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Showing 1501–1530 of 1709 reference entries

Earth & cosmosComparative planetary heat transport and observed phase curves

Planetary circulation couples radiation and dynamics; comparing advective, radiative and rotation time scales organizes possible regimes.

MethodEarth & cosmosSurvey selection in observed and intrinsic coordinates

A survey's detected density is the underlying observable distribution multiplied by a selection probability.

MethodEarth & cosmosActual 3D inversion: synthetic secret-model comparisons

The Dublin Secret Model 1 benchmark compares three actual 3D magnetotelluric reconstructions of a spiral conductor from synthetic impedance data.

MethodEarth & cosmosQAMT 2025: synthetic recovery and a field application

Schiffler and colleagues validate a 3D EM inversion implementation using five synthetic anomalies and compare recovered structures across survey observables.

MethodEarth & cosmosDark-sector models: compare evidence at each scale

GR with cold dark matter is the established cosmological comparison baseline.

MethodEarth & cosmosAcceleration models: fit and independent gravity tests

A constant Λ is the minimal GR acceleration baseline; dynamical fields require additional assumptions and parameters.

MethodEarth & cosmosRadiation–matter exchange in relativistic accretion

Radiation-GRMHD couples the matter-plus-magnetic stress tensor to a radiation tensor with opposite four-force sources, so their sum is covariantly conserved.

MethodMathematics & logicReal-number arithmetic laws

Real arithmetic permits regrouping, distributing and taking nonzero reciprocals.

Mathematical statementMathematics & logicFinite decimal place value

A decimal digit's position determines its power-of-ten weight.

DefinitionMathematics & logicRational numbers and recurring decimals

Terminating or eventually recurring decimals represent exactly the rational numbers.

Mathematical statementMathematics & logicArithmetic with fractions

Fraction arithmetic preserves value by using common denominators or reciprocals.

MethodMathematics & logicAbsolute value and distance intervals

Absolute value measures distance from zero on the real line.

DefinitionMathematics & logicPercent and relative change

A percentage states a ratio per hundred and needs an explicit base.

MethodMathematics & logicOrder rules for real inequalities

Scaling an inequality depends on the factor's sign.

Mathematical statementMathematics & logicInteger powers and positive-base rational powers

Negative exponents use reciprocals; rational exponents use principal roots.

DefinitionMathematics & logicReal logarithms and their laws

A logarithm is the inverse of exponentiation for an admissible positive base.

DefinitionMathematics & logicQuadratic formula and discriminant

The discriminant classifies the real solutions of a quadratic equation.

Mathematical statementMathematics & logicFunction composition and inverses

Composition feeds one output into another function; inversion requires unique preimages.

DefinitionMathematics & logicEuclidean triangle congruence tests

Specified matching sides and angles can establish the same triangle shape and size.

Mathematical statementMathematics & logicEuclidean triangle similarity tests

Triangle similarity keeps corresponding angles and one common positive side scale.

Mathematical statementMathematics & logicAcute-angle right-triangle ratios

Opposite and adjacent legs depend on the chosen acute angle.

DefinitionMathematics & logicRadian angle, circular arc and sector

Radian measure is arc length divided by radius.

DefinitionMathematics & logicPlane area and perimeter formulas

Area measures covered surface; perimeter measures boundary length.

Mathematical statementMathematics & logicRight-prism and cylinder volumes

Volume and surface area measure different aspects of a solid.

Mathematical statementMathematics & logicDescriptive sample center and spread

Mean, median, mode and sample variance summarize observed values.

MethodInformationTwo's-complement signed integers

A signed interpretation assigns a negative weight to the highest bit.

DefinitionInformationOctets and multi-byte order

Byte order describes the significance assigned to successive stored octets.

DefinitionInformationUnicode scalar values and UTF-8

UTF-8 encodes scalar values with a variable number of octets.

DefinitionInformationText boundaries and Unicode normalization

Text equality and character counts depend on the declared representation and convention.

MethodInformationIP addresses and datagram delivery

IP addresses identify network destinations; IP does not guarantee delivery.

Definition
MethodSource checked

Arithmetic with fractions

Mathematics & logic

Fraction arithmetic preserves value by using common denominators or reciprocals.

Key knowledge

Add or subtract fractions after converting them to a common denominator; multiply their numerators and denominators. To divide by a nonzero fraction, multiply by its reciprocal. A product denominator works even when it is not the least common denominator.

Values and equations

a/b ± c/d = (ad ± bc)/(bd); (a/b)(c/d) = ac/(bd); (a/b)/(c/d) = ad/(bc)

a, c
Integer numerators; c≠0 for division
b, d
Nonzero integer denominators

Conditions and scope

  • The formulas use integer numerators and nonzero integer denominators.
  • Division additionally requires the divisor's numerator c≠0.

Denominator and divisor restrictions retained. Product denominator is a valid common denominator; it need not be least. Directly inspected mathematical section only; historical narrative and the number-line recipe for negative denominators excluded.

Sources · 3

References

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

  1. Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis / OpenStax2 sources
    1. Prealgebra 2e, 4.2: Fraction multiplication and divisionLynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis / OpenStaxCourse notesopenstax.orgRead source
    2. Prealgebra 2e, 4.5: Adding unlike fractionsLynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis / OpenStaxCourse notesopenstax.orgRead source
  2. Ganita Manjari, Grade 9 Part I, 3.4: Fractions and Rational NumbersNCERT, IndiaReference guide · PDFncert.nic.inRead source

Evidence and review

Checked the stated arithmetic with fractions and restrictions against the listed primary sections; exact inspected boundaries are in the author receipt.

2026-10-10 · Independent Physics and Measurement reference editor (AI editorial bounded review). Independent AI editorial review of this entry's statements, equations, applicability conditions and source coverage. Source checks apply to these claims within their stated limits.

Related reference entries

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

Showing 1–2 of 2 relationships

Scientific relationships · 2 / 2

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

  1. Uses definition
    Relationship sources · 2

    Linked statements

    • Add or subtract fractions after converting them to a common denominator; multiply their numerators and denominators. To divide by a nonzero fraction, multiply by its reciprocal. A product denominator works even when it is not the least common denominator.

    Operations use nonzero denominators and value-preserving equivalent fractions.

  2. Related to
    Relationship sources · 1

    Linked statements

    • For positive integers a and b, their least common multiple is the smallest positive integer divisible by both. In prime factorizations, retain the largest multiplicity of each prime appearing in either number. This supplies a common denominator for fractions with positive denominators.

    The least common multiple provides a shared positive denominator.

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