RealNumberSystem

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The real number system category is intended as a model for the real numbers. The real numbers form an ordered normed field. Note that we have purposely not included DifferentialRing or the elementary functions (see TranscendentalFunctionCategory) in the definition.

* : (%, %) -> %

from Magma

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

from RightModule(Fraction(Integer))

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

from LeftModule(Fraction(Integer))

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

/ : (%, %) -> %

from Field

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

< : (%, %) -> Boolean

from PartialOrder

<= : (%, %) -> Boolean

from PartialOrder

= : (%, %) -> Boolean

from BasicType

> : (%, %) -> Boolean

from PartialOrder

>= : (%, %) -> Boolean

from PartialOrder

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

from RadicalCategory

^ : (%, Integer) -> %

from DivisionRing

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

abs : % -> %

from OrderedRing

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associates? : (%, %) -> Boolean

from EntireRing

associator : (%, %, %) -> %

from NonAssociativeRng

ceiling : % -> %

ceiling x returns the small integer >= x.

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coerce : % -> %

from Algebra(%)

coerce : Fraction(Integer) -> %

from Algebra(Fraction(Integer))

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

convert : % -> DoubleFloat

from ConvertibleTo(DoubleFloat)

convert : % -> Float

from ConvertibleTo(Float)

convert : % -> Pattern(Float)

from ConvertibleTo(Pattern(Float))

divide : (%, %) -> Record(quotient : %, remainder : %)

from EuclideanDomain

euclideanSize : % -> NonNegativeInteger

from EuclideanDomain

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

floor : % -> %

floor x returns the largest integer <= x.

fractionPart : % -> %

fractionPart x returns the fractional part of x.

gcd : (%, %) -> %

from GcdDomain

gcd : List(%) -> %

from GcdDomain

gcdPolynomial : (SparseUnivariatePolynomial(%), SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%)

from GcdDomain

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

max : (%, %) -> %

from OrderedSet

min : (%, %) -> %

from OrderedSet

multiEuclidean : (List(%), %) -> Union(List(%), "failed")

from EuclideanDomain

negative? : % -> Boolean

from OrderedRing

norm : % -> %

norm x returns the same as absolute value.

nthRoot : (%, Integer) -> %

from RadicalCategory

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

patternMatch : (%, Pattern(Float), PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %)

from PatternMatchable(Float)

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

positive? : % -> Boolean

from OrderedRing

prime? : % -> Boolean

from UniqueFactorizationDomain

principalIdeal : List(%) -> Record(coef : List(%), generator : %)

from PrincipalIdealDomain

quo : (%, %) -> %

from EuclideanDomain

recip : % -> Union(%, "failed")

from MagmaWithUnit

rem : (%, %) -> %

from EuclideanDomain

retract : % -> Fraction(Integer)

from RetractableTo(Fraction(Integer))

retract : % -> Integer

from RetractableTo(Integer)

retractIfCan : % -> Union(Fraction(Integer), "failed")

from RetractableTo(Fraction(Integer))

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

from RetractableTo(Integer)

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

round : % -> %

round x computes the integer closest to x.

sample : () -> %

from AbelianMonoid

sign : % -> Integer

from OrderedRing

sizeLess? : (%, %) -> Boolean

from EuclideanDomain

smaller? : (%, %) -> Boolean

from Comparable

sqrt : % -> %

from RadicalCategory

squareFree : % -> Factored(%)

from UniqueFactorizationDomain

squareFreePart : % -> %

from UniqueFactorizationDomain

subtractIfCan : (%, %) -> Union(%, "failed")

from CancellationAbelianMonoid

truncate : % -> %

truncate x returns the integer between x and 0 closest to x.

unit? : % -> Boolean

from EntireRing

unitCanonical : % -> %

from EntireRing

unitNormal : % -> Record(unit : %, canonical : %, associate : %)

from EntireRing

wholePart : % -> Integer

wholePart x returns the integer part of x.

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

IntegralDomain

Comparable

Module(Fraction(Integer))

noZeroDivisors

OrderedAbelianSemiGroup

LeftModule(Fraction(Integer))

RightModule(%)

GcdDomain

Monoid

AbelianMonoid

Algebra(%)

ConvertibleTo(DoubleFloat)

EuclideanDomain

EntireRing

NonAssociativeSemiRng

NonAssociativeAlgebra(Fraction(Integer))

CancellationAbelianMonoid

OrderedSet

MagmaWithUnit

NonAssociativeRing

RightModule(Fraction(Integer))

CoercibleFrom(Fraction(Integer))

RetractableTo(Integer)

RealConstant

unitsKnown

LeftModule(%)

canonicalUnitNormal

RadicalCategory

CommutativeStar

Module(%)

SetCategory

LeftOreRing

CoercibleTo(OutputForm)

Algebra(Fraction(Integer))

Rng

Field

PatternMatchable(Float)

ConvertibleTo(Float)

CommutativeRing

TwoSidedRecip

Magma

UniqueFactorizationDomain

SemiGroup

OrderedAbelianMonoid

DivisionRing

PartialOrder

CoercibleFrom(Integer)

AbelianGroup

AbelianSemiGroup

OrderedCancellationAbelianMonoid

RetractableTo(Fraction(Integer))

NonAssociativeSemiRing

ConvertibleTo(Pattern(Float))

SemiRng

canonicalsClosed

NonAssociativeAlgebra(%)

OrderedAbelianGroup

OrderedRing

PrincipalIdealDomain

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

NonAssociativeRng

Ring

CharacteristicZero

BasicType

BiModule(%, %)

SemiRing