# Fraction SΒΆ

`Canonical`

means that equal elements are in fact identical.

- 0: %
- from AbelianMonoid
- 1: %
- from MagmaWithUnit
- *: (%, %) -> %
- from Magma
- *: (%, Fraction Integer) -> %
- from RightModule Fraction Integer
- *: (%, S) -> %
- from RightModule S
- *: (Fraction Integer, %) -> %
- from LeftModule Fraction Integer
- *: (Integer, %) -> %
- from AbelianGroup
- *: (NonNegativeInteger, %) -> %
- from AbelianMonoid
- *: (PositiveInteger, %) -> %
- from AbelianSemiGroup
- *: (S, %) -> %
- from LeftModule S
- +: (%, %) -> %
- from AbelianSemiGroup
- -: % -> %
- from AbelianGroup
- -: (%, %) -> %
- from AbelianGroup
- /: (%, %) -> %
- from Field
- /: (S, S) -> %
- from QuotientFieldCategory S
- <: (%, %) -> Boolean if S has OrderedSet
- from PartialOrder
- <=: (%, %) -> Boolean if S has OrderedSet
- from PartialOrder
- =: (%, %) -> Boolean
- from BasicType
- >: (%, %) -> Boolean if S has OrderedSet
- from PartialOrder
- >=: (%, %) -> Boolean if S has OrderedSet
- from PartialOrder
- ^: (%, Integer) -> %
- from DivisionRing
- ^: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
- from Magma
- ~=: (%, %) -> Boolean
- from BasicType
- abs: % -> % if S has OrderedIntegralDomain
- from OrderedRing
- annihilate?: (%, %) -> Boolean
- from Rng
- antiCommutator: (%, %) -> %
- from NonAssociativeSemiRng
- associates?: (%, %) -> Boolean
- from EntireRing
- associator: (%, %, %) -> %
- from NonAssociativeRng
- ceiling: % -> S if S has IntegerNumberSystem
- from QuotientFieldCategory S
- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- charthRoot: % -> Union(%, failed) if % has CharacteristicNonZero and S has PolynomialFactorizationExplicit or S has CharacteristicNonZero
- from PolynomialFactorizationExplicit
- coerce: % -> %
- from Algebra %
- coerce: % -> OutputForm
- from CoercibleTo OutputForm
- coerce: Fraction Integer -> %
- from RetractableTo Fraction Integer
- coerce: Integer -> %
- from NonAssociativeRing
- coerce: S -> %
- from RetractableTo S
- coerce: Symbol -> % if S has RetractableTo Symbol
- from RetractableTo Symbol
- commutator: (%, %) -> %
- from NonAssociativeRng
- conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and S has PolynomialFactorizationExplicit
- from PolynomialFactorizationExplicit
- convert: % -> DoubleFloat if S has RealConstant
- from ConvertibleTo DoubleFloat
- convert: % -> Float if S has RealConstant
- from ConvertibleTo Float
- convert: % -> InputForm if S has ConvertibleTo InputForm
- from ConvertibleTo InputForm
- convert: % -> Pattern Float if S has ConvertibleTo Pattern Float
- from ConvertibleTo Pattern Float
- convert: % -> Pattern Integer if S has ConvertibleTo Pattern Integer
- from ConvertibleTo Pattern Integer
- D: % -> % if S has DifferentialRing
- from DifferentialRing
- D: (%, List Symbol) -> % if S has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- D: (%, List Symbol, List NonNegativeInteger) -> % if S has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- D: (%, NonNegativeInteger) -> % if S has DifferentialRing
- from DifferentialRing
- D: (%, S -> S) -> %
- from DifferentialExtension S
- D: (%, S -> S, NonNegativeInteger) -> %
- from DifferentialExtension S
- D: (%, Symbol) -> % if S has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- D: (%, Symbol, NonNegativeInteger) -> % if S has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- denom: % -> S
- from QuotientFieldCategory S
- denominator: % -> %
- from QuotientFieldCategory S
- differentiate: % -> % if S has DifferentialRing
- from DifferentialRing
- differentiate: (%, List Symbol) -> % if S has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- differentiate: (%, List Symbol, List NonNegativeInteger) -> % if S has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- differentiate: (%, NonNegativeInteger) -> % if S has DifferentialRing
- from DifferentialRing
- differentiate: (%, S -> S) -> %
- from DifferentialExtension S
- differentiate: (%, S -> S, NonNegativeInteger) -> %
- from DifferentialExtension S
- differentiate: (%, Symbol) -> % if S has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- differentiate: (%, Symbol, NonNegativeInteger) -> % if S has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- divide: (%, %) -> Record(quotient: %, remainder: %)
- from EuclideanDomain
- elt: (%, S) -> % if S has Eltable(S, S)
- from Eltable(S, %)
- euclideanSize: % -> NonNegativeInteger
- from EuclideanDomain
- eval: (%, Equation S) -> % if S has Evalable S
- from Evalable S
- eval: (%, List Equation S) -> % if S has Evalable S
- from Evalable S
- eval: (%, List S, List S) -> % if S has Evalable S
- from InnerEvalable(S, S)
- eval: (%, List Symbol, List S) -> % if S has InnerEvalable(Symbol, S)
- from InnerEvalable(Symbol, S)
- eval: (%, S, S) -> % if S has Evalable S
- from InnerEvalable(S, S)
- eval: (%, Symbol, S) -> % if S has InnerEvalable(Symbol, S)
- from InnerEvalable(Symbol, S)
- expressIdealMember: (List %, %) -> Union(List %, failed)
- from PrincipalIdealDomain
- exquo: (%, %) -> Union(%, failed)
- from EntireRing
- extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %)
- from EuclideanDomain
- extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed)
- from EuclideanDomain
- factor: % -> Factored %
- from UniqueFactorizationDomain
- factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if S has PolynomialFactorizationExplicit
- from PolynomialFactorizationExplicit
- factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if S has PolynomialFactorizationExplicit
- from PolynomialFactorizationExplicit
- floor: % -> S if S has IntegerNumberSystem
- from QuotientFieldCategory S
- fractionPart: % -> % if S has EuclideanDomain
- from QuotientFieldCategory S
- gcd: (%, %) -> %
- from GcdDomain
- gcd: List % -> %
- from GcdDomain
- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
- from PolynomialFactorizationExplicit
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- init: % if S has StepThrough
- from StepThrough
- inv: % -> %
- from DivisionRing
- latex: % -> String
- from SetCategory
- lcm: (%, %) -> %
- from GcdDomain
- lcm: List % -> %
- from GcdDomain
- lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %)
- from LeftOreRing
- leftPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- leftPower: (%, PositiveInteger) -> %
- from Magma
- leftRecip: % -> Union(%, failed)
- from MagmaWithUnit
- map: (S -> S, %) -> %
- from FullyEvalableOver S
- max: (%, %) -> % if S has OrderedSet
- from OrderedSet
- min: (%, %) -> % if S has OrderedSet
- from OrderedSet
- multiEuclidean: (List %, %) -> Union(List %, failed)
- from EuclideanDomain
- negative?: % -> Boolean if S has OrderedIntegralDomain
- from OrderedRing
- nextItem: % -> Union(%, failed) if S has StepThrough
- from StepThrough
- numer: % -> S
- from QuotientFieldCategory S
- numerator: % -> %
- from QuotientFieldCategory S
- OMwrite: % -> String if S has OpenMath and S has IntegerNumberSystem
- from OpenMath
- OMwrite: (%, Boolean) -> String if S has OpenMath and S has IntegerNumberSystem
- from OpenMath
- OMwrite: (OpenMathDevice, %) -> Void if S has OpenMath and S has IntegerNumberSystem
- from OpenMath
- OMwrite: (OpenMathDevice, %, Boolean) -> Void if S has OpenMath and S has IntegerNumberSystem
- from OpenMath
- one?: % -> Boolean
- from MagmaWithUnit
- opposite?: (%, %) -> Boolean
- from AbelianMonoid
- patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if S has PatternMatchable Float
- from PatternMatchable Float
- patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if S has PatternMatchable Integer
- from PatternMatchable Integer
- positive?: % -> Boolean if S has OrderedIntegralDomain
- from OrderedRing
- prime?: % -> Boolean
- from UniqueFactorizationDomain
- principalIdeal: List % -> Record(coef: List %, generator: %)
- from PrincipalIdealDomain
- quo: (%, %) -> %
- from EuclideanDomain
- recip: % -> Union(%, failed)
- from MagmaWithUnit
- reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if S has LinearlyExplicitOver Integer
- from LinearlyExplicitOver Integer
- reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix S, vec: Vector S)
- from LinearlyExplicitOver S
- reducedSystem: Matrix % -> Matrix Integer if S has LinearlyExplicitOver Integer
- from LinearlyExplicitOver Integer
- reducedSystem: Matrix % -> Matrix S
- from LinearlyExplicitOver S
- rem: (%, %) -> %
- from EuclideanDomain
- retract: % -> Fraction Integer if S has RetractableTo Integer
- from RetractableTo Fraction Integer
- retract: % -> Integer if S has RetractableTo Integer
- from RetractableTo Integer
- retract: % -> S
- from RetractableTo S
- retract: % -> Symbol if S has RetractableTo Symbol
- from RetractableTo Symbol
- retractIfCan: % -> Union(Fraction Integer, failed) if S has RetractableTo Integer
- from RetractableTo Fraction Integer
- retractIfCan: % -> Union(Integer, failed) if S has RetractableTo Integer
- from RetractableTo Integer
- retractIfCan: % -> Union(S, failed)
- from RetractableTo S
- retractIfCan: % -> Union(Symbol, failed) if S has RetractableTo Symbol
- from RetractableTo Symbol
- rightPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- rightPower: (%, PositiveInteger) -> %
- from Magma
- rightRecip: % -> Union(%, failed)
- from MagmaWithUnit
- sample: %
- from AbelianMonoid
- sign: % -> Integer if S has OrderedIntegralDomain
- from OrderedRing
- sizeLess?: (%, %) -> Boolean
- from EuclideanDomain
- smaller?: (%, %) -> Boolean if S has Comparable
- from Comparable
- solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if S has PolynomialFactorizationExplicit
- from PolynomialFactorizationExplicit
- squareFree: % -> Factored %
- from UniqueFactorizationDomain
- squareFreePart: % -> %
- from UniqueFactorizationDomain
- squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if S has PolynomialFactorizationExplicit
- from PolynomialFactorizationExplicit
- subtractIfCan: (%, %) -> Union(%, failed)
- from CancellationAbelianMonoid
- unit?: % -> Boolean
- from EntireRing
- unitCanonical: % -> %
- from EntireRing
- unitNormal: % -> Record(unit: %, canonical: %, associate: %)
- from EntireRing
- wholePart: % -> S if S has EuclideanDomain
- from QuotientFieldCategory S
- zero?: % -> Boolean
- from AbelianMonoid

Algebra %

Algebra S

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(S, S)

Canonical if S has Canonical and S has GcdDomain and S has canonicalUnitNormal

CharacteristicNonZero if S has CharacteristicNonZero

CharacteristicZero if S has CharacteristicZero

Comparable if S has Comparable

ConvertibleTo DoubleFloat if S has RealConstant

ConvertibleTo Float if S has RealConstant

ConvertibleTo InputForm if S has ConvertibleTo InputForm

ConvertibleTo Pattern Float if S has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if S has ConvertibleTo Pattern Integer

DifferentialRing if S has DifferentialRing

Eltable(S, %) if S has Eltable(S, S)

Evalable S if S has Evalable S

InnerEvalable(S, S) if S has Evalable S

InnerEvalable(Symbol, S) if S has InnerEvalable(Symbol, S)

LinearlyExplicitOver Integer if S has LinearlyExplicitOver Integer

Module %

Module S

OpenMath if S has OpenMath and S has IntegerNumberSystem

OrderedAbelianGroup if S has OrderedIntegralDomain

OrderedAbelianMonoid if S has OrderedIntegralDomain

OrderedAbelianSemiGroup if S has OrderedIntegralDomain

OrderedCancellationAbelianMonoid if S has OrderedIntegralDomain

OrderedIntegralDomain if S has OrderedIntegralDomain

OrderedRing if S has OrderedIntegralDomain

OrderedSet if S has OrderedSet

PartialDifferentialRing Symbol if S has PartialDifferentialRing Symbol

PartialOrder if S has OrderedSet

PatternMatchable Float if S has PatternMatchable Float

PatternMatchable Integer if S has PatternMatchable Integer

PolynomialFactorizationExplicit if S has PolynomialFactorizationExplicit

RealConstant if S has RealConstant

RetractableTo Fraction Integer if S has RetractableTo Integer

RetractableTo Integer if S has RetractableTo Integer

RetractableTo Symbol if S has RetractableTo Symbol

StepThrough if S has StepThrough