The rationals form a very important field. On the one hand it is the quotient field of the integers (see chapter Integers). On the other hand it is the prime field of the fields of characteristic zero (see chapter Abelian Number Fields).
The former comment suggests the representation actually used. A rational is represented as a pair of integers, called numerator and denominator. Numerator and denominator are reduced, i.e., their greatest common divisor is 1. If the denominator is 1, the rational is in fact an integer and is represented as such. The numerator holds the sign of the rational, thus the denominator is always positive.
Because the underlying integer arithmetic can compute with arbitrary size integers, the rational arithmetic is always exact, even for rationals whose numerators and denominators have thousands of digits.
gap> 2/3; 2/3 gap> 66/123; # numerator and denominator are made relatively prime 22/41 gap> 17/-13; # the numerator carries the sign; -17/13 gap> 121/11; # rationals with denominator 1 (after cancelling) are integers 11
Rationals V
IsRationals( obj ) P
Rationals is the field Q of rational integers,
as a set of cyclotomic numbers,
see Chapter Cyclotomic Numbers for basic operations,
Functions for the field Rationals can be found in the
chapters Fields and Division Rings and Abelian Number Fields.
IsRationals returns true for a prime field that consists of
cyclotomic numbers --for example the GAP object Rationals--
and false for all other GAP objects.
gap> Size( Rationals ); 2/3 in Rationals; infinity true
IsRat( obj ) C
Every rational number lies in the category IsRat,
which is a subcategory of IsCyc (see Cyclotomic Numbers).
gap> IsRat( 2/3 ); true gap> IsRat( 17/-13 ); true gap> IsRat( 11 ); true gap> IsRat( IsRat ); # `IsRat' is a function, not a rational false
IsPosRat( obj ) C
Every positive rational number lies in the category IsPosRat.
IsNegRat( obj ) C
Every negative rational number lies in the category IsNegRat.
NumeratorRat( rat ) F
NumeratorRat returns the numerator of the rational rat.
Because the numerator holds the sign of the rational it may be any
integer.
Integers are rationals with denominator 1, thus NumeratorRat is the
identity function for integers.
gap> NumeratorRat( 2/3 ); 2 gap> NumeratorRat( 66/123 ); # numerator and denominator are made relatively prime 22 gap> NumeratorRat( 17/-13 ); # the numerator holds the sign of the rational -17 gap> NumeratorRat( 11 ); # integers are rationals with denominator 1 11
DenominatorRat( rat ) F
DenominatorRat returns the denominator of the rational rat.
Because the numerator holds the sign of the rational the denominator is
always a positive integer.
Integers are rationals with the denominator 1, thus DenominatorRat
returns 1 for integers.
gap> DenominatorRat( 2/3 ); 3 gap> DenominatorRat( 66/123 ); # numerator and denominator are made relatively prime 41 gap> DenominatorRat( 17/-13 ); # the denominator holds the sign of the rational 13 gap> DenominatorRat( 11 ); # integers are rationals with denominator 1 1
Rat( elm ) A
Rat returns a rational number rat whose meaning depends on the type
of elm.
If elm is a string consisting of digits '0', '1', …,
'9' and '-' (at the first position), '/' and the decimal dot
'.' then rat is the rational described by this string.
The operation String (see String) can be used to compute a string for
rational numbers, in fact for all cyclotomics.
gap> Rat( "1/2" ); Rat( "35/14" ); Rat( "35/-27" ); Rat( "3.14159" ); 1/2 5/2 -35/27 314159/100000
Random( Rationals )
Random for rationals returns pseudo random rationals which are the
quotient of two random integers. See the description of Random for integers
(Random!for integers) for details. (Also see Random.)
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GAP 4 manual
March 2006