Fraction(S)

fraction.spad line 251 [edit on github]

• S : IntegralDomain

Fraction takes an IntegralDomain S and produces the domain of Fractions with numerators and denominators from S. If S is also a GcdDomain, then gcd's between numerator and denominator will be cancelled during all operations.

* : (%, %) -> %

from Magma

* : (%, S) -> %

from RightModule(S)

* : (%, Fraction(Integer)) -> %

from RightModule(Fraction(Integer))

* : (%, Integer) -> % if S has LinearlyExplicitOver(Integer)

from RightModule(Integer)

* : (S, %) -> %

from LeftModule(S)

* : (Fraction(Integer), %) -> %

from LeftModule(Fraction(Integer))

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

/ : (%, %) -> %

from Field

/ : (S, S) -> %

from QuotientFieldCategory(S)

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

< : (%, %) -> 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

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 : (%, Mapping(S, S)) -> %

from DifferentialExtension(S)

D : (%, Mapping(S, S), NonNegativeInteger) -> %

from DifferentialExtension(S)

D : (%, NonNegativeInteger) -> % if S has DifferentialRing

from DifferentialRing

D : (%, Symbol) -> % if S has PartialDifferentialRing(Symbol)

from PartialDifferentialRing(Symbol)

D : (%, Symbol, NonNegativeInteger) -> % if S has PartialDifferentialRing(Symbol)

from PartialDifferentialRing(Symbol)

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

^ : (%, Integer) -> %

from DivisionRing

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

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 S has CharacteristicNonZero or % has CharacteristicNonZero and S has PolynomialFactorizationExplicit

from CharacteristicNonZero

coerce : % -> %

from Algebra(%)

coerce : S -> %

from CoercibleFrom(S)

coerce : Fraction(Integer) -> %

from CoercibleFrom(Fraction(Integer))

coerce : Integer -> %

from NonAssociativeRing

coerce : Symbol -> % if S has RetractableTo(Symbol)

from CoercibleFrom(Symbol)

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

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))

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 : (%, Mapping(S, S)) -> %

from DifferentialExtension(S)

differentiate : (%, Mapping(S, S), NonNegativeInteger) -> %

from DifferentialExtension(S)

differentiate : (%, NonNegativeInteger) -> % if S has DifferentialRing

from DifferentialRing

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 : (%, S, S) -> % if S has Evalable(S)

from InnerEvalable(S, S)

eval : (%, 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(Equation(S))) -> % if S has Evalable(S)

from Evalable(S)

eval : (%, List(Symbol), List(S)) -> % if S has InnerEvalable(Symbol, S)

from InnerEvalable(Symbol, 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

factorFraction : % -> Fraction(Factored(S)) if S has UniqueFactorizationDomain

factorFraction(r) factors the numerator and the denominator of the fraction r.

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 if S has Hashable

from Hashable

hashUpdate! : (HashState, %) -> HashState if S has Hashable

from Hashable

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 : (Mapping(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)

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)

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

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(%) -> Matrix(S)

from LinearlyExplicitOver(S)

reducedSystem : Matrix(%) -> Matrix(Integer) if S has LinearlyExplicitOver(Integer)

from LinearlyExplicitOver(Integer)

reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(S), vec : Vector(S))

from LinearlyExplicitOver(S)

reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(Integer), vec : Vector(Integer)) if S has LinearlyExplicitOver(Integer)

from LinearlyExplicitOver(Integer)

rem : (%, %) -> %

from EuclideanDomain

retract : % -> S

from RetractableTo(S)

retract : % -> Fraction(Integer) if S has RetractableTo(Integer)

from RetractableTo(Fraction(Integer))

retract : % -> Integer if S has RetractableTo(Integer)

from RetractableTo(Integer)

retract : % -> Symbol if S has RetractableTo(Symbol)

from RetractableTo(Symbol)

retractIfCan : % -> Union(S, "failed")

from RetractableTo(S)

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(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

~= : (%, %) -> Boolean

from BasicType

LinearlyExplicitOver(S)

Module(Fraction(Integer))

ConvertibleTo(Float)

PrincipalIdealDomain

PartialOrder

NonAssociativeSemiRing

BiModule(%, %)

RetractableTo(Symbol)

ConvertibleTo(InputForm)

Field

canonicalUnitNormal

Rng

CoercibleFrom(Integer)

TwoSidedRecip

OrderedAbelianGroup

SemiRing

EntireRing

PatternMatchable(Float)

NonAssociativeAlgebra(Fraction(Integer))

CharacteristicNonZero

DifferentialExtension(S)

unitsKnown

InnerEvalable(Symbol, S)

NonAssociativeAlgebra(S)

SetCategory

Patternable(S)

noZeroDivisors

RetractableTo(Fraction(Integer))

FullyPatternMatchable(S)

CoercibleFrom(S)

SemiGroup

RightModule(Fraction(Integer))

Magma

GcdDomain

IntegralDomain

LeftModule(%)

NonAssociativeRing

UniqueFactorizationDomain

Eltable(S, %)

NonAssociativeAlgebra(%)

PartialDifferentialRing(Symbol)

CharacteristicZero

InnerEvalable(S, S)

RightModule(S)

OrderedIntegralDomain

Ring

CoercibleFrom(Fraction(Integer))

Algebra(%)

DifferentialRing

OrderedAbelianMonoid

DivisionRing

CommutativeRing

OpenMath

NonAssociativeSemiRng

CancellationAbelianMonoid

EuclideanDomain

canonicalsClosed

Module(S)

RetractableTo(Integer)

OrderedCancellationAbelianMonoid

OrderedRing

CommutativeStar

AbelianMonoid

MagmaWithUnit

Comparable

CoercibleFrom(Symbol)

RightModule(%)

Hashable

RealConstant

RetractableTo(S)

LeftModule(S)

ConvertibleTo(DoubleFloat)

QuotientFieldCategory(S)

OrderedAbelianSemiGroup

Module(%)

LinearlyExplicitOver(Integer)

CoercibleTo(OutputForm)

ConvertibleTo(Pattern(Integer))

SemiRng

Monoid

PolynomialFactorizationExplicit

LeftOreRing

OrderedSet

StepThrough

Algebra(Fraction(Integer))

Algebra(S)

BiModule(S, S)

BasicType

Evalable(S)

RightModule(Integer)

LeftModule(Fraction(Integer))

Canonical

AbelianSemiGroup

ConvertibleTo(Pattern(Float))

FullyLinearlyExplicitOver(S)

FullyEvalableOver(S)

NonAssociativeRng

PatternMatchable(Integer)

BiModule(Fraction(Integer), Fraction(Integer))

AbelianGroup