The rationals form a very important field. On the one hand it is the quotient field of the integers (see Integers). On the other hand it is the prime field of the fields of characteristic zero (see Subfields of Cyclotomic 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; 22/41 # numerator and denominator are made relatively prime gap> 17/-13; -17/13 # the numerator carries the sign gap> 121/11; 11 # rationals with denominator 1 (after cancelling) are integers
The first sections of this chapter describe the functions that test whether an object is a rational (see IsRat), and select the numerator and denominator of a rational (see Numerator, Denominator).
The next sections describe the rational operations (see Comparisons of Rationals, and Operations for Rationals).
The GAP3 object Rationals
is the field domain of all rationals. All
set theoretic functions are applicable to this domain (see chapter
Domains and Set Functions for Rationals). Since Rationals
is a
field all field functions are also applicable to this domain and its
elements (see chapter Fields and Field Functions for Rationals).
All external functions are defined in the file "LIBNAME/rational.g"
.
IsRat( obj )
IsRat
returns true
if obj, which can be an arbitrary object, is a
rational and false
otherwise. Integers are rationals with denominator
1, thus IsRat
returns true
for integers. IsRat
will signal an
error if obj is an unbound variable or a procedure call.
gap> IsRat( 2/3 );
true
gap> IsRat( 17/-13 );
true
gap> IsRat( 11 );
true
gap> IsRat( IsRat );
false # IsRat
is a function, not a rational
Numerator( rat )
Numerator
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 Numerator
is the identity
function for integers.
gap> Numerator( 2/3 ); 2 gap> Numerator( 66/123 ); 22 # numerator and denominator are made relatively prime gap> Numerator( 17/-13 ); -17 # the numerator holds the sign of the rational gap> Numerator( 11 ); 11 # integers are rationals with denominator 1
Denominator
(see Denominator) is the counterpart to Numerator
.
Denominator( rat )
Denominator
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
Denominator
returns 1 for integers.
gap> Denominator( 2/3 ); 3 gap> Denominator( 66/123 ); 41 # numerator and denominator are made relatively prime gap> Denominator( 17/-13 ); 13 # the denominator holds the sign of the rational gap> Denominator( 11 ); 1 # integers are rationals with denominator 1
Numerator
(see Numerator) is the counterpart to Denominator
.
Floor(r)
This function returns the largest integer smaller or equal to r.
gap> Floor(-2/3); -1 gap> Floor(2/3); 0
Mod1(r)
The argument should be a rational or a list. If r is a rational, it
returns (Numerator(r) mod Denominator(r))/Denominator(r)
. If r is a
list, it returns List(r,Mod1)
. This function is very useful for working
in ℚ/ℤ.
gap> Mod1([-2/3,-1,7/4,3]); [ 1/3, 0, 3/4, 0 ]
q1 = q2
q1 <> q2
The equality operator =
evaluates to true
if the two rationals q1
and q2 are equal and to false
otherwise. The inequality operator
<>
evaluates to true
if the two rationals q1 and q2 are not
equal and to false
otherwise.
gap> 2/3 = -4/-6; true gap> 66/123 <> 22/41; false gap> 17/13 = 11; false
q1 < q2
q1 <= q2
q1 > q2
q1 >= q2
The operators <
, <=
, >
, and =>
evaluate to true
if the
rational q1 is less than, less than or equal to, greater than, and
greater than or equal to the rational q2 and to false
otherwise.
One rational q1 = n1/d1 is less than another q2 = n2/d2 if and only if n1 d2 < n2 d2. This definition is of course only valid because the denominator of rationals is always defined to be positive. This definition also extends to the comparison of rationals with integers, which are interpreted as rationals with denominator 1. Rationals can also be compared with objects of other types. They are smaller than objects of any other type by definition.
gap> 2/3 < 22/41; false gap> -17/13 < 11; true
q1 + q2
q1 - q2
q1 * q2
q1 / q2
The operators +
, -
, *
and /
evaluate to the sum, difference,
product, and quotient of the two rationals q1 and q2. For the
quotient /
q2 must of course be nonzero, otherwise an error is
signalled. Either operand may also be an integer i, which is
interpreted as a rational with denominator 1. The result of those
operations is always reduced. If, after the reduction, the denominator
is 1, the rational is in fact an integer, and is represented as such.
gap> 2/3 + 4/5; 22/15 gap> 7/6 * 2/3; 7/9 # note how the result is cancelled gap> 67/6 - 1/6; 11 # the result is an integer
q ^ i
The powering operator ^
returns the i-th power of the rational q.
i must be an integer. If the exponent i is zero, q^i
is
defined as 1; if i is positive, q^i
is defined as the i-fold
product q*q*..*q
; finally, if i is negative, q^i
is
defined as (1/q)^-i
. In this case q must of course be nonzero.
gap> (2/3) ^ 3; 8/27 gap> (-17/13) ^ -1; -13/17 # note how the sign switched gap> (1/2) ^ -2; 4
12.8 Set Functions for Rationals
As was already mentioned in the introduction of this chapter the GAP3
object Rationals
is the domain of all rationals. All set theoretic
functions, e.g., Intersection
and Size
, are applicable to this
domain.
gap> Intersection( Rationals, [ E(4)^0, E(4)^1, E(4)^2, E(4)^3 ] );
[ -1, 1 ] # E(4)
is the complex square root of -1
gap> Size( Rationals );
"infinity"
This does not seem to be very useful.
12.9 Field Functions for Rationals
As was already mentioned in the introduction of this chapter the GAP3
object Rationals
is the field of all rationals. All field functions,
e.g., Norm
and MinPol
are applicable to this domain and its elements.
However, since the field of rationals is the prime field, all those
functions are trivial. Therefore, Conjugates( Rationals, q )
returns
[ q ]
, Norm( Rationals, q )
and Trace( Rationals, q )
return
q, and CharPol( Rationals, q )
and MinPol( Rationals, q )
both
return [ -q, 1 ]
.
gap3-jm