|
[_____]
A variety of different types of operations are provided for rational
elements including arithmetic operations, comparison and predicates
and converting to a sequence.
Parent(r) : FldRatElt -> FldRat
Category(r) : FldRatElt -> Cat
+ a : FldRatElt -> FldRatElt
- a : FldRatElt -> FldRatElt
a + b : FldRatElt, FldRatElt -> FldRatElt
a - b : FldRatElt, FldRatElt -> FldRatElt
a * b : FldRatElt, FldRatElt -> FldRatElt
a ^ k : FldRatElt, RngIntElt -> FldRatElt
a / b : FldRatElt, FldRatElt -> FldRatElt
a +:= b : FldRatElt, FldRatElt -> FldRatElt
a -:= b : FldRatElt, FldRatElt -> FldRatElt
a *:= b : FldRatElt, FldRatElt -> FldRatElt
a /:= b : FldRatElt, FldRatElt -> FldRatElt
a ^:= k : FldRatElt, RngIntElt -> FldRatElt
The (integer) numerator of the rational number q in reduced form.
The (integer) denominator of the rational number q in reduced form.
This will always be a positive integer.
Rational numbers are always immediately put in reduced form, that is,
the greatest common divisor of numerator and denominator is taken out,
and the denominator will be positive.
> Numerator(10/-4);
-5
> Denominator(10/-4);
2
a eq b : FldRatElt, FldRatElt -> BoolElt
a ne b : FldRatElt, FldRatElt -> BoolElt
a in R : FldRatElt, Rng -> BoolElt
a notin R : FldRatElt, Rng -> BoolElt
IsZero(a) : FldRatElt -> BoolElt
IsOne(a) : FldRatElt -> BoolElt
IsMinusOne(a) : FldRatElt -> BoolElt
IsNilpotent(a) : FldRatElt -> BoolElt
IsIdempotent(a) : FldRatElt -> BoolElt
IsUnit(a) : FldRatElt -> BoolElt
IsZeroDivisor(a) : FldRatElt -> BoolElt
IsRegular(a) : FldRatElt -> BoolElt
IsIrreducible(a) : FldRatElt -> BoolElt
IsPrime(a) : FldRatElt -> BoolElt
Returns true if the rational number q is an element of the ring
of integers, false otherwise.
a gt b : FldRatElt, FldRatElt -> BoolElt
a ge b : FldRatElt, FldRatElt -> BoolElt
a lt b : FldRatElt, FldRatElt -> BoolElt
a le b : FldRatElt, FldRatElt -> BoolElt
Maximum(a, b) : FldRatElt, FldRatElt -> FldRatElt
Maximum(Q) : [FldRatElt] -> FldRatElt
Minimum(a, b) : FldRatElt, FldRatElt -> FldRatElt
Minimum(Q) : [FldRatElt] -> FldRatElt
The complex conjugate of q, which will be the rational number q itself.
The conjugate of q, which will be the rational number q itself.
The norm (in Q) of q, which will be the rational number q itself.
The trace (in Q) of q, which will be the rational number q itself.
Returns the minimal polynomial of the rational number q, which
is the monic linear polynomial with constant coefficient q in a
univariate polynomial ring R over the rational field. (If R has
not been created before with a name for its indeterminate, $.1-q
will be returned.)
Abs(q) : FldRatElt -> FldRatElt
The absolute value |q|of a rational number q.
Returns the sign of the rational number q, which is one of the integers
-1, 0, 1, corresponding to the cases q<0, q=0, and q>0.
The height of q=r/s. For r and s coprime, the height is defined as
the maximum of the absolute value of r and s.
The ceiling of the rational number q, that is, the least integer
greater than or equal to q.
The floor of the rational number q, that is, the largest integer
less than or equal to q.
This function returns the integer value of the rational number q rounded
to the nearest integer.
In the case of a tie, rounding is done away from zero
(that is, i + (1/2) is rounded to i + 1, for non-negative integers i
and i - (1/2) is rounded to i - 1, for non-positive integers i).
This function returns the integer truncation of the rational number q, that
is the integral part of q. Thus the effect is that of rounding towards 0.
ContFrac: BoolElt Default: false
Finds a rational approximation d of q such that the denominator
of d is bounded by M. If ContFrac is given then an
optimal approximation is computed using the continued fraction process.
By default d is obtained by some rounding procedure which is faster but
gives worse results.
Given a rational r, return the sequence of partial quotients of
the continued fraction expansion of r.
Given a continued fraction expansion C, return the rational r such
that C equals ContinuedFraction(r).
HJContinuedFraction(r) : FldRatElt -> [ RngIntElt ]
Given a rational r, return the sequence of partial quotients of
the Hirzebruch-Jung continued fraction expansion of r.
HJContinuedFractionValue(C) : [ RngIntElt ] -> FldRatElt
Given a Hirzebruch-Jung continued fraction expansion C, return the
rational r so that HJContinuedFraction(r) equals C.
Under certain circumstances it is useful to have a partial inverse of
the function ψm:Q -> Z/mZ of taking residues
modulo m (where the obvious value of ψm is only defined for rational
numbers with denominator in smallest terms coprime to m); the partial
inverse of the function is sometimes referred to as `rational reconstruction'.
For s∈Z/mZ the value of ψ - 1(s) is the rational number r for
which ψm(r)=s and, in addition, the absolute values of both
the numerator and denominator of r are at most Sqrt(m/2);
such r does not always exist, but if r exists it is unique.
RationalReconstruction(s) : FldFinElt -> BoolElt, FldRatElt
Given an element s of a ring S of m elements, return
a Boolean flag indicating whether or not a rational number
r exists such that for the representation r=n/d in minimal
terms it holds that n.d - 1 ≡ s mod m,
|n|≤Sqrt(m/2) and 0 < d ≤Sqrt(m/2).
If the flag
is true, the element r is also returned.
The ring S is allowed to be a residue class ring Integers(m)
or a finite field of prime cardinality p=m: FiniteField(p).
In addition, s is allowed to be a matrix over a prime finite field,
in which case the existence (and, if possible, value) of a rational
reconstruction of the matrix is determined.
Valuation(x, I) : FldRatElt, RngIntElt -> RngIntElt, FldRatElt
The valuation v of the rational number x at the prime p
(the prime ideal I). This is the
difference of the valuations of the numerator and denominator of x.
The optional second return value is the rational u such that x = pv u.
Eltseq(a) : FldRatElt -> [FldRatElt]
The sequence [a] for compatibility with the other field types.
[Next][Prev] [_____] [Left] [Up] [Index] [Root]
|