IntegerNumberSystemΒΆ

si.spad line 1

An IntegerNumberSystem is a model for the integers.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
<: (%, %) -> Boolean
from PartialOrder
<=: (%, %) -> Boolean
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean
from PartialOrder
>=: (%, %) -> Boolean
from PartialOrder
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
abs: % -> %
from OrderedRing
addmod: (%, %, %) -> %
addmod(a, b, p), 0<=a, b<p>1, means a+b mod p.
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
associates?: (%, %) -> Boolean
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
base: () -> %
base() returns the base for the operations of IntegerNumberSystem.
binomial: (%, %) -> %
from CombinatorialFunctionCategory
bit?: (%, %) -> Boolean
bit?(n, i) returns true if and only if i-th bit of n is a 1.
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Integer -> %
from NonAssociativeRing
commutator: (%, %) -> %
from NonAssociativeRng
convert: % -> DoubleFloat
from ConvertibleTo DoubleFloat
convert: % -> Float
from ConvertibleTo Float
convert: % -> InputForm
from ConvertibleTo InputForm
convert: % -> Integer
from ConvertibleTo Integer
convert: % -> Pattern Integer
from ConvertibleTo Pattern Integer
copy: % -> %
copy(n) gives a copy of n.
D: % -> %
from DifferentialRing
D: (%, NonNegativeInteger) -> %
from DifferentialRing
dec: % -> %
dec(x) returns x - 1.
differentiate: % -> %
from DifferentialRing
differentiate: (%, NonNegativeInteger) -> %
from DifferentialRing
divide: (%, %) -> Record(quotient: %, remainder: %)
from EuclideanDomain
euclideanSize: % -> NonNegativeInteger
from EuclideanDomain
even?: % -> Boolean
even?(n) returns true if and only if n is even.
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
factorial: % -> %
from CombinatorialFunctionCategory
gcd: (%, %) -> %
from GcdDomain
gcd: List % -> %
from GcdDomain
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
from GcdDomain
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
inc: % -> %
inc(x) returns x + 1.
init: %
from StepThrough
invmod: (%, %) -> %
invmod(a, b), 0<=a<b>1, (a, b)=1 means 1/a mod b.
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
length: % -> %
length(a) length of a in digits.
mask: % -> %
mask(n) returns 2^n-1 (an n bit mask).
max: (%, %) -> %
from OrderedSet
min: (%, %) -> %
from OrderedSet
mulmod: (%, %, %) -> %
mulmod(a, b, p), 0<=a, b<p>1, means a*b mod p.
multiEuclidean: (List %, %) -> Union(List %, failed)
from EuclideanDomain
negative?: % -> Boolean
from OrderedRing
nextItem: % -> Union(%, failed)
from StepThrough
odd?: % -> Boolean
odd?(n) returns true if and only if n is odd.
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %)
from PatternMatchable Integer
permutation: (%, %) -> %
from CombinatorialFunctionCategory
positive?: % -> Boolean
from OrderedRing
positiveRemainder: (%, %) -> %
positiveRemainder(a, b) (where b > 1) yields r where 0 <= r < b and r == a rem b.
powmod: (%, %, %) -> %
powmod(a, b, p), 0<=a, b<p>1, means a^b mod p.
prime?: % -> Boolean
from UniqueFactorizationDomain
principalIdeal: List % -> Record(coef: List %, generator: %)
from PrincipalIdealDomain
quo: (%, %) -> %
from EuclideanDomain
random: % -> %
random(n) creates a random element from 0 to n-1.
rational: % -> Fraction Integer
rational(n) creates a rational number (see Fraction Integer).
rational?: % -> Boolean
rational?(n) tests if n is a rational number (see Fraction Integer).
rationalIfCan: % -> Union(Fraction Integer, failed)
rationalIfCan(n) creates a rational number, or returns “failed” if this is not possible.
recip: % -> Union(%, failed)
from MagmaWithUnit
rem: (%, %) -> %
from EuclideanDomain
retract: % -> Integer
from RetractableTo Integer
retractIfCan: % -> Union(Integer, failed)
from RetractableTo Integer
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
shift: (%, %) -> %
shift(a, i) shift a by i digits.
sign: % -> Integer
from OrderedRing
sizeLess?: (%, %) -> Boolean
from EuclideanDomain
smaller?: (%, %) -> Boolean
from Comparable
squareFree: % -> Factored %
from UniqueFactorizationDomain
squareFreePart: % -> %
from UniqueFactorizationDomain
submod: (%, %, %) -> %
submod(a, b, p), 0<=a, b<p>1, means a-b mod p.
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
symmetricRemainder: (%, %) -> %
symmetricRemainder(a, b) (where b > 1) yields r where `` -b/2 <= r < b/2 ``.
unit?: % -> Boolean
from EntireRing
unitCanonical: % -> %
from EntireRing
unitNormal: % -> Record(unit: %, canonical: %, associate: %)
from EntireRing
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

BasicType

BiModule(%, %)

CancellationAbelianMonoid

canonicalUnitNormal

CharacteristicZero

CoercibleTo OutputForm

CombinatorialFunctionCategory

CommutativeRing

CommutativeStar

Comparable

ConvertibleTo DoubleFloat

ConvertibleTo Float

ConvertibleTo InputForm

ConvertibleTo Integer

ConvertibleTo Pattern Integer

DifferentialRing

EntireRing

EuclideanDomain

GcdDomain

IntegralDomain

LeftModule %

LeftOreRing

Magma

MagmaWithUnit

Module %

Monoid

multiplicativeValuation

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianGroup

OrderedAbelianMonoid

OrderedAbelianSemiGroup

OrderedCancellationAbelianMonoid

OrderedIntegralDomain

OrderedRing

OrderedSet

PartialOrder

PatternMatchable Integer

PrincipalIdealDomain

RealConstant

RetractableTo Integer

RightModule %

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough

UniqueFactorizationDomain

unitsKnown