732806091531829659019827567549Approximate: 70 significant digits; calculation provenance | 0.9866130821723352224796314116859467015478326719390846817034098017243245Approximate: 70 significant digits; calculation provenance | -0.9732261643446704449592628233718934030956653438781693634068196034486490Approximate: 70 significant digits; calculation provenance | 0.05283083303496051389744929665550905236784646766469243656316088207633650Approximate: 70 significant digits; calculation provenance | -2.163477330416026301248113488248782362621159896370748971831472377875204Approximate: 70 significant digits; calculation provenance | -43/20Exact rational | 43/20Exact rational | -43/20Exact rational | false | true |
35 | -17/8Exact rational | 0.01406362704324547424372752312564798097473538148216727383528916283029646Approximate: 70 significant digits; calculation provenance | 0.9859363729567545257562724768743520190252646185178327261647108371697035Approximate: 70 significant digits; calculation provenance | -0.9718727459135090515125449537487040380505292370356654523294216743394071Approximate: 70 significant digits; calculation provenance | 0.05546336575053587521381428053334842284415803722670943959387184973634696Approximate: 70 significant digits; calculation provenance | -2.139163456931504975714938383770760116485397733617481604445285369593905Approximate: 70 significant digits; calculation provenance | -17/8Exact rational | 17/8Exact rational | -17/8Exact rational | false | true |
36 | -21/10Exact rational | 0.01477403169327305775053400086688419306797497594106549080548633369343096Approximate: 70 significant digits; calculation provenance | 0.9852259683067269422494659991331158069320250240589345091945136663065690Approximate: 70 significant digits; calculation provenance | -0.9704519366134538844989319982662316138640500481178690183890273326131381Approximate: 70 significant digits; calculation provenance | 0.05822303872319688390625643629354477317692114197532726370052111556975025Approximate: 70 significant digits; calculation provenance | -2.114884254671918127807191357432826018005623844933442979226939024275903Approximate: 70 significant digits; calculation provenance | -21/10Exact rational | 21/10Exact rational | -21/10Exact rational | false | true |
37 | -83/40Exact rational | 0.01551975657840889274968556895987744068702406874481030980655612427417089Approximate: 70 significant digits; calculation provenance | 0.9844802434215911072503144310401225593129759312551896901934438757258291Approximate: 70 significant digits; calculation provenance | -0.9689604868431822145006288620802451186259518625103793803868877514516582Approximate: 70 significant digits; calculation provenance | 0.06111557493662330659087257413601293682126735505946868365006433652878323Approximate: 70 significant digits; calculation provenance | -2.090641448730935860600411643487324066569541141229760743428454469788705Approximate: 70 significant digits; calculation provenance | -83/40Exact rational | 83/40Exact rational | -83/40Exact rational | false | true |
38 | -41/20Exact rational | 0.01630249937144094007490529972662115607872114311196143136053599195208372Approximate: 70 significant digits; calculation provenance | 0.9836975006285590599250947002733788439212788568880385686394640080479163Approximate: 70 significant digits; calculation provenance | -0.9673950012571181198501894005467576878425577137760771372789280160958326Approximate: 70 significant digits; calculation provenance | 0.06414691154274043131470608246046353139812055198227789153841106421171330Approximate: 70 significant digits; calculation provenance | -2.066436847252909523485825735724019227115343951368720722661647574587303Approximate: 70 significant digits; calculation provenance | -41/20Exact rational | 41/20Exact rational | -41/20Exact rational | false | true |
39 | -81/40Exact rational | 0.01712403331572773451979593704314534926396151438205102083665915839234000Approximate: 70 significant digits; calculation provenance | 0.9828759666842722654802040629568546507360384856179489791633408416076600Approximate: 70 significant digits; calculation provenance | -0.9657519333685445309604081259137093014720769712358979583266816832153200Approximate: 70 significant digits; calculation provenance | 0.06732320319491832452098994055902847813804244055189916465601331836200292Approximate: 70 significant digits; calculation provenance | -2.042272345143765470366628439688664986657495695475547519292944091749114Approximate: 70 significant digits; calculation provenance | -81/40Exact rational | 81/40Exact rational | -81/40Exact rational | false | true |
40 | -2Exact rational | 0.01798620996209155802679313794953842487248501187953261197586839129344603Approximate: 70 significant digits; calculation provenance | 0.9820137900379084419732068620504615751275149881204673880241316087065540Approximate: 70 significant digits; calculation provenance | -0.9640275800758168839464137241009231502550299762409347760482632174131079Approximate: 70 significant digits; calculation provenance | 0.07065082485316446568624765586105479231212725983765596112486380982686206Approximate: 70 significant digits; calculation provenance | -2.018149927917809740354983318287753884762148873963581175677534475091048Approximate: 70 significant digits; calculation provenance | -2Exact rational | 2Exact rational | -2Exact rational | false | true |
41 | -79/40Exact rational | 0.01889096193703905165632770839788465461262156716748775572342199065924643Approximate: 70 significant digits; calculation provenance | 0.9811090380629609483436722916021153453873784328325122442765780093407536Approximate: 70 significant digits; calculation provenance | -0.9622180761259218966873445832042306907747568656650244885531560186815071Approximate: 70 significant digits; calculation provenance | 0.07413637397652957367060871913506817660786945503632755832141743041467197Approximate: 70 significant digits; calculation provenance | -1.994071675682192492026318535190050501756017611297832060372767860251917Approximate: 70 significant digits; calculation provenance | -79/40Exact rational | 79/40Exact rational | -79/40Exact rational | false | true |
42 | -39/20Exact rational | 0.01984030573407750655241319468565658195880806302421628534149660202080676Approximate: 70 significant digits; calculation provenance | 0.9801596942659224934475868053143434180411919369757837146585033979791932Approximate: 70 significant digits; calculation provenance | -0.9603193885318449868951736106286868360823838739515674293170067959583865Approximate: 70 significant digits; calculation provenance | 0.07778667200982335106505862608909093503331500493628392707238446561518654Approximate: 70 significant digits; calculation provenance | -1.970039767260397607455752968097949867813616800392086530423946504923834Approximate: 70 significant digits; calculation provenance | -39/20Exact rational | 39/20Exact rational | -39/20Exact rational | false | true |
43 | -77/40Exact rational | 0.02083634451868042513794409919712106269043333860337448428197185013982372Approximate: 70 significant digits; calculation provenance | 0.9791636554813195748620559008028789373095666613966255157180281498601763Approximate: 70 significant digits; calculation provenance | -0.9583273109626391497241118016057578746191333227932510314360562997203526Approximate: 70 significant digits; calculation provenance | 0.08160876506311712536173525747423981252827228740080525134811426535396623Approximate: 70 significant digits; calculation provenance | -1.946056484455681376998457009211047344763684398054106785457144736120413Approximate: 70 significant digits; calculation provenance | -77/40Exact rational | 77/40Exact rational | -77/40Exact rational | false | true |
44 | -19/10Exact rational | 0.02188127093613047395225129698567348265973305563226792432112252373885990Approximate: 70 significant digits; calculation provenance | 0.9781187290638695260477487030143265173402669443677320756788774762611401Approximate: 70 significant digits; calculation provenance | -0.9562374581277390520954974060286530346805338887354641513577549525222802Approximate: 70 significant digits; calculation provenance | 0.08560992367340050305742670899687769366796266277325939033632392574245715Approximate: 70 significant digits; calculation provenance | -1.922124216454879193076514799413436093318705163779467953688502031120878Approximate: 70 significant digits; calculation provenance | -19/10Exact rational | 19/10Exact rational | -19/10Exact rational | false | true |
45 | -15/8Exact rational | 0.02297736991002561495390388669015859139170942989980242380177698943313830Approximate: 70 significant digits; calculation provenance | 0.9770226300899743850460961133098414086082905701001975761982230105668617Approximate: 70 significant digits; calculation provenance | -0.9540452601799487700921922266196828172165811402003951523964460211337234Approximate: 70 significant digits; calculation provenance | 0.08979764152817385766904692651159891548610297142722295683956647020475296Approximate: 70 significant digits; calculation provenance | -1.898245464372425028195870552274882442648579480000923092040701284398497Approximate: 70 significant digits; calculation provenance | -15/8Exact rational | 15/8Exact rational | -15/8Exact rational | false | true |
46 | -37/20Exact rational | 0.02412702141766920128038024813719466717230420866984161854651404424528761Approximate: 70 significant digits; calculation provenance | 0.9758729785823307987196197518628053328276957913301583814534859557547124Approximate: 70 significant digits; calculation provenance | -0.9517459571646615974392395037256106656553915826603167629069719115094248Approximate: 70 significant digits; calculation provenance | 0.09417963302072213170097105667768505772198026625520402797105572461535931Approximate: 70 significant digits; calculation provenance | -1.874422845933779159494329738188660388424823583012678237701184241482271Approximate: 70 significant digits; calculation provenance | -37/20Exact rational | 37/20Exact rational | -37/20Exact rational | false | true |
47 | -73/40Exact rational | 0.02533270322687172941652584344975821628146100330521860272852215006279448Approximate: 70 significant digits; calculation provenance | 0.9746672967731282705834741565502417837185389966947813972714778499372055Approximate: 70 significant digits; calculation provenance | -0.9493345935462565411669483131004835674370779933895627945429556998744110Approximate: 70 significant digits; calculation provenance | 0.09876382949636388833697854749375383169988500287439141521105342737022724Approximate: 70 significant digits; calculation provenance | -1.850659100296728858463986402142599168487957101849038603473991099364792Approximate: 70 significant digits; calculation provenance | -73/40Exact rational | 73/40Exact rational | -73/40Exact rational | false | true |
48 | -9/5Exact rational | 0.02659699357686585517680084260127811813030488396764518223394431698402956Approximate: 70 significant digits; calculation provenance | 0.9734030064231341448231991573987218818696951160323548177660556830159704Approximate: 70 significant digits; calculation provenance | -0.9468060128462682896463983147974437637393902320647096355321113660319409Approximate: 70 significant digits; calculation provenance | 0.1035583740381520464794954247451830513492124417342322554844778389714661Approximate: 70 significant digits; calculation provenance | -1.826957093008208060442969061093719005900269587573282563293715466410106Approximate: 70 significant digits; calculation provenance | -9/5Exact rational | 9/5Exact rational | -9/5Exact rational | false | true |
49 | -71/40Exact rational | 0.02792257378407301896910984831318627069972782763081479551909855771172258Approximate: 70 significant digits; calculation provenance | 0.9720774262159269810308901516868137293002721723691852044809014422882774Approximate: 70 significant digits; calculation provenance | -0.9441548524318539620617803033736274586005443447383704089618028845765548Approximate: 70 significant digits; calculation provenance | 0.1085716146293840685413945610750577506689910796460042700045401696654224Approximate: 70 significant digits; calculation provenance | -1.803319821093368606461859226303512841511516059896573667181618776594244Approximate: 70 significant digits; calculation provenance | -71/40Exact rational | 71/40Exact rational | -71/40Exact rational | false | true |
50 | -7/4Exact rational | 0.02931223075135631886528955811418158389418212979838005127657477015424853Approximate: 70 significant digits; calculation provenance | 0.9706877692486436811347104418858184161058178702016199487234252298457515Approximate: 70 significant digits; calculation provenance | -0.9413755384972873622694208837716368322116357404032398974468504596915029Approximate: 70 significant digits; calculation provenance | 0.1138120955189422393587513085092015706827119559983316401207377855625312Approximate: 70 significant digits; calculation provenance | -1.779750418272620565194811174081749828815391172841522460200582523441863Approximate: 70 significant digits; calculation provenance | -7/4Exact rational | 7/4Exact rational | -7/4Exact rational | false | true |
51 | -69/40Exact rational | 0.03076885935714800916653013859261532868076063576497449412378848068845621Approximate: 70 significant digits; calculation provenance | 0.9692311406428519908334698614073846713192393642350255058762115193115438Approximate: 70 significant digits; calculation provenance | -0.9384622812857039816669397228147693426384787284700510117524230386230876Approximate: 70 significant digits; calculation provenance | 0.1192885466040322182664963657351446152020045407746583792227947280911648Approximate: 70 significant digits; calculation provenance | -1.756252160301235314762219806312722994475562687544073529641077836648716Approximate: 70 significant digits; calculation provenance | -69/40Exact rational | 69/40Exact rational | -69/40Exact rational | false | true |
52 | -17/10Exact rational | 0.03229546469845050973877562112699701997971124371264500769924981455567723Approximate: 70 significant digits; calculation provenance | 0.9677045353015494902612243788730029800202887562873549923007501854443228Approximate: 70 significant digits; calculation provenance | -0.9354090706030989805224487577460059600405775125747099846015003708886455Approximate: 70 significant digits; calculation provenance | 0.1250098706334465866593678064810895113372531617800235200467952294771509Approximate: 70 significant digits; calculation provenance | -1.732828470424865341747239134360648684016082194864521100075579571838508Approximate: 70 significant digits; calculation provenance | -17/10Exact rational | 17/10Exact rational | -17/10Exact rational | false | true |
53 | -67/40Exact rational | 0.03389516415917814865965447372254520360187504334909566436799188782875483Approximate: 70 significant digits; calculation provenance | 0.9661048358408218513403455262774547963981249566509043356320081121712452Approximate: 70 significant digits; calculation provenance | -0.9322096716816437026806910525549095927962499133018086712640162243424903Approximate: 70 significant digits; calculation provenance | 0.1309851280232020549049842142860456957023772213280774911929341443239059Approximate: 70 significant digits; calculation provenance | -1.709482924942971955040031191245815340189309123522972394334716965567623Approximate: 70 significant digits; calculation provenance | -67/40Exact rational | 67/40Exact rational | -67/40Exact rational | false | true |
54 | -33/20Exact rational | 0.03557118927263617277735647404062813594777206254986794138976101352755918Approximate: 70 significant digits; calculation provenance | 0.9644288107273638272226435259593718640522279374501320586102389864724408Approximate: 70 significant digits; calculation provenance | -0.9288576214547276544452870519187437281044558749002641172204779729448816Approximate: 70 significant digits; calculation provenance | 0.1372235190654658641713039488679920416589949262585419732651662322077304Approximate: 70 significant digits; calculation provenance | -1.686219258870659275671226786055661309663649211230079867661478491790305Approximate: 70 significant digits; calculation provenance | -33/20Exact rational | 33/20Exact rational | -33/20Exact rational | false | true |
55 | -13/8Exact rational | 0.03732688734412946019771416603849814526022220211518720374581920131333861Approximate: 70 significant digits; calculation provenance | 0.9626731126558705398022858339615018547397777978848127962541807986866614Approximate: 70 significant digits; calculation provenance | -0.9253462253117410796045716679230037094795555957696255925083615973733228Approximate: 70 significant digits; calculation provenance | 0.1437343633013125125237877279779659639963923972703809244601636889818485Approximate: 70 significant digits; calculation provenance | -1.663041371687783128561208561610680358286802282251023153527884588912428Approximate: 70 significant digits; calculation provenance | -13/8Exact rational | 13/8Exact rational | -13/8Exact rational | false | true |
56 | -8/5Exact rational | 0.03916572279676435865836788845558746534592256287477300692655513529705145Approximate: 70 significant digits; calculation provenance | 0.9608342772032356413416321115444125346540774371252269930734448647029485Approximate: 70 significant digits; calculation provenance | -0.9216685544064712826832642230888250693081548742504539861468897294058971Approximate: 70 significant digits; calculation provenance | 0.1505270758182854851283623433654002757494587831390396389712535129653613Approximate: 70 significant digits; calculation provenance | -1.639953333162430357063348799134171653799273521482867056096408457711270Approximate: 70 significant digits; calculation provenance | -8/5Exact rational | 8/5Exact rational | -8/5Exact rational | false | true |
57 | -63/40Exact rational | 0.04109127820046501077000316727152140756206930358659539568786376476667578Approximate: 70 significant digits; calculation provenance | 0.9589087217995349892299968327284785924379306964134046043121362352333242Approximate: 70 significant digits; calculation provenance | -0.9178174435990699784599936654569571848758613928268092086242724704666484Approximate: 70 significant digits; calculation provenance | 0.1576111402252679990246822332561389857310456743849833802725515129540452Approximate: 70 significant digits; calculation provenance | -1.616959389233941642269206912630506652405874971989012235472545150172495Approximate: 70 significant digits; calculation provenance | -63/40Exact rational | 63/40Exact rational | -63/40Exact rational | false | true |
58 | -31/20Exact rational | 0.04310725494108612255377124367113945017543326254370535827741516763062234Approximate: 70 significant digits; calculation provenance | 0.9568927450589138774462287563288605498245667374562946417225848323693777Approximate: 70 significant digits; calculation provenance | -0.9137854901178277548924575126577210996491334749125892834451696647387553Approximate: 70 significant digits; calculation provenance | 0.1649960780501213145061082491607692000373033713438522013281455034850210Approximate: 70 significant digits; calculation provenance | -1.594063967938573859254143722314314082545703520469818851327848644953224Approximate: 70 significant digits; calculation provenance | -31/20Exact rational | 31/20Exact rational | -31/20Exact rational | false | true |
59 | -61/40Exact rational | 0.04521747348328748947736358905408310829642995438507441267959612373881131Approximate: 70 significant digits; calculation provenance | 0.9547825265167125105226364109459168917035700456149255873204038762611887Approximate: 70 significant digits; calculation provenance | -0.9095650530334250210452728218918337834071400912298511746408077525223774Approximate: 70 significant digits; calculation provenance | 0.1726914143003027289217433886811123580092109684043552398415875543792194Approximate: 70 significant digits; calculation provenance | -1.571271685358661957824396600243818042314316640329728315560630694995370Approximate: 70 significant digits; calculation provenance | -61/40Exact rational | 61/40Exact rational | -61/40Exact rational | false | true |
60 | -3/2Exact rational | 0.04742587317756678087884815177175220138617943242060910071788800877444871Approximate: 70 significant digits; calculation provenance | 0.9525741268224332191211518482282477986138205675793908992821119912255513Approximate: 70 significant digits; calculation provenance | -0.9051482536448664382423036964564955972276411351587817985642239824511026Approximate: 70 significant digits; calculation provenance | 0.1807066389236485305974401137426065519794398095899587698581913111392369Approximate: 70 significant digits; calculation provenance | -1.548587351573742058758925919854689997941881642828108620530846980487520Approximate: 70 significant digits; calculation provenance | -3/2Exact rational | 3/2Exact rational | -3/2Exact rational | true | true |
61 | -59/40Exact rational | 0.04973651155855672518397144142433188532696187247528073944338658132678605Approximate: 70 significant digits; calculation provenance | 0.9502634884414432748160285585756681146730381275247192605566134186732140Approximate: 70 significant digits; calculation provenance | -0.9005269768828865496320571171513362293460762550494385211132268373464279Approximate: 70 significant digits; calculation provenance | 0.1890511639061691138380507237436600467907108772761312424859360041680161Approximate: 70 significant digits; calculation provenance | -1.526015976589535040680222366025510802241052462267022728069589187060076Approximate: 70 significant digits; calculation provenance | -59/40Exact rational | 59/40Exact rational | -59/40Exact rational | false | true |
62 | -29/20Exact rational | 0.05215356307841773477995532498890506406991433814971297419544370674866888Approximate: 70 significant digits; calculation provenance | 0.9478464369215822652200446750110949359300856618502870258045562932513311Approximate: 70 significant digits; calculation provenance | -0.8956928738431645304400893500221898718601713237005740516091125865026622Approximate: 70 significant digits; calculation provenance | 0.1977342757465729489259144060115879197115135238464029596274780190343313Approximate: 70 significant digits; calculation provenance | -1.503562776217963173260831475323559556222954003130506337237796936451904Approximate: 70 significant digits; calculation provenance | -29/20Exact rational | 29/20Exact rational | -29/20Exact rational | false | true |
63 | -57/40Exact rational | 0.05468131721594075720062816568698962009395166385532265555314193827660736Approximate: 70 significant digits; calculation provenance | 0.9453186827840592427993718343130103799060483361446773444468580617233926Approximate: 70 significant digits; calculation provenance | -0.8906373655681184855987436686260207598120966722893546888937161234467853Approximate: 70 significant digits; calculation provenance | 0.2067650830538816726343303696499654828865047773281405960574869821568330Approximate: 70 significant digits; calculation provenance | -1.481233177878483152522616321616629002161295426363029610607893789829096Approximate: 70 significant digits; calculation provenance | -57/40Exact rational | 57/40Exact rational | -57/40Exact rational | false | true |
64 | -7/5Exact rational | 0.05732417589886874620829117398721303527216144785446360424929208974713673Approximate: 70 significant digits; calculation provenance | 0.9426758241011312537917088260127869647278385521455363957507079102528633Approximate: 70 significant digits; calculation provenance | -0.8853516482022625075834176520255739294556771042910727915014158205057265Approximate: 70 significant digits; calculation provenance | 0.2161524590255372071239072466398030836272924866539352039681529625277187Approximate: 70 significant digits; calculation provenance | -1.459032826287971399138171852604277883648122327295520366156354995283702Approximate: 70 significant digits; calculation provenance | -7/5Exact rational | 7/5Exact rational | -7/5Exact rational | false | true |
65 | -11/8Exact rational | 0.06008665017400762201972344559490871336252001116421100791634362621981223Approximate: 70 significant digits; calculation provenance | 0.9399133498259923779802765544050912866374799888357889920836563737801878Approximate: 70 significant digits; calculation provenance | -0.8798266996519847559605531088101825732749599776715779841673127475603755Approximate: 70 significant digits; calculation provenance | 0.2259049785794962073047278461177565053894799590474085725141825459907048Approximate: 70 significant digits; calculation provenance | -1.436967589003198624576485597894303647745849723995463673294254484297896Approximate: 70 significant digits; calculation provenance | -11/8Exact rational | 11/8Exact rational | -11/8Exact rational | false | true |
66 | -27/20Exact rational | 0.06297335605699648951415632850893890638644250916269336429060969427880706Approximate: 70 significant digits; calculation provenance | 0.9370266439430035104858436714910610936135574908373066357093903057211929Approximate: 70 significant digits; calculation provenance | -0.8740532878860070209716873429821221872271149816746132714187806114423859Approximate: 70 significant digits; calculation provenance | 0.2360308499356609322442290466492354248382291922071062348869784731620760Approximate: 70 significant digits; calculation provenance | -1.415043561776590548309753637919068996116779308821747734007900794856700Approximate: 70 significant digits; calculation provenance | -27/20Exact rational | 27/20Exact rational | -27/20Exact rational | false | true |
67 | -53/40Exact rational | 0.06598900949121877861048606761221738358270667038149921830145121384011651Approximate: 70 significant digits; calculation provenance | 0.9340109905087812213895139323877826164172933296185007816985487861598835Approximate: 70 significant digits; calculation provenance | -0.8680219810175624427790278647755652328345866592370015633970975723197670Approximate: 70 significant digits; calculation provenance | 0.2465378404703464662551897357302608644104978565313277112291784915089075Approximate: 70 significant digits; calculation provenance | -1.393267073682503959180741662935593674501532284687038912920738358690038Approximate: 70 significant digits; calculation provenance | -53/40Exact rational | 53/40Exact rational | -53/40Exact rational | false | true |
68 | -13/10Exact rational | 0.06913842034334681785252314203162410057661444766670660721198081273559173Approximate: 70 significant digits; calculation provenance | 0.9308615796566531821474768579683758994233855523332933927880191872644083Approximate: 70 significant digits; calculation provenance | -0.8617231593133063642949537159367517988467711046665867855760383745288165Approximate: 70 significant digits; calculation provenance | 0.2574331967030940189517344743806649427337928787943978188017380394382650Approximate: 70 significant digits; calculation provenance | -1.371644691967669814406403590902148677628789957962867152218692838435794Approximate: 70 significant digits; calculation provenance | -13/10Exact rational | 13/10Exact rational | -13/10Exact rational | false | true |
69 | -51/40Exact rational | 0.07242648536151771784392350614769284156307047148566544469067800855126713Approximate: 70 significant digits; calculation provenance | 0.9275735146384822821560764938523071584369295285143345553093219914487329Approximate: 70 significant digits; calculation provenance | -0.8551470292769645643121529877046143168738590570286691106186439828974657Approximate: 70 significant digits; calculation provenance | 0.2687235583187823103036565180572303462896585013792145541192349913068698Approximate: 70 significant digits; calculation provenance | -1.350183226575790057019486886367011038020169445520365140023040521978744Approximate: 70 significant digits; calculation provenance | -51/40Exact rational | 51/40Exact rational | -51/40Exact rational | false | true |
70 | -5/4Exact rational | 0.07585818002124355119330617664624777313071203966078423883632735848208332Approximate: 70 significant digits; calculation provenance | 0.9241418199787564488066938233537522268692879603392157611636726415179167Approximate: 70 significant digits; calculation provenance | -0.8482836399575128976133876467075044537385759206784315223273452830358334Approximate: 70 significant digits; calculation provenance | 0.2804148661804326277293312747706377322057927192456265986022007246598311Approximate: 70 significant digits; calculation provenance | -1.328889734292549623344043916717550997259079421066961476183422357091817Approximate: 70 significant digits; calculation provenance | -5/4Exact rational | 5/4Exact rational | -5/4Exact rational | false | true |
71 | -49/40Exact rational | 0.07943854918397837440162335820883119204395186129705106958334797351361907Approximate: 70 significant digits; calculation provenance | 0.9205614508160216255983766417911688079560481387029489304166520264863809Approximate: 70 significant digits; calculation provenance | -0.8411229016320432511967532835823376159120962774058978608333040529727619Approximate: 70 significant digits; calculation provenance | 0.2925122643500920925923516412805348532458645958323277859498379111373692Approximate: 70 significant digits; calculation provenance | -1.307771522453552522417881672566973000167092436651002968434036336641785Approximate: 70 significant digits; calculation provenance | -49/40Exact rational | 49/40Exact rational | -49/40Exact rational | false | true |
72 | -6/5Exact rational | 0.08317269649392237066295243909384281089021635257418277768389981699302972Approximate: 70 significant digits; calculation provenance | 0.9168273035060776293370475609061571891097836474258172223161001830069703Approximate: 70 significant digits; calculation provenance | -0.8336546070121552586740951218123143782195672948516344446322003660139406Approximate: 70 significant digits; calculation provenance | 0.3050199962074089762139800299764416938310900832609193254586195024148496Approximate: 70 significant digits; calculation provenance | -1.286836152153949678480990395226517825132790577206467223706689803126962Approximate: 70 significant digits; calculation provenance | -6/5Exact rational | 6/5Exact rational | -6/5Exact rational | false | true |
73 | -47/40Exact rational | 0.08706577244027125972766884375067475452272418494570341653285576394674985Approximate: 70 significant digits; calculation provenance | 0.9129342275597287402723311562493252454772758150542965834671442360532501Approximate: 70 significant digits; calculation provenance | -0.8258684551194574805446623124986504909545516301085931669342884721065003Approximate: 70 significant digits; calculation provenance | 0.3179412948386006451945042653019462839999663232922046906603801862139635Approximate: 70 significant digits; calculation provenance | -1.266091440894841611285156340979529644746285639052160118424711008181297Approximate: 70 significant digits; calculation provenance | -47/40Exact rational | 47/40Exact rational | -47/40Exact rational | false | true |
74 | -23/20Exact rational | 0.09112296101485613533930474697641075069735499412538281582404554213509480Approximate: 70 significant digits; calculation provenance | 0.9088770389851438646606952530235892493026450058746171841759544578649052Approximate: 70 significant digits; calculation provenance | -0.8177540779702877293213905060471784986052900117492343683519089157298104Approximate: 70 significant digits; calculation provenance | 0.3312782679629645770272626630699932872649235163333968824499428483084358Approximate: 70 significant digits; calculation provenance | -1.245545464597962942695928157963484281043240516276454399297256530874566Approximate: 70 significant digits; calculation provenance | -23/20Exact rational | 23/20Exact rational | -23/20Exact rational | false | true |
75 | -9/8Exact rational | 0.09534946489910949396961476322833650130170439240603489180122924935701094Approximate: 70 significant digits; calculation provenance | 0.9046505351008905060303852367716634986982956075939651081987707506429891Approximate: 70 significant digits; calculation provenance | -0.8093010702017810120607704735433269973965912151879302163975415012859781Approximate: 70 significant digits; calculation provenance | 0.3450317777702519220263555246456887399878264851737243615202901172103272Approximate: 70 significant digits; calculation provenance | -1.225206558916747211673152826784967298131435400418382604358125662701238Approximate: 70 significant digits; calculation provenance | -9/8Exact rational | 9/8Exact rational | -9/8Exact rational | false | true |
76 | -11/10Exact rational | 0.09975048911968514699426933469965177097642044906823783160512682876511098Approximate: 70 significant digits; calculation provenance | 0.9002495108803148530057306653003482290235795509317621683948731712348890Approximate: 70 significant digits; calculation provenance | -0.8004990217606297060114613306006964580471591018635243367897463424697780Approximate: 70 significant digits; calculation provenance | 0.3592013161602748884100572304824881937796786715955766009770588764284391Approximate: 70 significant digits; calculation provenance | -1.205083319768695917520491879219231837591287215137043902047306230238518Approximate: 70 significant digits; calculation provenance | -11/10Exact rational | 11/10Exact rational | -11/10Exact rational | false | true |
77 | -43/40Exact rational | 0.1043312231190012973699035524746277760963288377376151664109433805649316Approximate: 70 significant digits; calculation provenance | 0.8956687768809987026300964475253722239036711622623848335890566194350684Approximate: 70 significant digits; calculation provenance | -0.7913375537619974052601928950507444478073423245247696671781132388701368Approximate: 70 significant digits; calculation provenance | 0.3737848760059778662774592066218495089636934512248804703412976172348003Approximate: 70 significant digits; calculation provenance | -1.185184603011108871840814495071048675307276852600753041551481344282117Approximate: 70 significant digits; calculation provenance | -43/40Exact rational | 43/40Exact rational | -43/40Exact rational | false | true |
78 | -21/20Exact rational | 0.1090968211956129382991060724084703919157056198596737407764264499411157Approximate: 70 significant digits; calculation provenance | 0.8909031788043870617008939275915296080842943801403262592235735500588843Approximate: 70 significant digits; calculation provenance | -0.7818063576087741234017878551830592161685887602806525184471471001177686Approximate: 70 significant digits; calculation provenance | 0.3887788192025015913241536459813625846802510088548704639266704462920162Approximate: 70 significant digits; calculation provenance | -1.165519523179754965644273927488443324891055424040716590427872938537194Approximate: 70 significant digits; calculation provenance | -21/20Exact rational | 21/20Exact rational | -21/20Exact rational | false | true |
79 | -41/40Exact rational | 0.1140523812797908562368506288388715627718796022564432921291993498111459Approximate: 70 significant digits; calculation provenance | 0.8859476187202091437631493711611284372281203977435567078708006501888541Approximate: 70 significant digits; calculation provenance | -0.7718952374404182875262987423222568744562407954871134157416013003777082Approximate: 70 significant digits; calculation provenance | 0.4041777424168002739475387091371216486688056146628395244464182021374122Approximate: 70 significant digits; calculation provenance | -1.146097451208061624833917775402162200925666554728626436335095569985020Approximate: 70 significant digits; calculation provenance | -41/40Exact rational | 41/40Exact rational | -41/40Exact rational | false | true |
80 | -1Exact rational | 0.1192029220221175559402708586976032047936157013710317242015947499390234Approximate: 70 significant digits; calculation provenance | 0.8807970779778824440597291413023967952063842986289682757984052500609766Approximate: 70 significant digits; calculation provenance | -0.7615941559557648881194582826047935904127685972579365515968105001219532Approximate: 70 significant digits; calculation provenance | 0.4199743416140260693944967390417014449171867282307709547133114402445899Approximate: 70 significant digits; calculation provenance | -1.126928011042972496443726806358304431434333062808358357550141216787973Approximate: 70 significant digits; calculation provenance | -1Exact rational | 1Exact rational | -1Exact rational | false | true |
81 | -39/40Exact rational | 0.1245533581874164215836395589846995961213181763970005554236694809754631Approximate: 70 significant digits; calculation provenance | 0.8754466418125835784163604410153004038786818236029994445763305190245369Approximate: 70 significant digits; calculation provenance | -0.7508932836251671568327208820306008077573636472059988891526610380490739Approximate: 70 significant digits; calculation provenance | 0.4361592766066142729733706961564650781860541957654981076443021805371937Approximate: 70 significant digits; calculation provenance | -1.108021075072867232479514039033198250297835043668413313951281371996274Approximate: 70 significant digits; calculation provenance | -39/40Exact rational | 39/40Exact rational | -39/40Exact rational | false | true |
82 | -19/20Exact rational | 0.1301084743629978521553259910741266366155475554525898101345656045470355Approximate: 70 significant digits; calculation provenance | 0.8698915256370021478446740089258733633844524445474101898654343954529645Approximate: 70 significant digits; calculation provenance | -0.7397830512740042956893480178517467267689048890948203797308687909059289Approximate: 70 significant digits; calculation provenance | 0.4527210370477239312205775962494490208258410959433593543602364696865015Approximate: 70 significant digits; calculation provenance | -1.089386758282960585104792768183276506319006505692506172850167491514701Approximate: 70 significant digits; calculation provenance | -19/20Exact rational | 19/20Exact rational | -19/20Exact rational | false | true |
83 | -37/40Exact rational | 0.1358728970090942649637465191027426997379010381212270002786496632966118Approximate: 70 significant digits; calculation provenance | 0.8641271029909057350362534808972573002620989618787729997213503367033882Approximate: 70 significant digits; calculation provenance | -0.7282542059818114700725069617945146005241979237575459994427006734067765Approximate: 70 significant digits; calculation provenance | 0.4696458114698013108409765375968035589740456202796731827196321861849879Approximate: 70 significant digits; calculation provenance | -1.071035411054510086153469042118751038760047640465269275953081661361360Approximate: 70 significant digits; calculation provenance | -37/40Exact rational | 37/40Exact rational | -37/40Exact rational | false | true |
84 | -9/10Exact rational | 0.1418510649004877895942781084709525684125674176996191559505566675653068Approximate: 70 significant digits; calculation provenance | 0.8581489350995122104057218915290474315874325823003808440494433324346932Approximate: 70 significant digits; calculation provenance | -0.7162978701990244208114437830580948631748651646007616880988866648693863Approximate: 70 significant digits; calculation provenance | 0.4869173611483415625499475730621584692698779641582036888502490396166801Approximate: 70 significant digits; calculation provenance | -1.052977610526074134599376863819689551783095387594252033795913451416605Approximate: 70 significant digits; calculation provenance | -9/10Exact rational | 9/10Exact rational | -9/10Exact rational | false | true |
85 | -7/8Exact rational | 0.1480471980316894697061848651843231254462237534863856333987536969684058Approximate: 70 significant digits; calculation provenance | 0.8519528019683105302938151348156768745537762465136143666012463030315942Approximate: 70 significant digits; calculation provenance | -0.7039056039366210605876302696313537491075524930272287332024926060631883Approximate: 70 significant digits; calculation provenance | 0.5045169007466207652518701502918813306505570518478332779796687528123639Approximate: 70 significant digits; calculation provenance | -1.035224150438087226886947638647396192500601405696264831480236870991777Approximate: 70 significant digits; calculation provenance | -7/8Exact rational | 7/8Exact rational | -7/8Exact rational | false | true |
86 | -17/20Exact rational | 0.1544652650835347043339537653914042385440628875351577194395957977267325Approximate: 70 significant digits; calculation provenance | 0.8455347349164652956660462346085957614559371124648422805604042022732675Approximate: 70 significant digits; calculation provenance | -0.6910694698329305913320924692171915229118742249296845611208084045465350Approximate: 70 significant digits; calculation provenance | 0.5224229878648322353747763393015239827662447979769864786142706729892168Approximate: 70 significant digits; calculation provenance | -1.017786029386265917004277511513088651967798911693834108936276857787181Approximate: 70 significant digits; calculation provenance | -17/20Exact rational | 17/20Exact rational | -17/20Exact rational | false | true |
87 | -33/40Exact rational | 0.1611089495765852371914525967835320283499341507660779724688722538533422Approximate: 70 significant digits; calculation provenance | 0.8388910504234147628085474032164679716500658492339220275311277461466578Approximate: 70 significant digits; calculation provenance | -0.6777821008468295256170948064329359433001316984678440550622554922933156Approximate: 70 significant digits; calculation provenance | 0.5406114237716582108533631269367085053651654357047381504029653940668138Approximate: 70 significant digits; calculation provenance | -1.000674437414932412380631400546103626442253025579667965707582318225750Approximate: 70 significant digits; calculation provenance | -33/40Exact rational | 33/40Exact rational | -33/40Exact rational | false | true |
88 | -4/5Exact rational | 0.1679816148660755181575776717998785749626723536455424798409624618225564Approximate: 70 significant digits; calculation provenance | 0.8320183851339244818424223282001214250373276463544575201590375381774436Approximate: 70 significant digits; calculation provenance | -0.6640367702678489636848446564002428500746552927089150403180750763548872Approximate: 70 significant digits; calculation provenance | 0.5590551677322439785419938454774005897068583570510465372645745030074800Approximate: 70 significant digits; calculation provenance | -0.9839007408883388297074952119288453425649025245752130727019753837225414Approximate: 70 significant digits; calculation provenance | -4/5Exact rational | 4/5Exact rational | -4/5Exact rational | false | true |
89 | -31/40Exact rational | 0.1750862681640398181234376020625483318599835555874042817491061351796350Approximate: 70 significant digits; calculation provenance | 0.8249137318359601818765623979374516681400164444125957182508938648203650Approximate: 70 significant digits; calculation provenance | -0.6498274636719203637531247958749033362800328888251914365017877296407300Approximate: 70 significant digits; calculation provenance | 0.5777242674577190199170625359223058572054918778283332427078253104351313Approximate: 70 significant digits; calculation provenance | -0.9674764655865786937684008428131873077983022021440413704718960215536197Approximate: 70 significant digits; calculation provenance | -31/40Exact rational | 31/40Exact rational | -31/40Exact rational | false | true |
90 | -3/4Exact rational | 0.1824255238063563403927828213437517524537591564190953833899450385048858Approximate: 70 significant digits; calculation provenance | 0.8175744761936436596072171786562482475462408435809046166100549614951142Approximate: 70 significant digits; calculation provenance | -0.6351489523872873192144343573124964950924816871618092332201099229902284Approximate: 70 significant digits; calculation provenance | 0.5965858082813314260061336417360454535406025074814188237601855682426663Approximate: 70 significant digits; calculation provenance | -0.9514132779827524094994828090570911052543986289505390097295348213966711Approximate: 70 significant digits; calculation provenance | -3/4Exact rational | 3/4Exact rational | -3/4Exact rational | true | true |
91 | -29/40Exact rational | 0.1900015660153129751605924650112313999178219267055621166751052428597259Approximate: 70 significant digits; calculation provenance | 0.8099984339846870248394075349887686000821780732944378833248947571402741Approximate: 70 significant digits; calculation provenance | -0.6199968679693740496788150699775372001643561465888757666497895142805483Approximate: 70 significant digits; calculation provenance | 0.6156038837081665625563785524923765237708175593139090827411753136428732Approximate: 70 significant digits; calculation provenance | -0.9357229646697597712414249085641459477171191862113566591504281604105559Approximate: 70 significant digits; calculation provenance | -29/40Exact rational | 29/40Exact rational | -29/40Exact rational | false | true |
92 | -7/10Exact rational | 0.1978161114414182518456564084480867624922166762915695556937900032002307Approximate: 70 significant digits; calculation provenance | 0.8021838885585817481543435915519132375077833237084304443062099967997693Approximate: 70 significant digits; calculation provenance | -0.6043677771171634963086871831038264750155666474168608886124199935995386Approximate: 70 significant digits; calculation provenance | 0.6347395899824585873670147578510211110212312312177218059998864096509017Approximate: 70 significant digits; calculation provenance | -0.9204174099184509266048326564310639391679608718224685721070149380798202Approximate: 70 significant digits; calculation provenance | -7/10Exact rational | 7/10Exact rational | -7/10Exact rational | false | true |
93 | -27/40Exact rational | 0.2058703718009473289591818095496433703838833090651813113538112933892701Approximate: 70 significant digits; calculation provenance | 0.7941296281990526710408181904503566296161166909348186886461887066107299Approximate: 70 significant digits; calculation provenance | -0.5882592563981053420816363809007132592322333818696373772923774132214597Approximate: 70 significant digits; calculation provenance | 0.6539510472619481591563743192236908938265165802182742584476057612696071Approximate: 70 significant digits; calculation provenance | -0.9055085713638753831132184292055231145883364578105440907601100564504761Approximate: 70 significant digits; calculation provenance | -27/40Exact rational | 27/40Exact rational | -27/40Exact rational | false | true |
94 | -13/20Exact rational | 0.2141650169574413873979115277415728310257864131454974902803454170438009Approximate: 70 significant digits; calculation provenance | 0.7858349830425586126020884722584271689742135868545025097196545829561991Approximate: 70 significant digits; calculation provenance | -0.5716699660851172252041769445168543379484271737090050194393091659123982Approximate: 70 significant digits; calculation provenance | 0.6731934498762409215157827041136734135019448170815907703071500640622212Approximate: 70 significant digits; calculation provenance | -0.8910084538329922052680022037179210179006910073164205931021882661108321Approximate: 70 significant digits; calculation provenance | -13/20Exact rational | 13/20Exact rational | -13/20Exact rational | false | true |
95 | -5/8Exact rational | 0.2227001388253088530004804523384582019709507184521052349878431290596117Approximate: 70 significant digits; calculation provenance | 0.7772998611746911469995195476615417980290492815478947650121568709403883Approximate: 70 significant digits; calculation provenance | -0.5545997223493822939990390953230835960580985630957895300243137418807765Approximate: 70 significant digits; calculation provenance | 0.6924191479699880696307533115733416850475083228900575482323459198998405Approximate: 70 significant digits; calculation provenance | -0.8769290813453728889534187262141826880129861863803319136821729202532293Approximate: 70 significant digits; calculation provenance | -5/8Exact rational | 5/8Exact rational | -5/8Exact rational | false | true |
96 | -3/5Exact rational | 0.2314752165009823570690873475365516470085790061403839329138394934317377Approximate: 70 significant digits; calculation provenance | 0.7685247834990176429309126524634483529914209938596160670861605065682623Approximate: 70 significant digits; calculation provenance | -0.5370495669980352858618253049268967059828419877192321341723210131365245Approximate: 70 significant digits; calculation provenance | 0.7115777625872228087543671362170895712615728416429858006526814420830111Approximate: 70 significant digits; calculation provenance | -0.8632824673380311891877444624432650572400241918932592023074740649506396Approximate: 70 significant digits; calculation provenance | -3/5Exact rational | 3/5Exact rational | -3/5Exact rational | false | true |
97 | -23/40Exact rational | 0.2404890830508889254142806440745570911312699645459407062255121398558958Approximate: 70 significant digits; calculation provenance | 0.7595109169491110745857193559254429088687300354540592937744878601441042Approximate: 70 significant digits; calculation provenance | -0.5190218338982221491714387118508858177374600709081185875489757202882084Approximate: 70 significant digits; calculation provenance | 0.7306163359369262975848778677938305430180190370212981695061651134892074Approximate: 70 significant digits; calculation provenance | -0.8500805831863984325180050835714892292892156229458977331190933782005298Approximate: 70 significant digits; calculation provenance | -23/40Exact rational | 23/40Exact rational | -23/40Exact rational | false | true |
98 | -11/20Exact rational | 0.2497398944048823957900231575879370996627292060331703666191243928054977Approximate: 70 significant digits; calculation provenance | 0.7502601055951176042099768424120629003372707939668296333808756071945023Approximate: 70 significant digits; calculation provenance | -0.5005202111902352084199536848241258006745415879336592667617512143890046Approximate: 70 significant digits; calculation provenance | 0.7494795181900823456478421756006188953695398765420025122015558841965653Approximate: 70 significant digits; calculation provenance | -0.8373353251154308105756455497210148558540866582482033733255192006555983Approximate: 70 significant digits; calculation provenance | -11/20Exact rational | 11/20Exact rational | -11/20Exact rational | false | true |
99 | -21/40Exact rational | 0.2592251008178460062091043729357115918801826955417515045766577549590718Approximate: 70 significant digits; calculation provenance | 0.7407748991821539937908956270642884081198173044582484954233422450409282Approximate: 70 significant digits; calculation provenance | -0.4815497983643079875817912541285768162396346089164969908466844900818564Approximate: 70 significant digits; calculation provenance | 0.7681097916952943202076835163765397655945860873861411120732387650878945Approximate: 70 significant digits; calculation provenance | -0.8250584796176432012308926824229317880174760162413312206616415908275552Approximate: 70 significant digits; calculation provenance | -21/40Exact rational | 21/40Exact rational | -21/40Exact rational | false | true |
100 | -1/2Exact rational | 0.2689414213699951207488407581781637256348553598349434807236340920809595Approximate: 70 significant digits; calculation provenance | 0.7310585786300048792511592418218362743651446401650565192763659079190405Approximate: 70 significant digits; calculation provenance | -0.4621171572600097585023184836436725487302892803301130385527318158380809Approximate: 70 significant digits; calculation provenance | 0.7864477329659274101496989343436361024890691417092543103053073299473211Approximate: 70 significant digits; calculation provenance | -0.8132616875182228340489954949678556419152800856703483747190635148371949Approximate: 70 significant digits; calculation provenance | -1/2Exact rational | 1/2Exact rational | -1/2Exact rational | false | true |
101 | -19/40Exact rational | 0.2788848219771369082347031360139426488201499420231792425137483470436459Approximate: 70 significant digits; calculation provenance | 0.7211151780228630917652968639860573511798500579768207574862516529563541Approximate: 70 significant digits; calculation provenance | -0.4422303560457261835305937279721147023597001159536415149725033059127083Approximate: 70 significant digits; calculation provenance | 0.8044323121916702511553975137147755731159315796196328520821612499737944Approximate: 70 significant digits; calculation provenance | -0.8019564068509520465526896546333792624818734484189360493603518655043054Approximate: 70 significant digits; calculation provenance | -19/40Exact rational | 19/40Exact rational | -19/40Exact rational | false | true |
102 | -9/20Exact rational | 0.2890504973749960365369017631996680864885745643977816449761510968765987Approximate: 70 significant digits; calculation provenance | 0.7109495026250039634630982368003319135114254356022183550238489031234013Approximate: 70 significant digits; calculation provenance | -0.4218990052500079269261964736006638270228508712044367100476978062468027Approximate: 70 significant digits; calculation provenance | 0.8220012293690537837129460351808079323544628714062134336012867305302153Approximate: 70 significant digits; calculation provenance | -0.7911538747320878232261588166228443456011355160707116204634228959828445Approximate: 70 significant digits; calculation provenance | -9/20Exact rational | 9/20Exact rational | -9/20Exact rational | false | true |
103 | -17/40Exact rational | 0.2994328575260270498985122299639104673234898952749479713870521212941182Approximate: 70 significant digits; calculation provenance | 0.7005671424739729501014877700360895326765101047250520286129478787058818Approximate: 70 significant digits; calculation provenance | -0.4011342849479459002029755400721790653530202094501040572258957574117637Approximate: 70 significant digits; calculation provenance | 0.8390912854393001432040864744156366225350392940873570606889968141793510Approximate: 70 significant digits; calculation provenance | -0.7808650684421958434178801554739572411985272899415859156396059722247795Approximate: 70 significant digits; calculation provenance | -17/40Exact rational | 17/40Exact rational | -17/40Exact rational | false | true |
104 | -2/5Exact rational | 0.3100255188723875573661259380515633447434078128142999482577618337832492Approximate: 70 significant digits; calculation provenance | 0.6899744811276124426338740619484366552565921871857000517422381662167508Approximate: 70 significant digits; calculation provenance | -0.3799489622552248852677481238968733105131843743714001034844763324335017Approximate: 70 significant digits; calculation provenance | 0.8556387860811776954667602744359359879172442763650452898015018902623989Approximate: 70 significant digits; calculation provenance | -0.7711006659477777260061389149147307694816370898803023077552370114622026Approximate: 70 significant digits; calculation provenance | -2/5Exact rational | 2/5Exact rational | -2/5Exact rational | false | true |
105 | -3/8Exact rational | 0.3208213008246070268403198842234209787657787517804239780170482563194541Approximate: 70 significant digits; calculation provenance | 0.6791786991753929731596801157765790212342212482195760219829517436805459Approximate: 70 significant digits; calculation provenance | -0.3583573983507859463193602315531580424684424964391520439659034873610918Approximate: 70 significant digits; calculation provenance | 0.8715799750472561179210020006566964924541727557995941421255322570459127Approximate: 70 significant digits; calculation provenance | -0.7618710061148999436698402692140229086689359900071546992433165846441577Approximate: 70 significant digits; calculation provenance | -3/8Exact rational | 3/8Exact rational | -3/8Exact rational | false | true |
106 | -7/20Exact rational | 0.3318122278318338934692116703701277045130563411913805605387944250650336Approximate: 70 significant digits; calculation provenance | 0.6681877721681661065307883296298722954869436588086194394612055749349664Approximate: 70 significant digits; calculation provenance | -0.3363755443363322130615766592597445909738873176172388789224111498699328Approximate: 70 significant digits; calculation provenance | 0.8868514931724362016202571339447037546604569623239555938769136260835696Approximate: 70 significant digits; calculation provenance | -0.7531860488854578931909063259318878032793490764966581292717391855016017Approximate: 70 significant digits; calculation provenance | -7/20Exact rational | 7/20Exact rational | -7/20Exact rational | false | true |
107 | -13/40Exact rational | 0.3429895373265012140667666777646734274565653606638376753206767292441609Approximate: 70 significant digits; calculation provenance | 0.6570104626734987859332333222353265725434346393361623246793232707558391Approximate: 70 significant digits; calculation provenance | -0.3140209253469975718664666444706531450868692786723246493586465415116783Approximate: 70 significant digits; calculation provenance | 0.9013908584442153778990690207621350443328853172769259975841129883751682Approximate: 70 significant digits; calculation provenance | -0.7450553357027152118295760501284963782930763201967671933944194442660426Approximate: 70 significant digits; calculation provenance | -13/40Exact rational | 13/40Exact rational | -13/40Exact rational | false | true |
108 | -3/10Exact rational | 0.3543436937742045470908893635881170359232015975411939127928437902009042Approximate: 70 significant digits; calculation provenance | 0.6456563062257954529091106364118829640767984024588060872071562097990958Approximate: 70 significant digits; calculation provenance | -0.2913126124515909058182212728237659281535968049176121744143124195981917Approximate: 70 significant digits; calculation provenance | 0.9151369618266292031373615814217666677469839818217527892542448105836614Approximate: 70 significant digits; calculation provenance | -0.7374879504858856264513401532202476931981130233116905194522353644985003Approximate: 70 significant digits; calculation provenance | -3/10Exact rational | 3/10Exact rational | -3/10Exact rational | false | true |
109 | -11/40Exact rational | 0.3658644089891993172617809076054763580302556893657243581625608498043218Approximate: 70 significant digits; calculation provenance | 0.6341355910108006827382190923945236419697443106342756418374391501956782Approximate: 70 significant digits; calculation provenance | -0.2682711820216013654764381847890472839394886212685512836748783003913563Approximate: 70 significant digits; calculation provenance | 0.9280305728967328283011772510429690734812028815031173854359884451919717Approximate: 70 significant digits; calculation provenance | -0.7304924814633376379021455185763737878116710658294733658851958899543279Approximate: 70 significant digits; calculation provenance | -11/40Exact rational | 11/40Exact rational | -11/40Exact rational | false | true |
110 | -1/4Exact rational | 0.3775406687981454353610994342544915212467206346910898369405628373414550Approximate: 70 significant digits; calculation provenance | 0.6224593312018545646389005657455084787532793653089101630594371626585450Approximate: 70 significant digits; calculation provenance | -0.2449186624037091292778011314910169575065587306178203261188743253170900Approximate: 70 significant digits; calculation provenance | 0.9400148488063779562772107800840277767602862358092350573788919799963704Approximate: 70 significant digits; calculation provenance | -0.7240769841801066808729973550811707497555961946678628412008152750991935Approximate: 70 significant digits; calculation provenance | -1/4Exact rational | 1/4Exact rational | -1/4Exact rational | false | true |
111 | -9/40Exact rational | 0.3893607660507780117022307669820427198340657269023833089083589734155610Approximate: 70 significant digits; calculation provenance | 0.6106392339492219882977692330179572801659342730976166910916410265844390Approximate: 70 significant digits; calculation provenance | -0.2212784678984439765955384660359145603318685461952333821832820531688780Approximate: 70 significant digits; calculation provenance | 0.9510358396445172985399091593112780473664080964146782763250454187456551Approximate: 70 significant digits; calculation provenance | -0.7182489459974548696655945572464259262027711191834845540837086292171717Approximate: 70 significant digits; calculation provenance | -9/40Exact rational | 9/40Exact rational | -9/40Exact rational | false | true |
112 | -1/5Exact rational | 0.4013123398875479996309213405944921658053136580788234359572728755000027Approximate: 70 significant digits; calculation provenance | 0.5986876601124520003690786594055078341946863419211765640427271244999973Approximate: 70 significant digits; calculation provenance | -0.1973753202249040007381573188110156683893726838423531280854542489999946Approximate: 70 significant digits; calculation provenance | 0.9610429829661166017194701164306384580486458007735590694822039302905432Approximate: 70 significant digits; calculation provenance | -0.7130152523999526236683719508459124932686531650059861603611000564516561Approximate: 70 significant digits; calculation provenance | -1/5Exact rational | 1/5Exact rational | -1/5Exact rational | false | true |
113 | -7/40Exact rational | 0.4133824210826699422058227024802777262043132992140038543326782944894282Approximate: 70 significant digits; calculation provenance | 0.5866175789173300577941772975197222737956867007859961456673217055105718Approximate: 70 significant digits; calculation provenance | -0.1732351578346601155883545950394445475913734015719922913346434110211436Approximate: 70 significant digits; calculation provenance | 0.9699895800900003979721689801593753519514653531045975578399487662904875Approximate: 70 significant digits; calculation provenance | -0.7083821554187770321300536981423565488317240710598515348931408671419475Approximate: 70 significant digits; calculation provenance | -7/40Exact rational | 7/40Exact rational | -7/40Exact rational | false | true |
114 | -3/20Exact rational | 0.4255574831883410128479287347653482349739193173880957467199217224411976Approximate: 70 significant digits; calculation provenance | 0.5744425168116589871520712652346517650260806826119042532800782775588024Approximate: 70 significant digits; calculation provenance | -0.1488850336233179743041425304693035300521613652238085065601565551176049Approximate: 70 significant digits; calculation provenance | 0.9778332467629834762639438775381287889911131569080625370892274173719470Approximate: 70 significant digits; calculation provenance | -0.7043552444685271188145884355756765719720195237455736616571399777016886Approximate: 70 significant digits; calculation provenance | -3/20Exact rational | 3/20Exact rational | -3/20Exact rational | false | true |
115 | -1/8Exact rational | 0.4378234991142018959726763620970536461620406037423306247926650045638719Approximate: 70 significant digits; calculation provenance | 0.5621765008857981040273236379029463538379593962576693752073349954361281Approximate: 70 significant digits; calculation provenance | -0.1243530017715962080546472758058927076759187925153387504146699908722562Approximate: 70 significant digits; calculation provenance | 0.9845363309503933903410077702298145710512672860886544027839591145602179Approximate: 70 significant digits; calculation provenance | -0.7009394198788435622078361316583179420901750894333243242235171025749700Approximate: 70 significant digits; calculation provenance | -1/8Exact rational | 1/8Exact rational | -1/8Exact rational | false | true |
116 | -1/10Exact rational | 0.4501660026875220914408474581608239082305189521116327815347617807280041Approximate: 70 significant digits; calculation provenance | 0.5498339973124779085591525418391760917694810478883672184652382192719959Approximate: 70 significant digits; calculation provenance | -0.09966799462495581711830508367835218353896209577673443693047643854399178Approximate: 70 significant digits; calculation provenance | 0.9900662908474397783478055699095446038173916531650000855906991787027516Approximate: 70 significant digits; calculation provenance | -0.6981388693815918396849437125412322904934726704225570311240970983452535Approximate: 70 significant digits; calculation provenance | -1/10Exact rational | 1/10Exact rational | -1/10Exact rational | false | true |
117 | -3/40Exact rational | 0.4625701546562504505549843063075810792650212380379037796374384224727559Approximate: 70 significant digits; calculation provenance | 0.5374298453437495494450156936924189207349787619620962203625615775272441Approximate: 70 significant digits; calculation provenance | -0.07485969068749909889003138738483784146995752392419244072512315505448811Approximate: 70 significant digits; calculation provenance | 0.9943960267101719606909708015255971001448946105878298724062713721217914Approximate: 70 significant digits; calculation provenance | -0.6959570477895320775138418585710563047017043479946890498551201021393751Approximate: 70 significant digits; calculation provenance | -3/40Exact rational | 3/40Exact rational | -3/40Exact rational | false | true |
118 | -1/20Exact rational | 0.4750208125210600139008068173958578401786001031094907800164077069348779Approximate: 70 significant digits; calculation provenance | 0.5249791874789399860991931826041421598213998968905092199835922930651221Approximate: 70 significant digits; calculation provenance | -0.04995837495787997219838636520828431964279979378101843996718458613024430Approximate: 70 significant digits; calculation provenance | 0.9975041607715678712860727824020664521807554565548070172490615977926435Approximate: 70 significant digits; calculation provenance | -0.6943966600735708948300991083156580419055490232449290767188755390958803Approximate: 70 significant digits; calculation provenance | -1/20Exact rational | 1/20Exact rational | -1/20Exact rational | false | true |
119 | -1/40Exact rational | 0.4875026035157896563384986678705528697489419236176071678117367963425427Approximate: 70 significant digits; calculation provenance | 0.5124973964842103436615013321294471302510580763823928321882632036574573Approximate: 70 significant digits; calculation provenance | -0.02499479296842068732300266425889426050211615276478566437652640731491453Approximate: 70 significant digits; calculation provenance | 0.9993752603244657877658903955476493144643234100536313678200992324449276Approximate: 70 significant digits; calculation provenance | -0.6934596480132862941518864747650886822353675527614739128785603680887442Approximate: 70 significant digits; calculation provenance | -1/40Exact rational | 1/40Exact rational | -1/40Exact rational | false | true |
120At u=0 both probabilities are exactly1/2; mean0,variance1. T>0 remains fixed. | 0Exact rational | 1/2Exact rational | 1/2Exact rational | 0Exact rational | 1Exact rational | -0.6931471805599453094172321214581765680755001343602552541206800094933936Approximate: 70 significant digits; calculation provenance | 0Exact rational | 0Exact rational | 0Exact rational | true | true |
121 | 1/40Exact rational | 0.5124973964842103436615013321294471302510580763823928321882632036574573Approximate: 70 significant digits; calculation provenance | 0.4875026035157896563384986678705528697489419236176071678117367963425427Approximate: 70 significant digits; calculation provenance | 0.02499479296842068732300266425889426050211615276478566437652640731491453Approximate: 70 significant digits; calculation provenance | 0.9993752603244657877658903955476493144643234100536313678200992324449276Approximate: 70 significant digits; calculation provenance | -0.6934596480132862941518864747650886822353675527614739128785603680887442Approximate: 70 significant digits; calculation provenance | 1/40Exact rational | -1/40Exact rational | 1/40Exact rational | false | true |
122 | 1/20Exact rational | 0.5249791874789399860991931826041421598213998968905092199835922930651221Approximate: 70 significant digits; calculation provenance | 0.4750208125210600139008068173958578401786001031094907800164077069348779Approximate: 70 significant digits; calculation provenance | 0.04995837495787997219838636520828431964279979378101843996718458613024430Approximate: 70 significant digits; calculation provenance | 0.9975041607715678712860727824020664521807554565548070172490615977926435Approximate: 70 significant digits; calculation provenance | -0.6943966600735708948300991083156580419055490232449290767188755390958803Approximate: 70 significant digits; calculation provenance | 1/20Exact rational | -1/20Exact rational | 1/20Exact rational | false | true |
123 | 3/40Exact rational | 0.5374298453437495494450156936924189207349787619620962203625615775272441Approximate: 70 significant digits; calculation provenance | 0.4625701546562504505549843063075810792650212380379037796374384224727559Approximate: 70 significant digits; calculation provenance | 0.07485969068749909889003138738483784146995752392419244072512315505448811Approximate: 70 significant digits; calculation provenance | 0.9943960267101719606909708015255971001448946105878298724062713721217914Approximate: 70 significant digits; calculation provenance | -0.6959570477895320775138418585710563047017043479946890498551201021393751Approximate: 70 significant digits; calculation provenance | 3/40Exact rational | -3/40Exact rational | 3/40Exact rational | false | true |
124 | 1/10Exact rational | 0.5498339973124779085591525418391760917694810478883672184652382192719959Approximate: 70 significant digits; calculation provenance | 0.4501660026875220914408474581608239082305189521116327815347617807280041Approximate: 70 significant digits; calculation provenance | 0.09966799462495581711830508367835218353896209577673443693047643854399178Approximate: 70 significant digits; calculation provenance | 0.9900662908474397783478055699095446038173916531650000855906991787027516Approximate: 70 significant digits; calculation provenance | -0.6981388693815918396849437125412322904934726704225570311240970983452535Approximate: 70 significant digits; calculation provenance | 1/10Exact rational | -1/10Exact rational | 1/10Exact rational | false | true |
125 | 1/8Exact rational | 0.5621765008857981040273236379029463538379593962576693752073349954361281Approximate: 70 significant digits; calculation provenance | 0.4378234991142018959726763620970536461620406037423306247926650045638719Approximate: 70 significant digits; calculation provenance | 0.1243530017715962080546472758058927076759187925153387504146699908722562Approximate: 70 significant digits; calculation provenance | 0.9845363309503933903410077702298145710512672860886544027839591145602179Approximate: 70 significant digits; calculation provenance | -0.7009394198788435622078361316583179420901750894333243242235171025749700Approximate: 70 significant digits; calculation provenance | 1/8Exact rational | -1/8Exact rational | 1/8Exact rational | false | true |
126 | 3/20Exact rational | 0.5744425168116589871520712652346517650260806826119042532800782775588024Approximate: 70 significant digits; calculation provenance | 0.4255574831883410128479287347653482349739193173880957467199217224411976Approximate: 70 significant digits; calculation provenance | 0.1488850336233179743041425304693035300521613652238085065601565551176049Approximate: 70 significant digits; calculation provenance | 0.9778332467629834762639438775381287889911131569080625370892274173719470Approximate: 70 significant digits; calculation provenance | -0.7043552444685271188145884355756765719720195237455736616571399777016886Approximate: 70 significant digits; calculation provenance | 3/20Exact rational | -3/20Exact rational | 3/20Exact rational | false | true |
127 | 7/40Exact rational | 0.5866175789173300577941772975197222737956867007859961456673217055105718Approximate: 70 significant digits; calculation provenance | 0.4133824210826699422058227024802777262043132992140038543326782944894282Approximate: 70 significant digits; calculation provenance | 0.1732351578346601155883545950394445475913734015719922913346434110211436Approximate: 70 significant digits; calculation provenance | 0.9699895800900003979721689801593753519514653531045975578399487662904875Approximate: 70 significant digits; calculation provenance | -0.7083821554187770321300536981423565488317240710598515348931408671419475Approximate: 70 significant digits; calculation provenance | 7/40Exact rational | -7/40Exact rational | 7/40Exact rational | false | true |
128 | 1/5Exact rational | 0.5986876601124520003690786594055078341946863419211765640427271244999973Approximate: 70 significant digits; calculation provenance | 0.4013123398875479996309213405944921658053136580788234359572728755000027Approximate: 70 significant digits; calculation provenance | 0.1973753202249040007381573188110156683893726838423531280854542489999946Approximate: 70 significant digits; calculation provenance | 0.9610429829661166017194701164306384580486458007735590694822039302905432Approximate: 70 significant digits; calculation provenance | -0.7130152523999526236683719508459124932686531650059861603611000564516561Approximate: 70 significant digits; calculation provenance | 1/5Exact rational | -1/5Exact rational | 1/5Exact rational | false | true |
129 | 9/40Exact rational | 0.6106392339492219882977692330179572801659342730976166910916410265844390Approximate: 70 significant digits; calculation provenance | 0.3893607660507780117022307669820427198340657269023833089083589734155610Approximate: 70 significant digits; calculation provenance | 0.2212784678984439765955384660359145603318685461952333821832820531688780Approximate: 70 significant digits; calculation provenance | 0.9510358396445172985399091593112780473664080964146782763250454187456551Approximate: 70 significant digits; calculation provenance | -0.7182489459974548696655945572464259262027711191834845540837086292171717Approximate: 70 significant digits; calculation provenance | 9/40Exact rational | -9/40Exact rational | 9/40Exact rational | false | true |
130 | 1/4Exact rational | 0.6224593312018545646389005657455084787532793653089101630594371626585450Approximate: 70 significant digits; calculation provenance | 0.3775406687981454353610994342544915212467206346910898369405628373414550Approximate: 70 significant digits; calculation provenance | 0.2449186624037091292778011314910169575065587306178203261188743253170900Approximate: 70 significant digits; calculation provenance | 0.9400148488063779562772107800840277767602862358092350573788919799963704Approximate: 70 significant digits; calculation provenance | -0.7240769841801066808729973550811707497555961946678628412008152750991935Approximate: 70 significant digits; calculation provenance | 1/4Exact rational | -1/4Exact rational | 1/4Exact rational | false | true |
131 | 11/40Exact rational | 0.6341355910108006827382190923945236419697443106342756418374391501956782Approximate: 70 significant digits; calculation provenance | 0.3658644089891993172617809076054763580302556893657243581625608498043218Approximate: 70 significant digits; calculation provenance | 0.2682711820216013654764381847890472839394886212685512836748783003913563Approximate: 70 significant digits; calculation provenance | 0.9280305728967328283011772510429690734812028815031173854359884451919717Approximate: 70 significant digits; calculation provenance | -0.7304924814633376379021455185763737878116710658294733658851958899543279Approximate: 70 significant digits; calculation provenance | 11/40Exact rational | -11/40Exact rational | 11/40Exact rational | false | true |
132 | 3/10Exact rational | 0.6456563062257954529091106364118829640767984024588060872071562097990958Approximate: 70 significant digits; calculation provenance | 0.3543436937742045470908893635881170359232015975411939127928437902009042Approximate: 70 significant digits; calculation provenance | 0.2913126124515909058182212728237659281535968049176121744143124195981917Approximate: 70 significant digits; calculation provenance | 0.9151369618266292031373615814217666677469839818217527892542448105836614Approximate: 70 significant digits; calculation provenance | -0.7374879504858856264513401532202476931981130233116905194522353644985003Approximate: 70 significant digits; calculation provenance | 3/10Exact rational | -3/10Exact rational | 3/10Exact rational | false | true |
133 | 13/40Exact rational | 0.6570104626734987859332333222353265725434346393361623246793232707558391Approximate: 70 significant digits; calculation provenance | 0.3429895373265012140667666777646734274565653606638376753206767292441609Approximate: 70 significant digits; calculation provenance | 0.3140209253469975718664666444706531450868692786723246493586465415116783Approximate: 70 significant digits; calculation provenance | 0.9013908584442153778990690207621350443328853172769259975841129883751682Approximate: 70 significant digits; calculation provenance | -0.7450553357027152118295760501284963782930763201967671933944194442660426Approximate: 70 significant digits; calculation provenance | 13/40Exact rational | -13/40Exact rational | 13/40Exact rational | false | true |
134 | 7/20Exact rational | 0.6681877721681661065307883296298722954869436588086194394612055749349664Approximate: 70 significant digits; calculation provenance | 0.3318122278318338934692116703701277045130563411913805605387944250650336Approximate: 70 significant digits; calculation provenance | 0.3363755443363322130615766592597445909738873176172388789224111498699328Approximate: 70 significant digits; calculation provenance | 0.8868514931724362016202571339447037546604569623239555938769136260835696Approximate: 70 significant digits; calculation provenance | -0.7531860488854578931909063259318878032793490764966581292717391855016017Approximate: 70 significant digits; calculation provenance | 7/20Exact rational | -7/20Exact rational | 7/20Exact rational | false | true |
135 | 3/8Exact rational | 0.6791786991753929731596801157765790212342212482195760219829517436805459Approximate: 70 significant digits; calculation provenance | 0.3208213008246070268403198842234209787657787517804239780170482563194541Approximate: 70 significant digits; calculation provenance | 0.3583573983507859463193602315531580424684424964391520439659034873610918Approximate: 70 significant digits; calculation provenance | 0.8715799750472561179210020006566964924541727557995941421255322570459127Approximate: 70 significant digits; calculation provenance | -0.7618710061148999436698402692140229086689359900071546992433165846441577Approximate: 70 significant digits; calculation provenance | 3/8Exact rational | -3/8Exact rational | 3/8Exact rational | false | true |
136 | 2/5Exact rational | 0.6899744811276124426338740619484366552565921871857000517422381662167508Approximate: 70 significant digits; calculation provenance | 0.3100255188723875573661259380515633447434078128142999482577618337832492Approximate: 70 significant digits; calculation provenance | 0.3799489622552248852677481238968733105131843743714001034844763324335017Approximate: 70 significant digits; calculation provenance | 0.8556387860811776954667602744359359879172442763650452898015018902623989Approximate: 70 significant digits; calculation provenance | -0.7711006659477777260061389149147307694816370898803023077552370114622026Approximate: 70 significant digits; calculation provenance | 2/5Exact rational | -2/5Exact rational | 2/5Exact rational | false | true |
137 | 17/40Exact rational | 0.7005671424739729501014877700360895326765101047250520286129478787058818Approximate: 70 significant digits; calculation provenance | 0.2994328575260270498985122299639104673234898952749479713870521212941182Approximate: 70 significant digits; calculation provenance | 0.4011342849479459002029755400721790653530202094501040572258957574117637Approximate: 70 significant digits; calculation provenance | 0.8390912854393001432040864744156366225350392940873570606889968141793510Approximate: 70 significant digits; calculation provenance | -0.7808650684421958434178801554739572411985272899415859156396059722247795Approximate: 70 significant digits; calculation provenance | 17/40Exact rational | -17/40Exact rational | 17/40Exact rational | false | true |
138 | 9/20Exact rational | 0.7109495026250039634630982368003319135114254356022183550238489031234013Approximate: 70 significant digits; calculation provenance | 0.2890504973749960365369017631996680864885745643977816449761510968765987Approximate: 70 significant digits; calculation provenance | 0.4218990052500079269261964736006638270228508712044367100476978062468027Approximate: 70 significant digits; calculation provenance | 0.8220012293690537837129460351808079323544628714062134336012867305302153Approximate: 70 significant digits; calculation provenance | -0.7911538747320878232261588166228443456011355160707116204634228959828445Approximate: 70 significant digits; calculation provenance | 9/20Exact rational | -9/20Exact rational | 9/20Exact rational | false | true |
139 | 19/40Exact rational | 0.7211151780228630917652968639860573511798500579768207574862516529563541Approximate: 70 significant digits; calculation provenance | 0.2788848219771369082347031360139426488201499420231792425137483470436459Approximate: 70 significant digits; calculation provenance | 0.4422303560457261835305937279721147023597001159536415149725033059127083Approximate: 70 significant digits; calculation provenance | 0.8044323121916702511553975137147755731159315796196328520821612499737944Approximate: 70 significant digits; calculation provenance | -0.8019564068509520465526896546333792624818734484189360493603518655043054Approximate: 70 significant digits; calculation provenance | 19/40Exact rational | -19/40Exact rational | 19/40Exact rational | false | true |
140 | 1/2Exact rational | 0.7310585786300048792511592418218362743651446401650565192763659079190405Approximate: 70 significant digits; calculation provenance | 0.2689414213699951207488407581781637256348553598349434807236340920809595Approximate: 70 significant digits; calculation provenance | 0.4621171572600097585023184836436725487302892803301130385527318158380809Approximate: 70 significant digits; calculation provenance | 0.7864477329659274101496989343436361024890691417092543103053073299473211Approximate: 70 significant digits; calculation provenance | -0.8132616875182228340489954949678556419152800856703483747190635148371949Approximate: 70 significant digits; calculation provenance | 1/2Exact rational | -1/2Exact rational | 1/2Exact rational | false | true |
141 | 21/40Exact rational | 0.7407748991821539937908956270642884081198173044582484954233422450409282Approximate: 70 significant digits; calculation provenance | 0.2592251008178460062091043729357115918801826955417515045766577549590718Approximate: 70 significant digits; calculation provenance | 0.4815497983643079875817912541285768162396346089164969908466844900818564Approximate: 70 significant digits; calculation provenance | 0.7681097916952943202076835163765397655945860873861411120732387650878945Approximate: 70 significant digits; calculation provenance | -0.8250584796176432012308926824229317880174760162413312206616415908275552Approximate: 70 significant digits; calculation provenance | 21/40Exact rational | -21/40Exact rational | 21/40Exact rational | false | true |
142 | 11/20Exact rational | 0.7502601055951176042099768424120629003372707939668296333808756071945023Approximate: 70 significant digits; calculation provenance | 0.2497398944048823957900231575879370996627292060331703666191243928054977Approximate: 70 significant digits; calculation provenance | 0.5005202111902352084199536848241258006745415879336592667617512143890046Approximate: 70 significant digits; calculation provenance | 0.7494795181900823456478421756006188953695398765420025122015558841965653Approximate: 70 significant digits; calculation provenance | -0.8373353251154308105756455497210148558540866582482033733255192006555983Approximate: 70 significant digits; calculation provenance | 11/20Exact rational | -11/20Exact rational | 11/20Exact rational | false | true |
143 | 23/40Exact rational | 0.7595109169491110745857193559254429088687300354540592937744878601441042Approximate: 70 significant digits; calculation provenance | 0.2404890830508889254142806440745570911312699645459407062255121398558958Approximate: 70 significant digits; calculation provenance | 0.5190218338982221491714387118508858177374600709081185875489757202882084Approximate: 70 significant digits; calculation provenance | 0.7306163359369262975848778677938305430180190370212981695061651134892074Approximate: 70 significant digits; calculation provenance | -0.8500805831863984325180050835714892292892156229458977331190933782005298Approximate: 70 significant digits; calculation provenance | 23/40Exact rational | -23/40Exact rational | 23/40Exact rational | false | true |
144 | 3/5Exact rational | 0.7685247834990176429309126524634483529914209938596160670861605065682623Approximate: 70 significant digits; calculation provenance | 0.2314752165009823570690873475365516470085790061403839329138394934317377Approximate: 70 significant digits; calculation provenance | 0.5370495669980352858618253049268967059828419877192321341723210131365245Approximate: 70 significant digits; calculation provenance | 0.7115777625872228087543671362170895712615728416429858006526814420830111Approximate: 70 significant digits; calculation provenance | -0.8632824673380311891877444624432650572400241918932592023074740649506396Approximate: 70 significant digits; calculation provenance | 3/5Exact rational | -3/5Exact rational | 3/5Exact rational | false | true |
145 | 5/8Exact rational | 0.7772998611746911469995195476615417980290492815478947650121568709403883Approximate: 70 significant digits; calculation provenance | 0.2227001388253088530004804523384582019709507184521052349878431290596117Approximate: 70 significant digits; calculation provenance | 0.5545997223493822939990390953230835960580985630957895300243137418807765Approximate: 70 significant digits; calculation provenance | 0.6924191479699880696307533115733416850475083228900575482323459198998405Approximate: 70 significant digits; calculation provenance | -0.8769290813453728889534187262141826880129861863803319136821729202532293Approximate: 70 significant digits; calculation provenance | 5/8Exact rational | -5/8Exact rational | 5/8Exact rational | false | true |
146 | 13/20Exact rational | 0.7858349830425586126020884722584271689742135868545025097196545829561991Approximate: 70 significant digits; calculation provenance | 0.2141650169574413873979115277415728310257864131454974902803454170438009Approximate: 70 significant digits; calculation provenance | 0.5716699660851172252041769445168543379484271737090050194393091659123982Approximate: 70 significant digits; calculation provenance | 0.6731934498762409215157827041136734135019448170815907703071500640622212Approximate: 70 significant digits; calculation provenance | -0.8910084538329922052680022037179210179006910073164205931021882661108321Approximate: 70 significant digits; calculation provenance | 13/20Exact rational | -13/20Exact rational | 13/20Exact rational | false | true |
147 | 27/40Exact rational | 0.7941296281990526710408181904503566296161166909348186886461887066107299Approximate: 70 significant digits; calculation provenance | 0.2058703718009473289591818095496433703838833090651813113538112933892701Approximate: 70 significant digits; calculation provenance | 0.5882592563981053420816363809007132592322333818696373772923774132214597Approximate: 70 significant digits; calculation provenance | 0.6539510472619481591563743192236908938265165802182742584476057612696071Approximate: 70 significant digits; calculation provenance | -0.9055085713638753831132184292055231145883364578105440907601100564504761Approximate: 70 significant digits; calculation provenance | 27/40Exact rational | -27/40Exact rational | 27/40Exact rational | false | true |
148 | 7/10Exact rational | 0.8021838885585817481543435915519132375077833237084304443062099967997693Approximate: 70 significant digits; calculation provenance | 0.1978161114414182518456564084480867624922166762915695556937900032002307Approximate: 70 significant digits; calculation provenance | 0.6043677771171634963086871831038264750155666474168608886124199935995386Approximate: 70 significant digits; calculation provenance | 0.6347395899824585873670147578510211110212312312177218059998864096509017Approximate: 70 significant digits; calculation provenance | -0.9204174099184509266048326564310639391679608718224685721070149380798202Approximate: 70 significant digits; calculation provenance | 7/10Exact rational | -7/10Exact rational | 7/10Exact rational | false | true |
149 | 29/40Exact rational | 0.8099984339846870248394075349887686000821780732944378833248947571402741Approximate: 70 significant digits; calculation provenance | 0.1900015660153129751605924650112313999178219267055621166751052428597259Approximate: 70 significant digits; calculation provenance | 0.6199968679693740496788150699775372001643561465888757666497895142805483Approximate: 70 significant digits; calculation provenance | 0.6156038837081665625563785524923765237708175593139090827411753136428732Approximate: 70 significant digits; calculation provenance | -0.9357229646697597712414249085641459477171191862113566591504281604105559Approximate: 70 significant digits; calculation provenance | 29/40Exact rational | -29/40Exact rational | 29/40Exact rational | false | true |
150 | 3/4Exact rational | 0.8175744761936436596072171786562482475462408435809046166100549614951142Approximate: 70 significant digits; calculation provenance | 0.1824255238063563403927828213437517524537591564190953833899450385048858Approximate: 70 significant digits; calculation provenance | 0.6351489523872873192144343573124964950924816871618092332201099229902284Approximate: 70 significant digits; calculation provenance | 0.5965858082813314260061336417360454535406025074814188237601855682426663Approximate: 70 significant digits; calculation provenance | -0.9514132779827524094994828090570911052543986289505390097295348213966711Approximate: 70 significant digits; calculation provenance | 3/4Exact rational | -3/4Exact rational | 3/4Exact rational | true | true |
151 | 31/40Exact rational | 0.8249137318359601818765623979374516681400164444125957182508938648203650Approximate: 70 significant digits; calculation provenance | 0.1750862681640398181234376020625483318599835555874042817491061351796350Approximate: 70 significant digits; calculation provenance | 0.6498274636719203637531247958749033362800328888251914365017877296407300Approximate: 70 significant digits; calculation provenance | 0.5777242674577190199170625359223058572054918778283332427078253104351313Approximate: 70 significant digits; calculation provenance | -0.9674764655865786937684008428131873077983022021440413704718960215536197Approximate: 70 significant digits; calculation provenance | 31/40Exact rational | -31/40Exact rational | 31/40Exact rational | false | true |
152 | 4/5Exact rational | 0.8320183851339244818424223282001214250373276463544575201590375381774436Approximate: 70 significant digits; calculation provenance | 0.1679816148660755181575776717998785749626723536455424798409624618225564Approximate: 70 significant digits; calculation provenance | 0.6640367702678489636848446564002428500746552927089150403180750763548872Approximate: 70 significant digits; calculation provenance | 0.5590551677322439785419938454774005897068583570510465372645745030074800Approximate: 70 significant digits; calculation provenance | -0.9839007408883388297074952119288453425649025245752130727019753837225414Approximate: 70 significant digits; calculation provenance | 4/5Exact rational | -4/5Exact rational | 4/5Exact rational | false | true |
153 | 33/40Exact rational | 0.8388910504234147628085474032164679716500658492339220275311277461466578Approximate: 70 significant digits; calculation provenance | 0.1611089495765852371914525967835320283499341507660779724688722538533422Approximate: 70 significant digits; calculation provenance | 0.6777821008468295256170948064329359433001316984678440550622554922933156Approximate: 70 significant digits; calculation provenance | 0.5406114237716582108533631269367085053651654357047381504029653940668138Approximate: 70 significant digits; calculation provenance | -1.000674437414932412380631400546103626442253025579667965707582318225750Approximate: 70 significant digits; calculation provenance | 33/40Exact rational | -33/40Exact rational | 33/40Exact rational | false | true |
154 | 17/20Exact rational | 0.8455347349164652956660462346085957614559371124648422805604042022732675Approximate: 70 significant digits; calculation provenance | 0.1544652650835347043339537653914042385440628875351577194395957977267325Approximate: 70 significant digits; calculation provenance | 0.6910694698329305913320924692171915229118742249296845611208084045465350Approximate: 70 significant digits; calculation provenance | 0.5224229878648322353747763393015239827662447979769864786142706729892168Approximate: 70 significant digits; calculation provenance | -1.017786029386265917004277511513088651967798911693834108936276857787181Approximate: 70 significant digits; calculation provenance | 17/20Exact rational | -17/20Exact rational | 17/20Exact rational | false | true |
155 | 7/8Exact rational | 0.8519528019683105302938151348156768745537762465136143666012463030315942Approximate: 70 significant digits; calculation provenance | 0.1480471980316894697061848651843231254462237534863856333987536969684058Approximate: 70 significant digits; calculation provenance | 0.7039056039366210605876302696313537491075524930272287332024926060631883Approximate: 70 significant digits; calculation provenance | 0.5045169007466207652518701502918813306505570518478332779796687528123639Approximate: 70 significant digits; calculation provenance | -1.035224150438087226886947638647396192500601405696264831480236870991777Approximate: 70 significant digits; calculation provenance | 7/8Exact rational | -7/8Exact rational | 7/8Exact rational | false | true |
156 | 9/10Exact rational | 0.8581489350995122104057218915290474315874325823003808440494433324346932Approximate: 70 significant digits; calculation provenance | 0.1418510649004877895942781084709525684125674176996191559505566675653068Approximate: 70 significant digits; calculation provenance | 0.7162978701990244208114437830580948631748651646007616880988866648693863Approximate: 70 significant digits; calculation provenance | 0.4869173611483415625499475730621584692698779641582036888502490396166801Approximate: 70 significant digits; calculation provenance | -1.052977610526074134599376863819689551783095387594252033795913451416605Approximate: 70 significant digits; calculation provenance | 9/10Exact rational | -9/10Exact rational | 9/10Exact rational | false | true |
157 | 37/40Exact rational | 0.8641271029909057350362534808972573002620989618787729997213503367033882Approximate: 70 significant digits; calculation provenance | 0.1358728970090942649637465191027426997379010381212270002786496632966118Approximate: 70 significant digits; calculation provenance | 0.7282542059818114700725069617945146005241979237575459994427006734067765Approximate: 70 significant digits; calculation provenance | 0.4696458114698013108409765375968035589740456202796731827196321861849879Approximate: 70 significant digits; calculation provenance | -1.071035411054510086153469042118751038760047640465269275953081661361360Approximate: 70 significant digits; calculation provenance | 37/40Exact rational | -37/40Exact rational | 37/40Exact rational | false | true |
158 | 19/20Exact rational | 0.8698915256370021478446740089258733633844524445474101898654343954529645Approximate: 70 significant digits; calculation provenance | 0.1301084743629978521553259910741266366155475554525898101345656045470355Approximate: 70 significant digits; calculation provenance | 0.7397830512740042956893480178517467267689048890948203797308687909059289Approximate: 70 significant digits; calculation provenance | 0.4527210370477239312205775962494490208258410959433593543602364696865015Approximate: 70 significant digits; calculation provenance | -1.089386758282960585104792768183276506319006505692506172850167491514701Approximate: 70 significant digits; calculation provenance | 19/20Exact rational | -19/20Exact rational | 19/20Exact rational | false | true |
159 | 39/40Exact rational | 0.8754466418125835784163604410153004038786818236029994445763305190245369Approximate: 70 significant digits; calculation provenance | 0.1245533581874164215836395589846995961213181763970005554236694809754631Approximate: 70 significant digits; calculation provenance | 0.7508932836251671568327208820306008077573636472059988891526610380490739Approximate: 70 significant digits; calculation provenance | 0.4361592766066142729733706961564650781860541957654981076443021805371937Approximate: 70 significant digits; calculation provenance | -1.108021075072867232479514039033198250297835043668413313951281371996274Approximate: 70 significant digits; calculation provenance | 39/40Exact rational | -39/40Exact rational | 39/40Exact rational | false | true |
160 | 1Exact rational | 0.8807970779778824440597291413023967952063842986289682757984052500609766Approximate: 70 significant digits; calculation provenance | 0.1192029220221175559402708586976032047936157013710317242015947499390234Approximate: 70 significant digits; calculation provenance | 0.7615941559557648881194582826047935904127685972579365515968105001219532Approximate: 70 significant digits; calculation provenance | 0.4199743416140260693944967390417014449171867282307709547133114402445899Approximate: 70 significant digits; calculation provenance | -1.126928011042972496443726806358304431434333062808358357550141216787973Approximate: 70 significant digits; calculation provenance | 1Exact rational | -1Exact rational | 1Exact rational | false | true |
161 | 41/40Exact rational | 0.8859476187202091437631493711611284372281203977435567078708006501888541Approximate: 70 significant digits; calculation provenance | 0.1140523812797908562368506288388715627718796022564432921291993498111459Approximate: 70 significant digits; calculation provenance | 0.7718952374404182875262987423222568744562407954871134157416013003777082Approximate: 70 significant digits; calculation provenance | 0.4041777424168002739475387091371216486688056146628395244464182021374122Approximate: 70 significant digits; calculation provenance | -1.146097451208061624833917775402162200925666554728626436335095569985020Approximate: 70 significant digits; calculation provenance | 41/40Exact rational | -41/40Exact rational | 41/40Exact rational | false | true |
162 | 21/20Exact rational | 0.8909031788043870617008939275915296080842943801403262592235735500588843Approximate: 70 significant digits; calculation provenance | 0.1090968211956129382991060724084703919157056198596737407764264499411157Approximate: 70 significant digits; calculation provenance | 0.7818063576087741234017878551830592161685887602806525184471471001177686Approximate: 70 significant digits; calculation provenance | 0.3887788192025015913241536459813625846802510088548704639266704462920162Approximate: 70 significant digits; calculation provenance | -1.165519523179754965644273927488443324891055424040716590427872938537194Approximate: 70 significant digits; calculation provenance | 21/20Exact rational | -21/20Exact rational | 21/20Exact rational | false | true |
163 | 43/40Exact rational | 0.8956687768809987026300964475253722239036711622623848335890566194350684Approximate: 70 significant digits; calculation provenance | 0.1043312231190012973699035524746277760963288377376151664109433805649316Approximate: 70 significant digits; calculation provenance | 0.7913375537619974052601928950507444478073423245247696671781132388701368Approximate: 70 significant digits; calculation provenance | 0.3737848760059778662774592066218495089636934512248804703412976172348003Approximate: 70 significant digits; calculation provenance | -1.185184603011108871840814495071048675307276852600753041551481344282117Approximate: 70 significant digits; calculation provenance | 43/40Exact rational | -43/40Exact rational | 43/40Exact rational | false | true |
164 | 11/10Exact rational | 0.9002495108803148530057306653003482290235795509317621683948731712348890Approximate: 70 significant digits; calculation provenance | 0.09975048911968514699426933469965177097642044906823783160512682876511098Approximate: 70 significant digits; calculation provenance | 0.8004990217606297060114613306006964580471591018635243367897463424697780Approximate: 70 significant digits; calculation provenance | 0.3592013161602748884100572304824881937796786715955766009770588764284391Approximate: 70 significant digits; calculation provenance | -1.205083319768695917520491879219231837591287215137043902047306230238518Approximate: 70 significant digits; calculation provenance | 11/10Exact rational | -11/10Exact rational | 11/10Exact rational | false | true |
165 | 9/8Exact rational | 0.9046505351008905060303852367716634986982956075939651081987707506429891Approximate: 70 significant digits; calculation provenance | 0.09534946489910949396961476322833650130170439240603489180122924935701094Approximate: 70 significant digits; calculation provenance | 0.8093010702017810120607704735433269973965912151879302163975415012859781Approximate: 70 significant digits; calculation provenance | 0.3450317777702519220263555246456887399878264851737243615202901172103272Approximate: 70 significant digits; calculation provenance | -1.225206558916747211673152826784967298131435400418382604358125662701238Approximate: 70 significant digits; calculation provenance | 9/8Exact rational | -9/8Exact rational | 9/8Exact rational | false | true |
166 | 23/20Exact rational | 0.9088770389851438646606952530235892493026450058746171841759544578649052Approximate: 70 significant digits; calculation provenance | 0.09112296101485613533930474697641075069735499412538281582404554213509480Approximate: 70 significant digits; calculation provenance | 0.8177540779702877293213905060471784986052900117492343683519089157298104Approximate: 70 significant digits; calculation provenance | 0.3312782679629645770272626630699932872649235163333968824499428483084358Approximate: 70 significant digits; calculation provenance | -1.245545464597962942695928157963484281043240516276454399297256530874566Approximate: 70 significant digits; calculation provenance | 23/20Exact rational | -23/20Exact rational | 23/20Exact rational | false | true |
167 | 47/40Exact rational | 0.9129342275597287402723311562493252454772758150542965834671442360532501Approximate: 70 significant digits; calculation provenance | 0.08706577244027125972766884375067475452272418494570341653285576394674985Approximate: 70 significant digits; calculation provenance | 0.8258684551194574805446623124986504909545516301085931669342884721065003Approximate: 70 significant digits; calculation provenance | 0.3179412948386006451945042653019462839999663232922046906603801862139635Approximate: 70 significant digits; calculation provenance | -1.266091440894841611285156340979529644746285639052160118424711008181297Approximate: 70 significant digits; calculation provenance | 47/40Exact rational | -47/40Exact rational | 47/40Exact rational | false | true |
168 | 6/5Exact rational | 0.9168273035060776293370475609061571891097836474258172223161001830069703Approximate: 70 significant digits; calculation provenance | 0.08317269649392237066295243909384281089021635257418277768389981699302972Approximate: 70 significant digits; calculation provenance | 0.8336546070121552586740951218123143782195672948516344446322003660139406Approximate: 70 significant digits; calculation provenance | 0.3050199962074089762139800299764416938310900832609193254586195024148496Approximate: 70 significant digits; calculation provenance | -1.286836152153949678480990395226517825132790577206467223706689803126962Approximate: 70 significant digits; calculation provenance | 6/5Exact rational | -6/5Exact rational | 6/5Exact rational | false | true |
169 | 49/40Exact rational | 0.9205614508160216255983766417911688079560481387029489304166520264863809Approximate: 70 significant digits; calculation provenance | 0.07943854918397837440162335820883119204395186129705106958334797351361907Approximate: 70 significant digits; calculation provenance | 0.8411229016320432511967532835823376159120962774058978608333040529727619Approximate: 70 significant digits; calculation provenance | 0.2925122643500920925923516412805348532458645958323277859498379111373692Approximate: 70 significant digits; calculation provenance | -1.307771522453552522417881672566973000167092436651002968434036336641785Approximate: 70 significant digits; calculation provenance | 49/40Exact rational | -49/40Exact rational | 49/40Exact rational | false | true |
170 | 5/4Exact rational | 0.9241418199787564488066938233537522268692879603392157611636726415179167Approximate: 70 significant digits; calculation provenance | 0.07585818002124355119330617664624777313071203966078423883632735848208332Approximate: 70 significant digits; calculation provenance | 0.8482836399575128976133876467075044537385759206784315223273452830358334Approximate: 70 significant digits; calculation provenance | 0.2804148661804326277293312747706377322057927192456265986022007246598311Approximate: 70 significant digits; calculation provenance | -1.328889734292549623344043916717550997259079421066961476183422357091817Approximate: 70 significant digits; calculation provenance | 5/4Exact rational | -5/4Exact rational | 5/4Exact rational | false | true |
171 | 51/40Exact rational | 0.9275735146384822821560764938523071584369295285143345553093219914487329Approximate: 70 significant digits; calculation provenance | 0.07242648536151771784392350614769284156307047148566544469067800855126713Approximate: 70 significant digits; calculation provenance | 0.8551470292769645643121529877046143168738590570286691106186439828974657Approximate: 70 significant digits; calculation provenance | 0.2687235583187823103036565180572303462896585013792145541192349913068698Approximate: 70 significant digits; calculation provenance | -1.350183226575790057019486886367011038020169445520365140023040521978744Approximate: 70 significant digits; calculation provenance | 51/40Exact rational | -51/40Exact rational | 51/40Exact rational | false | true |
172 | 13/10Exact rational | 0.9308615796566531821474768579683758994233855523332933927880191872644083Approximate: 70 significant digits; calculation provenance | 0.06913842034334681785252314203162410057661444766670660721198081273559173Approximate: 70 significant digits; calculation provenance | 0.8617231593133063642949537159367517988467711046665867855760383745288165Approximate: 70 significant digits; calculation provenance | 0.2574331967030940189517344743806649427337928787943978188017380394382650Approximate: 70 significant digits; calculation provenance | -1.371644691967669814406403590902148677628789957962867152218692838435794Approximate: 70 significant digits; calculation provenance | 13/10Exact rational | -13/10Exact rational | 13/10Exact rational | false | true |
173 | 53/40Exact rational | 0.9340109905087812213895139323877826164172933296185007816985487861598835Approximate: 70 significant digits; calculation provenance | 0.06598900949121877861048606761221738358270667038149921830145121384011651Approximate: 70 significant digits; calculation provenance | 0.8680219810175624427790278647755652328345866592370015633970975723197670Approximate: 70 significant digits; calculation provenance | 0.2465378404703464662551897357302608644104978565313277112291784915089075Approximate: 70 significant digits; calculation provenance | -1.393267073682503959180741662935593674501532284687038912920738358690038Approximate: 70 significant digits; calculation provenance | 53/40Exact rational | -53/40Exact rational | 53/40Exact rational | false | true |
174 | 27/20Exact rational | 0.9370266439430035104858436714910610936135574908373066357093903057211929Approximate: 70 significant digits; calculation provenance | 0.06297335605699648951415632850893890638644250916269336429060969427880706Approximate: 70 significant digits; calculation provenance | 0.8740532878860070209716873429821221872271149816746132714187806114423859Approximate: 70 significant digits; calculation provenance | 0.2360308499356609322442290466492354248382291922071062348869784731620760Approximate: 70 significant digits; calculation provenance | -1.415043561776590548309753637919068996116779308821747734007900794856700Approximate: 70 significant digits; calculation provenance | 27/20Exact rational | -27/20Exact rational | 27/20Exact rational | false | true |
175 | 11/8Exact rational | 0.9399133498259923779802765544050912866374799888357889920836563737801878Approximate: 70 significant digits; calculation provenance | 0.06008665017400762201972344559490871336252001116421100791634362621981223Approximate: 70 significant digits; calculation provenance | 0.8798266996519847559605531088101825732749599776715779841673127475603755Approximate: 70 significant digits; calculation provenance | 0.2259049785794962073047278461177565053894799590474085725141825459907048Approximate: 70 significant digits; calculation provenance | -1.436967589003198624576485597894303647745849723995463673294254484297896Approximate: 70 significant digits; calculation provenance | 11/8Exact rational | -11/8Exact rational | 11/8Exact rational | false | true |
176 | 7/5Exact rational | 0.9426758241011312537917088260127869647278385521455363957507079102528633Approximate: 70 significant digits; calculation provenance | 0.05732417589886874620829117398721303527216144785446360424929208974713673Approximate: 70 significant digits; calculation provenance | 0.8853516482022625075834176520255739294556771042910727915014158205057265Approximate: 70 significant digits; calculation provenance | 0.2161524590255372071239072466398030836272924866539352039681529625277187Approximate: 70 significant digits; calculation provenance | -1.459032826287971399138171852604277883648122327295520366156354995283702Approximate: 70 significant digits; calculation provenance | 7/5Exact rational | -7/5Exact rational | 7/5Exact rational | false | true |
177 | 57/40Exact rational | 0.9453186827840592427993718343130103799060483361446773444468580617233926Approximate: 70 significant digits; calculation provenance | 0.05468131721594075720062816568698962009395166385532265555314193827660736Approximate: 70 significant digits; calculation provenance | 0.8906373655681184855987436686260207598120966722893546888937161234467853Approximate: 70 significant digits; calculation provenance | 0.2067650830538816726343303696499654828865047773281405960574869821568330Approximate: 70 significant digits; calculation provenance | -1.481233177878483152522616321616629002161295426363029610607893789829096Approximate: 70 significant digits; calculation provenance | 57/40Exact rational | -57/40Exact rational | 57/40Exact rational | false | true |
178 | 29/20Exact rational | 0.9478464369215822652200446750110949359300856618502870258045562932513311Approximate: 70 significant digits; calculation provenance | 0.05215356307841773477995532498890506406991433814971297419544370674866888Approximate: 70 significant digits; calculation provenance | 0.8956928738431645304400893500221898718601713237005740516091125865026622Approximate: 70 significant digits; calculation provenance | 0.1977342757465729489259144060115879197115135238464029596274780190343313Approximate: 70 significant digits; calculation provenance | -1.503562776217963173260831475323559556222954003130506337237796936451904Approximate: 70 significant digits; calculation provenance | 29/20Exact rational | -29/20Exact rational | 29/20Exact rational | false | true |
179 | 59/40Exact rational | 0.9502634884414432748160285585756681146730381275247192605566134186732140Approximate: 70 significant digits; calculation provenance | 0.04973651155855672518397144142433188532696187247528073944338658132678605Approximate: 70 significant digits; calculation provenance | 0.9005269768828865496320571171513362293460762550494385211132268373464279Approximate: 70 significant digits; calculation provenance | 0.1890511639061691138380507237436600467907108772761312424859360041680161Approximate: 70 significant digits; calculation provenance | -1.526015976589535040680222366025510802241052462267022728069589187060076Approximate: 70 significant digits; calculation provenance | 59/40Exact rational | -59/40Exact rational | 59/40Exact rational | false | true |
180 | 3/2Exact rational | 0.9525741268224332191211518482282477986138205675793908992821119912255513Approximate: 70 significant digits; calculation provenance | 0.04742587317756678087884815177175220138617943242060910071788800877444871Approximate: 70 significant digits; calculation provenance | 0.9051482536448664382423036964564955972276411351587817985642239824511026Approximate: 70 significant digits; calculation provenance | 0.1807066389236485305974401137426065519794398095899587698581913111392369Approximate: 70 significant digits; calculation provenance | -1.548587351573742058758925919854689997941881642828108620530846980487520Approximate: 70 significant digits; calculation provenance | 3/2Exact rational | -3/2Exact rational | 3/2Exact rational | true | true |
181 | 61/40Exact rational | 0.9547825265167125105226364109459168917035700456149255873204038762611887Approximate: 70 significant digits; calculation provenance | 0.04521747348328748947736358905408310829642995438507441267959612373881131Approximate: 70 significant digits; calculation provenance | 0.9095650530334250210452728218918337834071400912298511746408077525223774Approximate: 70 significant digits; calculation provenance | 0.1726914143003027289217433886811123580092109684043552398415875543792194Approximate: 70 significant digits; calculation provenance | -1.571271685358661957824396600243818042314316640329728315560630694995370Approximate: 70 significant digits; calculation provenance | 61/40Exact rational | -61/40Exact rational | 61/40Exact rational | false | true |
182 | 31/20Exact rational | 0.9568927450589138774462287563288605498245667374562946417225848323693777Approximate: 70 significant digits; calculation provenance | 0.04310725494108612255377124367113945017543326254370535827741516763062234Approximate: 70 significant digits; calculation provenance | 0.9137854901178277548924575126577210996491334749125892834451696647387553Approximate: 70 significant digits; calculation provenance | 0.1649960780501213145061082491607692000373033713438522013281455034850210Approximate: 70 significant digits; calculation provenance | -1.594063967938573859254143722314314082545703520469818851327848644953224Approximate: 70 significant digits; calculation provenance | 31/20Exact rational | -31/20Exact rational | 31/20Exact rational | false | true |
183 | 63/40Exact rational | 0.9589087217995349892299968327284785924379306964134046043121362352333242Approximate: 70 significant digits; calculation provenance | 0.04109127820046501077000316727152140756206930358659539568786376476667578Approximate: 70 significant digits; calculation provenance | 0.9178174435990699784599936654569571848758613928268092086242724704666484Approximate: 70 significant digits; calculation provenance | 0.1576111402252679990246822332561389857310456743849833802725515129540452Approximate: 70 significant digits; calculation provenance | -1.616959389233941642269206912630506652405874971989012235472545150172495Approximate: 70 significant digits; calculation provenance | 63/40Exact rational | -63/40Exact rational | 63/40Exact rational | false | true |
184 | 8/5Exact rational | 0.9608342772032356413416321115444125346540774371252269930734448647029485Approximate: 70 significant digits; calculation provenance | 0.03916572279676435865836788845558746534592256287477300692655513529705145Approximate: 70 significant digits; calculation provenance | 0.9216685544064712826832642230888250693081548742504539861468897294058971Approximate: 70 significant digits; calculation provenance | 0.1505270758182854851283623433654002757494587831390396389712535129653613Approximate: 70 significant digits; calculation provenance | -1.639953333162430357063348799134171653799273521482867056096408457711270Approximate: 70 significant digits; calculation provenance | 8/5Exact rational | -8/5Exact rational | 8/5Exact rational | false | true |
185 | 13/8Exact rational | 0.9626731126558705398022858339615018547397777978848127962541807986866614Approximate: 70 significant digits; calculation provenance | 0.03732688734412946019771416603849814526022220211518720374581920131333861Approximate: 70 significant digits; calculation provenance | 0.9253462253117410796045716679230037094795555957696255925083615973733228Approximate: 70 significant digits; calculation provenance | 0.1437343633013125125237877279779659639963923972703809244601636889818485Approximate: 70 significant digits; calculation provenance | -1.663041371687783128561208561610680358286802282251023153527884588912428Approximate: 70 significant digits; calculation provenance | 13/8Exact rational | -13/8Exact rational | 13/8Exact rational | false | true |
186 | 33/20Exact rational | 0.9644288107273638272226435259593718640522279374501320586102389864724408Approximate: 70 significant digits; calculation provenance | 0.03557118927263617277735647404062813594777206254986794138976101352755918Approximate: 70 significant digits; calculation provenance | 0.9288576214547276544452870519187437281044558749002641172204779729448816Approximate: 70 significant digits; calculation provenance | 0.1372235190654658641713039488679920416589949262585419732651662322077304Approximate: 70 significant digits; calculation provenance | -1.686219258870659275671226786055661309663649211230079867661478491790305Approximate: 70 significant digits; calculation provenance | 33/20Exact rational | -33/20Exact rational | 33/20Exact rational | false | true |
187 | 67/40Exact rational | 0.9661048358408218513403455262774547963981249566509043356320081121712452Approximate: 70 significant digits; calculation provenance | 0.03389516415917814865965447372254520360187504334909566436799188782875483Approximate: 70 significant digits; calculation provenance | 0.9322096716816437026806910525549095927962499133018086712640162243424903Approximate: 70 significant digits; calculation provenance | 0.1309851280232020549049842142860456957023772213280774911929341443239059Approximate: 70 significant digits; calculation provenance | -1.709482924942971955040031191245815340189309123522972394334716965567623Approximate: 70 significant digits; calculation provenance | 67/40Exact rational | -67/40Exact rational | 67/40Exact rational | false | true |
188 | 17/10Exact rational | 0.9677045353015494902612243788730029800202887562873549923007501854443228Approximate: 70 significant digits; calculation provenance | 0.03229546469845050973877562112699701997971124371264500769924981455567723Approximate: 70 significant digits; calculation provenance | 0.9354090706030989805224487577460059600405775125747099846015003708886455Approximate: 70 significant digits; calculation provenance | 0.1250098706334465866593678064810895113372531617800235200467952294771509Approximate: 70 significant digits; calculation provenance | -1.732828470424865341747239134360648684016082194864521100075579571838508Approximate: 70 significant digits; calculation provenance | 17/10Exact rational | -17/10Exact rational | 17/10Exact rational | false | true |
189 | 69/40Exact rational | 0.9692311406428519908334698614073846713192393642350255058762115193115438Approximate: 70 significant digits; calculation provenance | 0.03076885935714800916653013859261532868076063576497449412378848068845621Approximate: 70 significant digits; calculation provenance | 0.9384622812857039816669397228147693426384787284700510117524230386230876Approximate: 70 significant digits; calculation provenance | 0.1192885466040322182664963657351446152020045407746583792227947280911648Approximate: 70 significant digits; calculation provenance | -1.756252160301235314762219806312722994475562687544073529641077836648716Approximate: 70 significant digits; calculation provenance | 69/40Exact rational | -69/40Exact rational | 69/40Exact rational | false | true |
190 | 7/4Exact rational | 0.9706877692486436811347104418858184161058178702016199487234252298457515Approximate: 70 significant digits; calculation provenance | 0.02931223075135631886528955811418158389418212979838005127657477015424853Approximate: 70 significant digits; calculation provenance | 0.9413755384972873622694208837716368322116357404032398974468504596915029Approximate: 70 significant digits; calculation provenance | 0.1138120955189422393587513085092015706827119559983316401207377855625312Approximate: 70 significant digits; calculation provenance | -1.779750418272620565194811174081749828815391172841522460200582523441863Approximate: 70 significant digits; calculation provenance | 7/4Exact rational | -7/4Exact rational | 7/4Exact rational | false | true |
191 | 71/40Exact rational | 0.9720774262159269810308901516868137293002721723691852044809014422882774Approximate: 70 significant digits; calculation provenance | 0.02792257378407301896910984831318627069972782763081479551909855771172258Approximate: 70 significant digits; calculation provenance | 0.9441548524318539620617803033736274586005443447383704089618028845765548Approximate: 70 significant digits; calculation provenance | 0.1085716146293840685413945610750577506689910796460042700045401696654224Approximate: 70 significant digits; calculation provenance | -1.803319821093368606461859226303512841511516059896573667181618776594244Approximate: 70 significant digits; calculation provenance | 71/40Exact rational | -71/40Exact rational | 71/40Exact rational | false | true |
192 | 9/5Exact rational | 0.9734030064231341448231991573987218818696951160323548177660556830159704Approximate: 70 significant digits; calculation provenance | 0.02659699357686585517680084260127811813030488396764518223394431698402956Approximate: 70 significant digits; calculation provenance | 0.9468060128462682896463983147974437637393902320647096355321113660319409Approximate: 70 significant digits; calculation provenance | 0.1035583740381520464794954247451830513492124417342322554844778389714661Approximate: 70 significant digits; calculation provenance | -1.826957093008208060442969061093719005900269587573282563293715466410106Approximate: 70 significant digits; calculation provenance | 9/5Exact rational | -9/5Exact rational | 9/5Exact rational | false | true |
193 | 73/40Exact rational | 0.9746672967731282705834741565502417837185389966947813972714778499372055Approximate: 70 significant digits; calculation provenance | 0.02533270322687172941652584344975821628146100330521860272852215006279448Approximate: 70 significant digits; calculation provenance | 0.9493345935462565411669483131004835674370779933895627945429556998744110Approximate: 70 significant digits; calculation provenance | 0.09876382949636388833697854749375383169988500287439141521105342737022724Approximate: 70 significant digits; calculation provenance | -1.850659100296728858463986402142599168487957101849038603473991099364792Approximate: 70 significant digits; calculation provenance | 73/40Exact rational | -73/40Exact rational | 73/40Exact rational | false | true |
194 | 37/20Exact rational | 0.9758729785823307987196197518628053328276957913301583814534859557547124Approximate: 70 significant digits; calculation provenance | 0.02412702141766920128038024813719466717230420866984161854651404424528761Approximate: 70 significant digits; calculation provenance | 0.9517459571646615974392395037256106656553915826603167629069719115094248Approximate: 70 significant digits; calculation provenance | 0.09417963302072213170097105667768505772198026625520402797105572461535931Approximate: 70 significant digits; calculation provenance | -1.874422845933779159494329738188660388424823583012678237701184241482271Approximate: 70 significant digits; calculation provenance | 37/20Exact rational | -37/20Exact rational | 37/20Exact rational | false | true |
195 | 15/8Exact rational | 0.9770226300899743850460961133098414086082905701001975761982230105668617Approximate: 70 significant digits; calculation provenance | 0.02297736991002561495390388669015859139170942989980242380177698943313830Approximate: 70 significant digits; calculation provenance | 0.9540452601799487700921922266196828172165811402003951523964460211337234Approximate: 70 significant digits; calculation provenance | 0.08979764152817385766904692651159891548610297142722295683956647020475296Approximate: 70 significant digits; calculation provenance | -1.898245464372425028195870552274882442648579480000923092040701284398497Approximate: 70 significant digits; calculation provenance | 15/8Exact rational | -15/8Exact rational | 15/8Exact rational | false | true |
196 | 19/10Exact rational | 0.9781187290638695260477487030143265173402669443677320756788774762611401Approximate: 70 significant digits; calculation provenance | 0.02188127093613047395225129698567348265973305563226792432112252373885990Approximate: 70 significant digits; calculation provenance | 0.9562374581277390520954974060286530346805338887354641513577549525222802Approximate: 70 significant digits; calculation provenance | 0.08560992367340050305742670899687769366796266277325939033632392574245715Approximate: 70 significant digits; calculation provenance | -1.922124216454879193076514799413436093318705163779467953688502031120878Approximate: 70 significant digits; calculation provenance | 19/10Exact rational | -19/10Exact rational | 19/10Exact rational | false | true |
197 | 77/40Exact rational | 0.9791636554813195748620559008028789373095666613966255157180281498601763Approximate: 70 significant digits; calculation provenance | 0.02083634451868042513794409919712106269043333860337448428197185013982372Approximate: 70 significant digits; calculation provenance | 0.9583273109626391497241118016057578746191333227932510314360562997203526Approximate: 70 significant digits; calculation provenance | 0.08160876506311712536173525747423981252827228740080525134811426535396623Approximate: 70 significant digits; calculation provenance | -1.946056484455681376998457009211047344763684398054106785457144736120413Approximate: 70 significant digits; calculation provenance | 77/40Exact rational | -77/40Exact rational | 77/40Exact rational | false | true |
198 | 39/20Exact rational | 0.9801596942659224934475868053143434180411919369757837146585033979791932Approximate: 70 significant digits; calculation provenance | 0.01984030573407750655241319468565658195880806302421628534149660202080676Approximate: 70 significant digits; calculation provenance | 0.9603193885318449868951736106286868360823838739515674293170067959583865Approximate: 70 significant digits; calculation provenance | 0.07778667200982335106505862608909093503331500493628392707238446561518654Approximate: 70 significant digits; calculation provenance | -1.970039767260397607455752968097949867813616800392086530423946504923834Approximate: 70 significant digits; calculation provenance | 39/20Exact rational | -39/20Exact rational | 39/20Exact rational | false | true |
199 | 79/40Exact rational | 0.9811090380629609483436722916021153453873784328325122442765780093407536Approximate: 70 significant digits; calculation provenance | 0.01889096193703905165632770839788465461262156716748775572342199065924643Approximate: 70 significant digits; calculation provenance | 0.9622180761259218966873445832042306907747568656650244885531560186815071Approximate: 70 significant digits; calculation provenance | 0.07413637397652957367060871913506817660786945503632755832141743041467197Approximate: 70 significant digits; calculation provenance | -1.994071675682192492026318535190050501756017611297832060372767860251917Approximate: 70 significant digits; calculation provenance | 79/40Exact rational | -79/40Exact rational | 79/40Exact rational | false | true |
200 | 2Exact rational | 0.9820137900379084419732068620504615751275149881204673880241316087065540Approximate: 70 significant digits; calculation provenance | 0.01798620996209155802679313794953842487248501187953261197586839129344603Approximate: 70 significant digits; calculation provenance | 0.9640275800758168839464137241009231502550299762409347760482632174131079Approximate: 70 significant digits; calculation provenance | 0.07065082485316446568624765586105479231212725983765596112486380982686206Approximate: 70 significant digits; calculation provenance | -2.018149927917809740354983318287753884762148873963581175677534475091048Approximate: 70 significant digits; calculation provenance | 2Exact rational | -2Exact rational | 2Exact rational | false | true |
201 | 81/40Exact rational | 0.9828759666842722654802040629568546507360384856179489791633408416076600Approximate: 70 significant digits; calculation provenance | 0.01712403331572773451979593704314534926396151438205102083665915839234000Approximate: 70 significant digits; calculation provenance | 0.9657519333685445309604081259137093014720769712358979583266816832153200Approximate: 70 significant digits; calculation provenance | 0.06732320319491832452098994055902847813804244055189916465601331836200292Approximate: 70 significant digits; calculation provenance | -2.042272345143765470366628439688664986657495695475547519292944091749114Approximate: 70 significant digits; calculation provenance | 81/40Exact rational | -81/40Exact rational | 81/40Exact rational | false | true |
202 | 41/20Exact rational | 0.9836975006285590599250947002733788439212788568880385686394640080479163Approximate: 70 significant digits; calculation provenance | 0.01630249937144094007490529972662115607872114311196143136053599195208372Approximate: 70 significant digits; calculation provenance | 0.9673950012571181198501894005467576878425577137760771372789280160958326Approximate: 70 significant digits; calculation provenance | 0.06414691154274043131470608246046353139812055198227789153841106421171330Approximate: 70 significant digits; calculation provenance | -2.066436847252909523485825735724019227115343951368720722661647574587303Approximate: 70 significant digits; calculation provenance | 41/20Exact rational | -41/20Exact rational | 41/20Exact rational | false | true |
203 | 83/40Exact rational | 0.9844802434215911072503144310401225593129759312551896901934438757258291Approximate: 70 significant digits; calculation provenance | 0.01551975657840889274968556895987744068702406874481030980655612427417089Approximate: 70 significant digits; calculation provenance | 0.9689604868431822145006288620802451186259518625103793803868877514516582Approximate: 70 significant digits; calculation provenance | 0.06111557493662330659087257413601293682126735505946868365006433652878323Approximate: 70 significant digits; calculation provenance | -2.090641448730935860600411643487324066569541141229760743428454469788705Approximate: 70 significant digits; calculation provenance | 83/40Exact rational | -83/40Exact rational | 83/40Exact rational | false | true |
204 | 21/10Exact rational | 0.9852259683067269422494659991331158069320250240589345091945136663065690Approximate: 70 significant digits; calculation provenance | 0.01477403169327305775053400086688419306797497594106549080548633369343096Approximate: 70 significant digits; calculation provenance | 0.9704519366134538844989319982662316138640500481178690183890273326131381Approximate: 70 significant digits; calculation provenance | 0.05822303872319688390625643629354477317692114197532726370052111556975025Approximate: 70 significant digits; calculation provenance | -2.114884254671918127807191357432826018005623844933442979226939024275903Approximate: 70 significant digits; calculation provenance | 21/10Exact rational | -21/10Exact rational | 21/10Exact rational | false | true |
205 | 17/8Exact rational | 0.9859363729567545257562724768743520190252646185178327261647108371697035Approximate: 70 significant digits; calculation provenance | 0.01406362704324547424372752312564798097473538148216727383528916283029646Approximate: 70 significant digits; calculation provenance | 0.9718727459135090515125449537487040380505292370356654523294216743394071Approximate: 70 significant digits; calculation provenance | 0.05546336575053587521381428053334842284415803722670943959387184973634696Approximate: 70 significant digits; calculation provenance | -2.139163456931504975714938383770760116485397733617481604445285369593905Approximate: 70 significant digits; calculation provenance | 17/8Exact rational | -17/8Exact rational | 17/8Exact rational | false | true |
206 | 43/20Exact rational | 0.9866130821723352224796314116859467015478326719390846817034098017243245Approximate: 70 significant digits; calculation provenance | 0.01338691782766477752036858831405329845216732806091531829659019827567549Approximate: 70 significant digits; calculation provenance | 0.9732261643446704449592628233718934030956653438781693634068196034486490Approximate: 70 significant digits; calculation provenance | 0.05283083303496051389744929665550905236784646766469243656316088207633650Approximate: 70 significant digits; calculation provenance | -2.163477330416026301248113488248782362621159896370748971831472377875204Approximate: 70 significant digits; calculation provenance | 43/20Exact rational | -43/20Exact rational | 43/20Exact rational | false | true |
207 | 87/40Exact rational | 0.9872576505358883976759304464935930838093756591507766720136360820402697Approximate: 70 significant digits; calculation provenance | 0.01274234946411160232406955350640691619062434084922332798636391795973025Approximate: 70 significant digits; calculation provenance | 0.9745153010717767953518608929871861676187513183015533440272721640805395Approximate: 70 significant digits; calculation provenance | 0.05031992797698422834057985356994404594722972522867628930466470540732502Approximate: 70 significant digits; calculation provenance | -2.187824229505431179550563771528029172833784612918315531964549179186845Approximate: 70 significant digits; calculation provenance | 87/40Exact rational | -87/40Exact rational | 87/40Exact rational | false | true |
208 | 11/5Exact rational | 0.9878715650157257602071533340150573263906525170289943746575408369586672Approximate: 70 significant digits; calculation provenance | 0.01212843498427423979284666598494267360934748297100562534245916304133280Approximate: 70 significant digits; calculation provenance | 0.9757431300314515204143066680301146527813050340579887493150816739173344Approximate: 70 significant digits; calculation provenance | 0.04792534419642589005438183177647843978959740278429874376297693893828274Approximate: 70 significant digits; calculation provenance | -2.212202584607696098784946568941603330001682446155611976678025089663730Approximate: 70 significant digits; calculation provenance | 11/5Exact rational | -11/5Exact rational | 11/5Exact rational | false | true |
209 | 89/40Exact rational | 0.9884562475160777100145368919842492834888421926537405091925685315511661Approximate: 70 significant digits; calculation provenance | 0.01154375248392228998546310801575071651115780734625949080743146844883387Approximate: 70 significant digits; calculation provenance | 0.9769124950321554200290737839684985669776843853074810183851370631023323Approximate: 70 significant digits; calculation provenance | 0.04564197705004891178221514329123996758997351435807668397223220139938829Approximate: 70 significant digits; calculation provenance | -2.236610898842103741884889908335484168247435006266814082094457663177413Approximate: 70 significant digits; calculation provenance | 89/40Exact rational | -89/40Exact rational | 89/40Exact rational | false | true |
210 | 9/4Exact rational | 0.9890130573694068199613646215030915829323766155856412867874143107558973Approximate: 70 significant digits; calculation provenance | 0.01098694263059318003863537849690841706762338441435871321258568924410270Approximate: 70 significant digits; calculation provenance | 0.9780261147388136399227292430061831658647532311712825735748286215117946Approximate: 70 significant digits; calculation provenance | 0.04346491888890093700651012184112977910849967105229201430856291891240252Approximate: 70 significant digits; calculation provenance | -2.261047744848593816321644218716262933425798414904328961905696305324554Approximate: 70 significant digits; calculation provenance | 9/4Exact rational | -9/4Exact rational | 9/4Exact rational | false | true |
211 | 91/40Exact rational | 0.9895432937680819306088174338396464842729677205360013463422902815404751Approximate: 70 significant digits; calculation provenance | 0.01045670623191806939118256616035351572703227946399865365770971845952493Approximate: 70 significant digits; calculation provenance | 0.9790865875361638612176348676792929685459354410720026926845805630809501Approximate: 70 significant digits; calculation provenance | 0.04138945410678974080789674030756565077285084139568256165698378917621376Approximate: 70 significant digits; calculation provenance | -2.285511761720224447004342689608316839619116789362933632028241714537530Approximate: 70 significant digits; calculation provenance | 91/40Exact rational | -91/40Exact rational | 91/40Exact rational | false | true |
212 | 23/10Exact rational | 0.9900481981330956784270509615505628187833830908345643531878837220042642Approximate: 70 significant digits; calculation provenance | 0.009951801866904321572949038449437181216616909165435646812116277995735790Approximate: 70 significant digits; calculation provenance | 0.9800963962661913568541019231011256375667661816691287063757674440085284Approximate: 70 significant digits; calculation provenance | 0.03941105402602480493103177624000740328073769971724618794406231752561093Approximate: 70 significant digits; calculation provenance | -2.310001652055651831254680117182091813836876559564107587209782541401929Approximate: 70 significant digits; calculation provenance | 23/10Exact rational | -23/10Exact rational | 23/10Exact rational | false | true |
213 | 93/40Exact rational | 0.9905289564180538896977952877781656327137174253850029152230090797738115Approximate: 70 significant digits; calculation provenance | 0.009471043581946110302204712221834367286282574614997084776990920226188461Approximate: 70 significant digits; calculation provenance | 0.9810579128361077793955905755563312654274348507700058304460181595476231Approximate: 70 significant digits; calculation provenance | 0.03752537166165995077933435710982926451892240465651946234925713826332787Approximate: 70 significant digits; calculation provenance | -2.334516179128433890997617766135538998466585480699536860001770059584107Approximate: 70 significant digits; calculation provenance | 93/40Exact rational | -93/40Exact rational | 93/40Exact rational | false | true |
214 | 47/20Exact rational | 0.9909867013471521754819807718349318455912268609464037879863296333521349Approximate: 70 significant digits; calculation provenance | 0.009013298652847824518019228165068154408773139053596212013670366647865082Approximate: 70 significant digits; calculation provenance | 0.9819734026943043509639615436698636911824537218928075759726592667042698Approximate: 70 significant digits; calculation provenance | 0.03572823640096958443897487536317688565195252494626068086123104102527095Approximate: 70 significant digits; calculation provenance | -2.359054164169887486004273513617968604531616212752084297942504324530114Approximate: 70 significant digits; calculation provenance | 47/20Exact rational | -47/20Exact rational | 47/20Exact rational | false | true |
215 | 19/8Exact rational | 0.9914225145862880150117690081007943282349799960493598378773283858391139Approximate: 70 significant digits; calculation provenance | 0.008577485413711984988230991899205671765020003950640162122671614160886069Approximate: 70 significant digits; calculation provenance | 0.9828450291725760300235380162015886564699599920987196757546567716782279Approximate: 70 significant digits; calculation provenance | 0.03401564863075817250383932942880915939057652961384290936617588721635706Approximate: 70 significant digits; calculation provenance | -2.383614483762175557829804739019990758022824225207226103099087749411652Approximate: 70 significant digits; calculation provenance | 19/8Exact rational | -19/8Exact rational | 19/8Exact rational | false | true |
216 | 12/5Exact rational | 0.9918374288468401048198645749669374910646966514884225961825477091485508Approximate: 70 significant digits; calculation provenance | 0.008162571153159895180135425033062508935303348511577403817452290851449158Approximate: 70 significant digits; calculation provenance | 0.9836748576936802096397291499338749821293933029768451923650954182971017Approximate: 70 significant digits; calculation provenance | 0.03238377434131798847662658059430935272768412155309234222172150138303010Approximate: 70 significant digits; calculation provenance | -2.408196067338267772800755888240396519764878990641158907654638223011920Approximate: 70 significant digits; calculation provenance | 12/5Exact rational | -12/5Exact rational | 12/5Exact rational | false | true |
217 | 97/40Exact rational | 0.9922324299219849990692818468491786425159812612561078667173829021028149Approximate: 70 significant digits; calculation provenance | 0.007767570078015000930718153150821357484018738743892133282617097897185118Approximate: 70 significant digits; calculation provenance | 0.9844648598439699981385636936983572850319625225122157334347658042056298Approximate: 70 significant digits; calculation provenance | 0.03082893973239250785229190388344780911262209144870056590903646081336027Approximate: 70 significant digits; calculation provenance | -2.432797894785403457543945579071933055164915618662570648625045945593596Approximate: 70 significant digits; calculation provenance | 97/40Exact rational | -97/40Exact rational | 97/40Exact rational | false | true |
218 | 49/20Exact rational | 0.9926084586557180269646320577718557224179943206729406852878277857431190Approximate: 70 significant digits; calculation provenance | 0.007391541344281973035367942228144277582005679327059314712172214256881033Approximate: 70 significant digits; calculation provenance | 0.9852169173114360539292641155437114448359886413458813705756555714862379Approximate: 70 significant digits; calculation provenance | 0.02934762584335097311333692832355415661967272111385268674285365634998205Approximate: 70 significant digits; calculation provenance | -2.457418994148686709973886823389828700691971382230745445191656595492077Approximate: 70 significant digits; calculation provenance | 49/20Exact rational | -49/20Exact rational | 49/20Exact rational | false | true |
219 | 99/40Exact rational | 0.9929664128450048389791697924765124094819908850413247241880395514382870Approximate: 70 significant digits; calculation provenance | 0.007033587154995161020830207523487590518009114958675275811960448561712974Approximate: 70 significant digits; calculation provenance | 0.9859328256900096779583395849530248189639817700826494483760791028765741Approximate: 70 significant digits; calculation provenance | 0.02793646322691299239027468571970199102953956924724894613503541038794434Approximate: 70 significant digits; calculation provenance | -2.482058439431458551383859151335636693997314795269051676369912207931352Approximate: 70 significant digits; calculation provenance | 99/40Exact rational | -99/40Exact rational | 99/40Exact rational | false | true |
220 | 5/2Exact rational | 0.9933071490757151444406380196186748196062559910927034697307877569401159Approximate: 70 significant digits; calculation provenance | 0.006692850924284855559361980381325180393744008907296530269212243059884084Approximate: 70 significant digits; calculation provenance | 0.9866142981514302888812760392373496392125119821854069394615755138802318Approximate: 70 significant digits; calculation provenance | 0.02659222668316061965599413801657694702051866597994826696310613928330415Approximate: 70 significant digits; calculation provenance | -2.506715348489118068616416687732642075114892172964320326936005156413407Approximate: 70 significant digits; calculation provenance | 5/2Exact rational | -5/2Exact rational | 5/2Exact rational | false | true |
221 | 101/40Exact rational | 0.9936314844931844534471870775244199084844274559682516725379026349286184Approximate: 70 significant digits; calculation provenance | 0.006368515506815546552812922475580091515572544031748327462097365071381622Approximate: 70 significant digits; calculation provenance | 0.9872629689863689068943741550488398169688549119365033450758052698572368Approximate: 70 significant digits; calculation provenance | 0.02531183006821998590016539787997788414520092927543268381186044260271151Approximate: 70 significant digits; calculation provenance | -2.531388881013102052336979588252781932051164896704553908555161932307090Approximate: 70 significant digits; calculation provenance | 101/40Exact rational | -101/40Exact rational | 101/40Exact rational | false | true |
222 | 51/20Exact rational | 0.9939401985084158858695499531613037103921174364216146586015618584285893Approximate: 70 significant digits; calculation provenance | 0.006059801491584114130450046838696289607882563578385341398438141571410681Approximate: 70 significant digits; calculation provenance | 0.9878803970168317717390999063226074207842348728432293172031237168571786Approximate: 70 significant digits; calculation provenance | 0.02409232118986683630404589969884237987609217296360337828171928682096901Approximate: 70 significant digits; calculation provenance | -2.556078236601779195130616868369368184630344962862353709422606608491073Approximate: 70 significant digits; calculation provenance | 51/20Exact rational | -51/20Exact rational | 51/20Exact rational | false | true |
223 | 103/40Exact rational | 0.9942340344410750948825587421324600708114390952126576277904463932909509Approximate: 70 significant digits; calculation provenance | 0.005765965558924905117441257867539929188560904787342372209553606709049077Approximate: 70 significant digits; calculation provenance | 0.9884680688821501897651174842649201416228781904253152555808927865819018Approximate: 70 significant digits; calculation provenance | 0.02293087680039278769589293576863518658308199828896721672569453564310053Approximate: 70 significant digits; calculation provenance | -2.580782652915069140754634405824594435197212717132404173580432300102199Approximate: 70 significant digits; calculation provenance | 103/40Exact rational | -103/40Exact rational | 103/40Exact rational | false | true |
224 | 13/5Exact rational | 0.9945137011005495946704812183790806587496514124063770577062394465209691Approximate: 70 significant digits; calculation provenance | 0.005486298899450405329518781620919341250348587593622942293760553479030875Approximate: 70 significant digits; calculation provenance | 0.9890274022010991893409624367581613174993028248127541154124788930419383Approximate: 70 significant digits; calculation provenance | 0.02182479769534517840316277403338055957421933728773555330119580288611362Approximate: 70 significant digits; calculation provenance | -2.605501403909657374081844759492338301303046114327419802138325257519104Approximate: 70 significant digits; calculation provenance | 13/5Exact rational | -13/5Exact rational | 13/5Exact rational | false | true |
225 | 21/8Exact rational | 0.9947798743064416028968085474802555281283474855012541229736983870774744Approximate: 70 significant digits; calculation provenance | 0.005220125693558397103191452519744471871652514498745877026301612922525574Approximate: 70 significant digits; calculation provenance | 0.9895597486128832057936170949605110562566949710025082459473967741549489Approximate: 70 significant digits; calculation provenance | 0.02077150392520739426724643106090801540461937975714340910874636913789263Approximate: 70 significant digits; calculation provenance | -2.630233798151743031782728942771398758285003162072799827312006926724049Approximate: 70 significant digits; calculation provenance | 21/8Exact rational | -21/8Exact rational | 21/8Exact rational | false | true |
226 | 53/20Exact rational | 0.9950331983499430397185919289373945509130451470908653308750435831777555Approximate: 70 significant digits; calculation provenance | 0.004966801650056960281408071062605449086954852909134669124956416822244517Approximate: 70 significant digits; calculation provenance | 0.9900663966998860794371838578747891018260902941817306617500871663555110Approximate: 70 significant digits; calculation provenance | 0.01976853012570380695227481469911063601864296689355636060612855171344093Approximate: 70 significant digits; calculation provenance | -2.654979177204327229610104627794477411895141749525834275046289850648567Approximate: 70 significant digits; calculation provenance | 53/20Exact rational | -53/20Exact rational | 53/20Exact rational | false | true |
227 | 107/40Exact rational | 0.9952742873976045199196760650235117580775080387245451886339805238484079Approximate: 70 significant digits; calculation provenance | 0.004725712602395480080323934976488241922491961275454811366019476151592069Approximate: 70 significant digits; calculation provenance | 0.9905485747952090398393521300470235161550160774490903772679610476968159Approximate: 70 significant digits; calculation provenance | 0.01881352097118016247808369995389228311726053467292184155270135531608162Approximate: 70 significant digits; calculation provenance | -2.679736914086123578331872455234968834681888715646503936984968473222179Approximate: 70 significant digits; calculation provenance | 107/40Exact rational | -107/40Exact rational | 107/40Exact rational | false | true |
228 | 27/10Exact rational | 0.9955037268390588197743436246250153818175650875426064391923885328684964Approximate: 70 significant digits; calculation provenance | 0.004496273160941180225656375374984618182434912457393560807611467131503573Approximate: 70 significant digits; calculation provenance | 0.9910074536781176395486872492500307636351301750852128783847770657369929Approximate: 70 significant digits; calculation provenance | 0.01790422675441352093312322618721838019388078601390980649415127660422605Approximate: 70 significant digits; calculation provenance | -2.704506411799249425423352946213323472595839270586710745037151428061255Approximate: 70 significant digits; calculation provenance | 27/10Exact rational | -27/10Exact rational | 27/10Exact rational | false | true |
229 | 109/40Exact rational | 0.9957220745802950265821134330529178862163443025521371021932090921411968Approximate: 70 significant digits; calculation provenance | 0.004277925419704973417886566947082113783655697447862897806790907858803211Approximate: 70 significant digits; calculation provenance | 0.9914441491605900531642268661058357724326886051042742043864181842823936Approximate: 70 significant digits; calculation provenance | 0.01703849909523366177966750980623620604254016810132469584070768036734183Approximate: 70 significant digits; calculation provenance | -2.729287101922935327494560735418711758436793382962302988937662510208796Approximate: 70 significant digits; calculation provenance | 109/40Exact rational | -109/40Exact rational | 109/40Exact rational | false | true |
230 | 11/4Exact rational | 0.9959298622841038726748353945714251654327522437366205933867660386212967Approximate: 70 significant digits; calculation provenance | 0.004070137715896127325164605428574834567247756263379406613233961378703273Approximate: 70 significant digits; calculation provenance | 0.9918597245682077453496707891428503308655044874732411867735320772425935Approximate: 70 significant digits; calculation provenance | 0.01621428677947906872276645330446422220413547634109097514793157050260235Approximate: 70 significant digits; calculation provenance | -2.754078443270570720891516424678363665743836656757163253421493949418678Approximate: 70 significant digits; calculation provenance | 11/4Exact rational | -11/4Exact rational | 11/4Exact rational | false | true |
231 | 111/40Exact rational | 0.9961275965593289688845149729714035292840353751022304964427211093394359Approximate: 70 significant digits; calculation provenance | 0.003872403440671031115485027028596470715964624897769503557278890660564056Approximate: 70 significant digits; calculation provenance | 0.9922551931186579377690299459428070585680707502044609928854422186788719Approximate: 70 significant digits; calculation provenance | 0.01542963172905484110193869693588184693255623936218789082704486434591069Approximate: 70 significant digits; calculation provenance | -2.778879920607485177684989946277863003800013489713684954959085612835163Approximate: 70 significant digits; calculation provenance | 111/40Exact rational | -111/40Exact rational | 111/40Exact rational | false | true |
232 | 14/5Exact rational | 0.9963157601005640122403068162880109617704745946282461275566940077226343Approximate: 70 significant digits; calculation provenance | 0.003684239899435987759693183711989038229525405371753872443305992277365704Approximate: 70 significant digits; calculation provenance | 0.9926315202011280244806136325760219235409491892564922551133880154452686Approximate: 70 significant digits; calculation provenance | 0.01468266510319756664997007649520936827850354570826105693693409332118851Approximate: 70 significant digits; calculation provenance | -2.803691043426946538491740763648030278942176489730812671802569774194763Approximate: 70 significant digits; calculation provenance | 14/5Exact rational | -14/5Exact rational | 14/5Exact rational | false | true |
233 | 113/40Exact rational | 0.9964948127799336612047190921950352661055479326572406949379047706398074Approximate: 70 significant digits; calculation provenance | 0.003505187220066338795280907804964733894452067342759305062095229360192615Approximate: 70 significant digits; calculation provenance | 0.9929896255598673224094381843900705322110958653144813898758095412796148Approximate: 70 significant digits; calculation provenance | 0.01397160353047448962834434294173221318664538036772587676469519534146454Approximate: 70 significant digits; calculation provenance | -2.828511344781939187856797524113716463701628833044059255609038372713170Approximate: 70 significant digits; calculation provenance | 113/40Exact rational | -113/40Exact rational | 113/40Exact rational | false | true |
234 | 57/20Exact rational | 0.9966651926925866550702861619992953126373056998514423231706727373176736Approximate: 70 significant digits; calculation provenance | 0.003334807307413344929713838000704687362694300148557676829327262682326375Approximate: 70 significant digits; calculation provenance | 0.9933303853851733101405723239985906252746113997028846463413454746353472Approximate: 70 significant digits; calculation provenance | 0.01329474547054306994432329971962495479953930399714121586480621332260334Approximate: 70 significant digits; calculation provenance | -2.853340380170367420773267272297273702481679805583579137435795629181163Approximate: 70 significant digits; calculation provenance | 57/20Exact rational | -57/20Exact rational | 57/20Exact rational | false | true |
235 | 23/8Exact rational | 0.9968273171575148104980368314134982714328189346513600409869958682925369Approximate: 70 significant digits; calculation provenance | 0.003172682842485189501963168586501728567181065348639959013004131707463052Approximate: 70 significant digits; calculation provenance | 0.9936546343150296209960736628269965428656378693027200819739917365850739Approximate: 70 significant digits; calculation provenance | 0.01265046770426475840073175614827287378220214854479127092432028735160467Approximate: 70 significant digits; calculation provenance | -2.878177726471409926664346345528242255156173268151804689970618026722808Approximate: 70 significant digits; calculation provenance | 23/8Exact rational | -23/8Exact rational | 23/8Exact rational | false | true |
236 | 29/10Exact rational | 0.9969815836752915760667993975046895053171450125335908692044468682323992Approximate: 70 significant digits; calculation provenance | 0.003018416324708423933200602495310494682854987466409130795553131767600816Approximate: 70 significant digits; calculation provenance | 0.9939631673505831521335987950093790106342900250671817384088937364647984Approximate: 70 significant digits; calculation provenance | 0.01203722195039663049398212265025761043980025152584939647448307835804373Approximate: 70 significant digits; calculation provenance | -2.903022980930831619273345223197070524139264109464995710695004586745200Approximate: 70 significant digits; calculation provenance | 29/10Exact rational | -29/10Exact rational | 29/10Exact rational | false | true |
237 | 117/40Exact rational | 0.9971283708442996012268747937514442353207555788788983409254854619381302Approximate: 70 significant digits; calculation provenance | 0.002871629155700398773125206248555764679244421121101659074514538061869767Approximate: 70 significant digits; calculation provenance | 0.9942567416885992024537495875028884706415111577577966818509709238762605Approximate: 70 significant digits; calculation provenance | 0.01145353160677012075210260217862755025646883862866094393805835459926523Approximate: 70 significant digits; calculation provenance | -2.927875760193138135709121651006142711642355275414223984028350968609581Approximate: 70 significant digits; calculation provenance | 117/40Exact rational | -117/40Exact rational | 117/40Exact rational | false | true |
238 | 59/20Exact rational | 0.9972680392369889398964971542071813319432774309729375805897243616860119Approximate: 70 significant digits; calculation provenance | 0.002731960763011060103502845792818668056722569027062419410275638313988061Approximate: 70 significant digits; calculation provenance | 0.9945360784739778797929943084143626638865548619458751611794487233720239Approximate: 70 significant digits; calculation provenance | 0.01089798861360171251918494673345255242580407102798406569025223779295274Approximate: 70 significant digits; calculation provenance | -2.952735699378536118575704789814597983757527754003024838293187314583929Approximate: 70 significant digits; calculation provenance | 59/20Exact rational | -59/20Exact rational | 59/20Exact rational | false | true |
239 | 119/40Exact rational | 0.9974009322376766528349364113058605369565393185133139706346570453091131Approximate: 70 significant digits; calculation provenance | 0.002599067762323347165063588694139463043460681486686029365342954690886876Approximate: 70 significant digits; calculation provenance | 0.9948018644753533056698728226117210739130786370266279412693140906182262Approximate: 70 significant digits; calculation provenance | 0.01036925043636079469613672963456150908132490889540545861834989267236642Approximate: 70 significant digits; calculation provenance | -2.977602451202738716159791544151885411176318200598736263610052074097091Approximate: 70 significant digits; calculation provenance | 119/40Exact rational | -119/40Exact rational | 119/40Exact rational | false | true |
240 | 3Exact rational | 0.9975273768433652256659400926277442375489069273501412459119394075653324Approximate: 70 significant digits; calculation provenance | 0.002472623156634774334059907372255762451093072649858754088060592434667649Approximate: 70 significant digits; calculation provenance | 0.9950547536867304513318801852554884750978138547002824918238788151306647Approximate: 70 significant digits; calculation provenance | 0.009866037165440191273156169683523487327321609118799727204249490046423324Approximate: 70 significant digits; calculation provenance | -3.002475685137730449530859682043522968776733924661974105498096348803967Approximate: 70 significant digits; calculation provenance | 3Exact rational | -3Exact rational | 3Exact rational | true | true |
Complete statistical definitions, exact worked populations and response limits (31 rows)
Localized noninteracting two-state moments ±μ_s at fixed T>0 in prescribed local B. u=μ_sB_loc/(k_BT); M/(nμ_s), probabilities and variance dimensionless. Free energy per spin/(k_BT). dM/dB has A/(m T), not dimensionless M/H; extra local-field assumptions are required.
| Quantity or source condition | Retained admitted value |
|---|---|
definitions · Assumptions and regime | Localized noninteracting two-state moments with projections +/-muS in a prescribed local B at fixedT; no exchange/domain/demagnetization solver or measured material curve. Spin is intrinsic quantum angular momentum. |
definitions · u=μ_s B_loc/(k_BT) | muS B_loc/(kB T) |
definitions · State energy / (k_BT) · 1 | -u |
definitions · State energy / (k_BT) · 2 | u |
definitions · partition | 2cosh(u) |
definitions · probabilities · 1 | exp(u)/(2cosh u) |
definitions · probabilities · 2 | exp(-u)/(2cosh u) |
definitions · variance Normalized | sech^2(u) |
definitions · Free energy per spin / (k_BT) | -ln(2cosh u) |
definitions · curve Grid | u=-3+i/40,i=0..240 |
definitions · controls · 1 | -3/2 |
definitions · controls · 2 | -3/4 |
definitions · controls · 3 | 0 |
definitions · controls · 4 | 3/4 |
definitions · controls · 5 | 3/2 |
definitions · controls · 6 | 3 |
definitions · exact Worked Population U | ln(3) |
definitions · weak Field | |u|<<1. M/B response is not dimensionless M/H. Curie chi_H approximation uses additional dilute/local-field conditions B_loc approximately mu0 H_int. |
definitions · Validity limits | No spontaneous order at zero field in this independent-spin model. Do not replace it with the classical continuous-orientation Langevin model or universal low-temperature Curie law. |
definitions · magnetization Over Number Density Mu S | tanh(u) |
exact Worked Population · u=μ_s B_loc/(k_BT) | ln(3)Symbolic expression |
exact Worked Population · Positive/negative Boltzmann weight ratio | 9Exact rational |
exact Worked Population · Positive-projection probability p₊ | 9/10Exact rational |
exact Worked Population · Negative-projection probability p₋ | 1/10Exact rational |
exact Worked Population · M / (n μ_s) | 4/5Exact rational |
exact Worked Population · Normalized moment variance | 9/25Exact rational |
exact Worked Population · Free energy per spin / (k_BT) | -ln(10/3)Symbolic expression |
susceptibility Regime · Local-B response and SI units | n muS^2 sech^2(u)/(kB T); units A/(m T) |
susceptibility Regime · Conditional dimensionless H susceptibility | mu0 n muS^2/(kB T), only with stated dilute/local-field approximation |
susceptibility Regime · Independent-spin zero-field mean | 0Exact rational |
susceptibility Regime · Ordered-material model? | false |
Focus a named plot or table region to scroll horizontally with the keyboard, or swipe within that region. Readable fixed-size diagrams remain locally contained. Full disclosures retain exact values, all records and source/domain conditions.
| u | p_+ | p_− | M/(nμ_s) | Normalized moment variance |
|---|---|---|---|---|
| -3/4 | 0.1824 | 0.8176 | -0.6351 | 0.5966 |
| 0 | 0.5000 | 0.5000 | 0.0000 | 1.0000 |
| 3/4 | 0.8176 | 0.1824 | 0.6351 | 0.5966 |
| 3/2 | 0.9526 | 0.0474 | 0.9051 | 0.1807 |
| 3 | 0.9975 | 0.0025 | 0.9951 | 0.0099 |
Use Curie’s approximation only with its limits
For |u| ≪ 1, tanh u ≈ u. The independent-moment response is then M ≈ nμ_s²B_loc/(k_BT). Only with the additional dilute/local-field approximation B_loc ≈ μ₀H_int does this become the dimensionless χ_m ≈ μ₀nμ_s²/(k_BT). That chain of assumptions explains the 1/T trend; it is not a definition of every paramagnet.
At larger |u|, populations approach saturation and the straight weak-field line ceases to be a useful approximation. A classical dipole with continuously variable orientation instead has the Langevin function coth u − 1/u and a different small-u coefficient for the same moment scale. Those state spaces cannot be exchanged without changing the model.
Degenerate conduction electrons give a further warning: Pauli paramagnetism does not inherit the simple localized-moment Curie trend universally at low temperature. Orbital diamagnetism, spin response, exchange and crystal structure can contribute differently to a real specimen. A statement about one response channel is not a complete microscopic explanation of a material.
Distinguish magnetic order from net moment
Ferromagnetic order correlates local moments so that a magnetized state can persist without an applied field in the appropriate regime. A bulk sample may nevertheless have little net moment because differently oriented domains average out. Zero sample-average magnetization does not prove the absence of local magnetic order.
Antiferromagnetic order can have compensating oppositely oriented sublattices. Ferrimagnetic sublattices have unequal contributions and can leave a net moment; temperature-dependent compensation can also cancel it. The Curie and Néel transition names distinguish relevant ordering transitions, while a compensation point need not be a loss of order.
A Curie–Weiss fit can indicate effective interactions within its applicable temperature range, but its intercept is not automatically an exact measured transition temperature. Dimensionality, correlations, anisotropy and frustration can invalidate a simple mean-field prediction. The independent spin-half model above contains no exchange, so it has no spontaneous ordered zero-field state of its own.
Connect domains to history
A magnetic domain is a region with a correlated magnetization direction. It is not the same object as a crystallographic grain. Domain arrangements reflect competition among exchange, domain-wall, anisotropy, stray-field and magnetoelastic energies, together with defects and boundaries. No one fixed domain size follows from the dipole field formula.
Changing the applied field can move walls, unpin them or rotate magnetization. Preparation and history therefore affect the path through a magnetization curve. Remanence records a remaining response at a specified zero-field condition; coercivity records a field needed to reach a specified zero of B or M. Those two coercivity definitions need not give the same value.
Magnetically soft and hard describe response and coercivity in context, not mechanical softness and hardness. A low-loss core and a stable permanent magnet serve different purposes. At finite frequency, eddy-current and relaxation losses add to the quasistatic picture, so a static loop alone cannot predict every operating loss.
Advanced return: a point source is a distribution
For an Ampèrian point-dipole convention, write the ordinary exterior field only for r > 0. Its extension over all space involves a spherical principal-value distribution together with +(2μ₀/3)mδ³(r). A Dirac distribution has meaning through integration against test functions; it is not an extra finite central field sample.
One route uses A = μ₀(m × r)/(4πr³), takes its curl distributionally, and uses ∂_i∂_j(1/r) = PV[(3r_ir_j − r²δ_ij)/r⁵] − (4π/3)δ_ijδ³(r), along with ∇²(1/r) = −4πδ³(r). The resulting contact coefficient is positive 2μ₀/3. A monopole-pair analogy that discards the current source’s internal contribution gives the wrong contact term.
The finite sphere supplies a complementary check. Its interior integral is ∫_inside B dV = (2μ₀/3)m. Shrinking that prescribed source while holding its total moment fixed concentrates this contribution. This mathematical limit does not claim that a real material can maintain arbitrarily large magnetization, and it does not justify using the singular exterior formula inside the finite sphere.
Advanced return: source constraints and half factors
To see why an induced moment needs different bookkeeping, consider an ideal reversible positive polarizability α with an internal cost m²/(2α) and external coupling −mB_ext along one direction. Minimizing their sum gives m = αB_ext and the effective mechanical potential −αB_ext²/2. Substituting m(B) into the fixed-m expression alone would double that result. This illustrative response model does not define the force density in arbitrary magnetic matter.
For a lossless magnetoquasistatic coil, use its current i and flux linkage λ as terminal variables. Let W_mag include the reversible magnetic storage of the chosen field–material system. At fixed geometry dW_mag = i dλ. Coenergy is W′ = iλ − W_mag; the appropriate mechanical force is a positive geometry derivative of W′ at fixed current, or a negative derivative of W_mag at fixed flux linkage. These signs follow from including terminal source work.
Material force calculations must also specify what is held fixed, the field regime and the constitutive variables. Products of macroscopic averaged current and field need not equal the average microscopic force. Permanent bias, motion, hysteresis or dispersion can require a more complete energy or stress description than this reversible one-coil example.
Advanced return: mean field and pair counting
A simple interacting two-state mean-field model illustrates what the independent-spin calculation leaves out. Let N_s be the number of interacting sites, s a dimensionless mean polarization with |s| ≤ 1, η a dimensionless ferromagnetic coupling parameter, and h a dimensionless applied bias. These symbols are separate from the vector moment m and from the earlier prescribed local-field parameter u. The self-consistency equation is s = tanh(ηs + h).
One corresponding mean-field free energy per N_s k_BT is [(1+s)/2]ln[(1+s)/2] + [(1−s)/2]ln[(1−s)/2] − ηs²/2 − hs. The 1/2 in the interaction term prevents counting each pair twice. For |s| < 1 its derivative is atanh s − ηs − h. At h = 0 and s = 0 the curvature is 1 − η, so this approximation changes local stability at η = 1.
Solving a stationary equation is not enough: stability and free-energy comparison select the equilibrium branch. Entropy uses its limiting convention at |s| = 1, while derivatives there require care. This mean-field example omits spatial fluctuations, domains and much microscopic physics; it is not a universal transition-temperature formula or a one-sublattice model of every antiferromagnet.
Advanced return: energy and dissipative cycles
For a reversible single-valued material relation under specified mechanical and thermal conditions, magnetic energy can be expressed through a constitutive path integral ∫H·dB from a stated reference. The familiar ½B·H is a linear reversible specialization, not a universal energy density for permanent or hysteretic matter.
For a passive hysteretic cycle returning to the same macroscopic operating state, ∮H·dB gives the cycle work per volume in the applicable uniform-field description. Its units are (A/m)T = J/m³. Follow the physical driving orientation and distinguish work from recoverable stored energy. A plotted M versus applied-H loop needs its own conversion and geometry assumptions before its area is used as an energy density.
At one operating point H = 0, the expression ½B·H vanishes. That does not prove that the magnet, its external vacuum field and its sources contain no energy. Hysteresis also prevents assigning a unique recoverable energy from an instantaneous B alone. State the reference, constitutive path and complete system before comparing energies or losses.
Select an application and a checkable model
A small maintained coil is a useful torque or force probe when its geometry, external field and perturbation limit are controlled. A permanent magnet’s exterior can sometimes be summarized by a dipole, while its near field and interior require its actual source and shape. The finite sphere is a useful solved example, not a universal replacement for that geometry.
Core design, magnetic storage and magnetic resonance emphasize different physics. A core needs constitutive response and loss mechanisms; stable storage depends on barriers and history; resonance needs dynamics and relaxation as well as a magnetic moment. None follows from a single tabulated scalar permeability without conditions.
Electrical conduction supplies a connection to eddy currents, induction and inductance. Maxwell’s equations and gauge potentials provide the larger field framework. Those connections explain where the present magnetostatic models fit while leaving the full dynamical developments to their own lessons.
Audit the complete magnetic description
Begin with the complete source, the observation domain and the field being used. Keep external test fields separate from self-fields, and state whether the moment is maintained, prescribed, induced or statistically averaged. Check dimensions before comparing a moment, a moment density and a susceptibility.
Then check the appropriate curl and divergence equations, the interface normal convention and every limiting side. In a material, distinguish internal from applied and microscopic local fields. For a constitutive approximation, record geometry, linearity, reversibility, anisotropy, temperature and frequency limits.
Finally match mechanical work, stored energy, free energy and dissipated cycle work to their constraints. Inspect signed components rather than only magnitudes, preserve excluded sources, and label limiting shapes. Finite samples and smooth curves support the stated analytic models; they are not by themselves proofs of a universal material law or a continuum error bound.
Worked examples
A loop turns without a net pull
Use m = m₀(−1, 2, 2) in uniform B_ext = B₀(1, 3, 2). Find force, torque and mechanical interaction potential.
- The field is uniform and the source is a complete closed test loop, so the net translational magnetic force is zero.
- The cross product is (−1, 2, 2) × (1, 3, 2) = (−2, 4, −5). Thus τ = m₀B₀(−2, 4, −5).
- The dot product is −1 + 6 + 4 = 9. At fixed moment U_mech = −9m₀B₀.
Zero net force coexists with nonzero torque. The stated potential is a mechanical coupling with maintained sources; it is not the total circuit energy.
Translate the held loop through the local gradient
Move the same fixed moment along the stated path from A to B in B_ext/B₀ = (1 + 4X, 3 + Y, 2 − 5Z). Find the mechanical work balance.
- The constant force is F/(m₀B₀/L₀) = (−4, 2, −10). The displacement is (1/2, −1/2, −2/5)L₀.
- At A the field vector is (0, 3, 1)B₀ and U_A = −8m₀B₀. At B it is (2, 5/2, 3)B₀ and U_B = −9m₀B₀.
- Therefore ΔU_mech = −m₀B₀. The force work is m₀B₀[−2 − 1 + 4] = m₀B₀, equal to −ΔU_mech.
For an ideal quasistatic translation with no kinetic-energy change, the external translating agent does −m₀B₀ of work. Orientation and current remain maintained; their apparatus is not omitted from a claim about total energy.
Read the equatorial sheet and the polar limit
For the prescribed sphere, explain the B and H limits at the positive-x equator and at the north pole.
- At the positive-x equator, n̂ = x̂ and K_b = M₀ ẑ × x̂ = M₀ ŷ.
- The equatorial B_z limits are 2μ₀M₀/3 inside and −μ₀M₀/3 outside. Then n̂ × (B_out − B_in)/μ₀ = M₀ŷ, exactly the bound sheet. H_z is −M₀/3 on both sides.
- At the north pole, normal B_z stays 2μ₀M₀/3 while normal H_z changes from −M₀/3 to 2M₀/3. This is consistent with the jump in M and does not violate ∇·B = 0.
The component’s direction relative to the interface matters. Normal B and tangential H satisfy different continuity statements; assigning one unqualified surface value would hide the distinction.
Recover an intrinsic susceptibility
A model sphere has χ_m = 7/3 and H_app = H₀ẑ. Find H_int, M, B and the applied-field susceptibility.
- Use N = 1/3. The denominator is 1 + (1/3)(7/3) = 16/9.
- Hence H_int/H₀ = 9/16, M/H₀ = 21/16, and B/(μ₀H₀) = 9/16 + 21/16 = 15/8.
- The apparent susceptibility is 21/16. Inverting with the same geometry gives (21/16)/[1 − (1/3)(21/16)] = 7/3.
The measured M/H_app is not the intrinsic M/H_int. The inversion is valid for this specified linear, uniformly magnetized ellipsoid model, not every specimen shape.
A population beyond the weak-field line
In the independent spin-half model take u = ln 3. Find the populations, magnetization and normalized variance.
- The Boltzmann weight ratio is exp(2u) = 9, so p_+ = 9/10 and p_− = 1/10.
- The mean projected moment is (9/10 − 1/10)μ_s = 4μ_s/5, giving M = 4nμ_s/5.
- The normalized variance is 1 − (4/5)² = 9/25. The exact result remains within the saturation bound; extending the line M/(nμ_s) ≈ u to ln 3 is outside its weak-field regime.
The populations and their variance follow the two-state equilibrium model. No exchange-driven order, demagnetization response or measured material susceptibility has been inferred.
Try it yourself
Reverse the maintained test-loop current in the original uniform field. What happens to m, τ, U_mech and the net force?
Show the answer and reasoning
m, τ and U_mech reverse sign; the uniform-field net force stays zero.
The loop’s area normal is defined consistently with conventional current. Both the cross and dot products are linear in the maintained moment; uniformity still makes the complete-loop forces sum to zero.
Why is the prescribed gradient field compatible with a source-free observation region?
Show the answer and reasoning
Its divergence is zero and its curl is zero throughout that local box.
The diagonal derivatives are 4, 1 and −5 in B₀/L₀ units, whose sum is zero. Off-diagonal derivatives vanish. This checks the local vacuum equations while leaving the distant field-maintaining apparatus outside the model.
For constant m, when can ∇(m·B) be replaced by (m·∇)B?
Show the answer and reasoning
When the additional m × (∇ × B) term vanishes, in particular in a source-free curl-free magnetostatic test region.
A fixed moment removes derivatives of m, but it does not remove the curl of the external field. The equality must follow from the stated region or vectors, not from a label attached to B.
Compare B_z at z = 2a on the axis and at x = 2a in the equatorial plane of the prescribed sphere.
Show the answer and reasoning
The values are μ₀M₀/12 and −μ₀M₀/24, respectively.
Both points lie outside. The axial dipole factor is +2 and the equatorial one is −1; at radius 2a their common normalized radial denominator is 3×2³. These are signed components, not two positive magnitudes.
The sphere has no free currents. Why is its interior H not zero?
Show the answer and reasoning
The magnetization and its boundary data still require H_in = −M₀ẑ/3.
∇ × H = 0 alone does not determine H. Its divergence and boundary conditions also matter. Normal B continuity and the prescribed M fix the nonzero demagnetizing field.
Use χ_m = −1/4 and N = 1/3. Find H_int/H₀, M/H₀ and B/(μ₀H₀).
Show the answer and reasoning
The values are 12/11, −3/11 and 9/11.
The denominator is 1 − 1/12 = 11/12. A negative M makes the demagnetizing contribution add to H_app, while B = μ₀(H_int + M) remains smaller than μ₀H₀ in this model.
Why are N = 0 and N = 1 marked as limits rather than two ordinary finite ellipsoids?
Show the answer and reasoning
They correspond to limiting aspect ratios along the selected axis.
Finite axisymmetric ellipsoids have positive principal factors with SI sum 1. The endpoints describe ideal limiting geometries; an arbitrary finite rod or sheet is not thereby given an exact uniform-field solution.
At u = 0, what are the spin probabilities, mean normalized magnetization and normalized variance?
Show the answer and reasoning
They are 1/2, 1/2, zero and one.
Opposite moments cancel in the mean, but each remains present and can fluctuate. Zero mean does not remove microscopic degrees of freedom or imply zero response to a small field.
Why is ∂M/∂B_loc not automatically the dimensionless susceptibility χ_m?
Show the answer and reasoning
Its units are A/(m T), and the fields used in the two derivatives differ.
Converting to an M/H response requires the applicable local/internal-field relation. In the dilute weak-field approximation B_loc ≈ μ₀H_int, a factor μ₀ enters; outside that approximation the full constitutive and local-field problem remains.
In the stated mean-field example at h = 0, what diagnoses the local stability of s = 0?
Show the answer and reasoning
The free-energy curvature 1 − η, with the marginal point at η = 1 in that approximation.
A stationary self-consistency root is not automatically a stable equilibrium. Compare extrema and free energies, and retain the model’s fluctuation, domain and microscopic limitations.
Can the Ampèrian contact term be plotted as an ordinary finite B at the point origin?
Show the answer and reasoning
No.
It is a distribution contributing under integration. The ordinary exterior point formula excludes r = 0, while the finite sphere has its own regular interior solution. Mixing these descriptions changes the source model.
Does H = 0 at one point of a permanent magnet’s operating curve prove that its complete energy is zero?
Show the answer and reasoning
No.
The half-product formula has specific linear reversible assumptions, and the external field and sources also store or exchange energy. In a hysteretic cycle, work is path dependent and must not be replaced by a value inferred from one point.
What to carry forward
- A complete closed current source has a directed moment. Uniform external B can produce torque without net translational force.
- Fixed-m mechanical potential excludes source/self energy; induced responses require their internal and source-work bookkeeping.
- A finite prescribed magnet can have a regular interior and a dipolar exterior. The point-source formula has a different origin domain.
- M gives equivalent bound currents; H removes that curl contribution but still depends on magnetization and boundary data.
- Normal B and tangential H continuity are distinct; one-sided sheet limits must remain visible.
- Intrinsic, apparent and differential susceptibilities differ. SI demagnetization factors and material regimes must be explicit.
- Independent spin-half response saturates. Its Curie limit is conditional and does not explain every paramagnet or magnetic order.
- Exchange, domains, anisotropy and pinning make order and history different from a single independent-moment curve.
- Energy, coenergy, free energy and hysteretic loss answer different constrained questions; a single operating point is insufficient.
References
Sources for the concepts and teaching approaches in this article.
Caltech6 sources
- Feynman II14: The Magnetic Field in Various SituationsRichard P. Feynman, Robert B. Leighton and Matthew SandsTextbook chapter
- Feynman II15: The Vector PotentialRichard P. Feynman, Robert B. Leighton and Matthew SandsTextbook chapter
- Feynman II34: The Magnetism of MatterRichard P. Feynman, Robert B. Leighton and Matthew SandsTextbook chapter
- Feynman II35: Paramagnetism and Magnetic ResonanceRichard P. Feynman, Robert B. Leighton and Matthew SandsTextbook chapter
- Feynman II36: FerromagnetismRichard P. Feynman, Robert B. Leighton and Matthew SandsTextbook chapter
- Feynman II37: Magnetic MaterialsRichard P. Feynman, Robert B. Leighton and Matthew SandsTextbook chapter
Massachusetts Institute of Technology10 sources
- Haus and Melcher:9.2 Laws and Continuity Conditions with MagnetizationHermann A. Haus and James R. MelcherTextbook chapter
- Haus and Melcher:9.3 Permanent MagnetizationHermann A. Haus and James R. MelcherTextbook chapter
- Haus and Melcher:9.4 Magnetization Constitutive LawsHermann A. Haus and James R. MelcherTextbook chapter
- Haus and Melcher:9.5 Fields in the Presence of Magnetically Linear Insulating MaterialsHermann A. Haus and James R. MelcherTextbook chapter
- Haus and Melcher:9.6 Fields in Piece-Wise Uniform Magnetically Linear MaterialsHermann A. Haus and James R. MelcherTextbook chapter
- MIT8.333 Lecture24: The Degenerate Fermi GasMIT OCWCourse notes · PDF
- Haus and Melcher:11.4 Energy StorageHermann A. Haus and James R. MelcherTextbook chapter
- Haus and Melcher:11.7 Macroscopic Magnetic ForcesHermann A. Haus and James R. MelcherTextbook chapter
- Haus and Melcher:11.6 Electrical Forces on Macroscopic MediaHermann A. Haus and James R. MelcherTextbook chapter
- MIT6.007 Lecture10: Magnetic MaterialsMIT OCWCourse notes · PDF
National Institute of Standards and Technology (NIST)3 sources
- The International System of Units,9th edition,v4.01BIPMReference guide · PDF