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Showing 31–60 of 1709 reference entries

Physics · MeasurementCaesium-133 defining frequency

ΔνCs denotes the unperturbed ground-state hyperfine transition frequency of caesium-133. Its numerical value is fixed exactly in SI units.

DefinitionMathematics & logic · InformationClassical propositions and connectives

Classical logic combines truth-valued propositions using defined connectives.

DefinitionMathematics & logicAxioms, deduction and mathematical proof

A mathematical proof derives a conclusion within stated assumptions and rules.

MethodMathematics & logic · InformationSets, membership and basic operations

A set specifies membership without counting duplicate listings.

DefinitionMathematics & logic · InformationFunctions, domain and codomain

A function assigns exactly one output to each input in its domain.

DefinitionMathematics & logic · InformationMathematical induction

A base case and general successor step establish a predicate on all nonnegative integers.

MethodMathematics & logic · MeasurementEuclidean right triangles and the Pythagorean theorem

In a Euclidean right triangle, squared leg lengths sum to the squared hypotenuse length.

Mathematical statementMathematics & logic · PhysicsSine, cosine and the unit circle

Sine and cosine are unit-circle coordinates whose squared values sum to one.

Mathematical statementMathematics & logic · InformationComplex numbers

A complex number has real coordinates a and b and an imaginary unit satisfying i²=−1.

DefinitionMathematics & logic · Information · PhysicsLinear maps and matrix representation

A linear map preserves linear combinations and has a matrix once bases are chosen.

DefinitionMathematics & logicLimit of a real function

A finite limit controls output error through sufficiently small input distance.

DefinitionMathematics & logic · PhysicsDerivative at a point

A derivative is a finite limit of local difference quotients.

DefinitionMathematics & logic · PhysicsDefinite integral and signed accumulation

A definite integral is the limiting signed accumulation of small contributions.

DefinitionMathematics & logic · PhysicsFundamental theorem of calculus

Continuity connects differentiation with integration.

Mathematical statementMathematics & logic · InformationProbability space and axioms

A probability assigns nonnegative, normalised, countably additive weights to events.

DefinitionMathematics & logic · InformationConditional probability and independence

Conditioning restricts attention to an event of positive probability.

DefinitionMathematics & logic · InformationExpected value of a discrete random variable

Expected value weights possible values by their probabilities.

DefinitionMathematics & logic · MeasurementConfidence interval for a population mean

A confidence level describes repeated-sampling coverage of an interval procedure.

MethodInformation · Mathematics & logicBits and unsigned binary encoding

A bit has two possible states; a fixed-width unsigned binary word encodes an integer.

DefinitionInformation · Mathematics & logicShannon entropy of a discrete distribution

Shannon entropy measures uncertainty in a specified probability distribution.

DefinitionInformationAlgorithm and decision procedure

An algorithm specifies effective steps; a decision procedure must terminate on every permitted input.

MethodInformation · Mathematics & logicUndecidability of the halting problem

No total algorithm decides halting for every program and input in a general computation model.

Mathematical statementInformation · Mathematics & logicAsymptotic upper bounds and big O

Big O bounds eventual growth up to a fixed constant factor.

DefinitionMeasurementSI base units

Seven named base units organize the SI quantity system.

DefinitionMeasurementSI decimal prefixes

Prefix symbols specify decimal scale factors.

DefinitionMeasurementMetrological traceability

Connect a result to a reference through documented calibrations.

DefinitionMeasurement · Mathematics & logicFirst-order uncertainty propagation

Combine input uncertainty through a measurement model.

MethodMeasurement · Physics · Mathematics & logicDimensional consistency

Use dimensions to detect impossible quantity equations.

MethodPhysics · Earth & cosmosNewtonian gravitation

An inverse-square force models attraction between point masses.

ModelPhysicsFrequency, wavelength and phase speed

Relate a periodic wave's spatial and temporal repetition.

Definition
DefinitionSource checked

Bits and unsigned binary encoding

InformationMathematics & logic

A bit has two possible states; a fixed-width unsigned binary word encodes an integer.

Key knowledge

A binary digit takes a value 0 or 1. In an n-bit unsigned positional encoding, bit i contributes bᵢ2ⁱ, with the rightmost bit at i=0. The n-bit word represents an integer from 0 to 2ⁿ−1.

Values and equations

N = Σᵢ₌₀ⁿ⁻¹ bᵢ2ⁱ; 0 ≤ N ≤ 2ⁿ−1

N
Encoded nonnegative integer
n
Positive word length bits
bᵢ
Digit at position i, either 0 or 1
i
Bit position, counted from zero at the right

Conditions and scope

  • n is a positive integer and bᵢ is 0 or 1.
  • This is unsigned positional interpretation; signed numbers, text and other data require an explicitly specified encoding.
  • Stored bit count does not by itself specify Shannon entropy.
Sources · 2

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Introduction to Computer Science, 5.3: Machine-level information representationOpenStaxTextbook chapteropenstax.orgRead source
  2. COMP1521 · Integers (23T2)UNSW SydneyTeaching resourcecgi.cse.unsw.edu.auRead source

Evidence and review

Unsigned positional encoding and its representable range.

2026-10-10 · Checked bit positions and the distinction between a bit pattern and its interpretation.

Foundational review

2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.

Sources from other places

Scope or convention differs

Binary digits carry powers-of-two positional weights.

The repository finite-width range and entropy distinction are additional explicit qualifications.

  • UNSW Sydney, School of Computer Science and Engineering

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–9 of 9 relationships

Scientific relationships · 9 / 9

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • Floating-point numbers use a sign, finite significand and exponent. For rounding to nearest in the normal range, fl(x)=x(1+δ) with |δ| bounded by unit roundoff; representation precision does not guarantee result accuracy.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • For positive integers a₁,…,aₙ and target T≥0, D(i,t)=D(i−1,t) or [t≥aᵢ and D(i−1,t−aᵢ)], with D(0,0)=true and other initial states false. Evaluation costs O(nT) bounded-cost state operations, which is pseudopolynomial because T is binary encoded.
  3. Uses definition
    Relationship sources · 1

    Linked statements

    • An n-element set has 2ⁿ subsets; its k-element subsets correspond to length-n binary strings with k ones. Their number is C(n,k), with C(n,0)=C(n,n)=1. For n≥2 and 1≤k≤n−1, Pascal’s recurrence is C(n,k)=C(n−1,k−1)+C(n−1,k). C(n,k) is also the coefficient of xᵏyⁿ⁻ᵏ in (x+y)ⁿ.
  4. Uses definition
    Relationship sources · 1

    Linked statements

    • A deterministic decider's time cost counts steps and its space cost counts used storage cells; worst-case bounds maximize over inputs of the same encoded length. The machine model and encoding are part of the statement: a number's magnitude is different from the length of its binary representation.
  5. Uses definition
    Relationship sources · 1

    Linked statements

    • Represent nonnegative integer keys with d base-B digits and stably sort each digit from least to most significant. After each pass the processed suffix is ordered, giving total O(d(n+B)) time with counting-sort passes. Choosing B=n makes bounded nᶜ keys linear-time for fixed c.
  6. Related to
    Relationship sources · 2

    Linked statements

    • An n-bit two's-complement integer assigns weight −2^(n−1) to its highest bit and ordinary positive binary weights to the others. Its range is −2^(n−1) through 2^(n−1)−1; the same bits interpreted as unsigned can have a different value.

    Two's complement changes the highest bit's weight from positive unsigned significance to negative signed significance.

  7. Related to
    Relationship sources · 1

    Linked statements

    • An octet contains eight bits and has 256 possible patterns. For a multi-octet integer, little-endian order places the least significant octet at the lowest address; big-endian order places the most significant there.

    Multi-octet ordering assigns stored octets to positions in an integer's binary representation.

  8. Related to
    Relationship sources · 1

    Linked statements

    • UTF-8 encodes Unicode scalar values in one through four octets. Scalar values are U+0000 through U+10FFFF excluding U+D800 through U+DFFF. U+0000 through U+007F use one octet with the same values as ASCII.

    UTF-8 distributes a scalar's binary code-position bits into specified octet patterns.

  9. Related to
    Relationship sources · 1

    Linked statements

    • A combinational device specifies an output logic value for each valid input combination and a maximum propagation delay. Once inputs remain stable and valid for that delay, outputs match the specification. An acyclic connection of such devices can itself be combinational.

    A physical digital signal implements abstract 0/1 values under valid-level and timing specifications.

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Record identity
info-binary-encoding
Revision
core.1
Structured reference record
Essential core · first collection