# 12 Rationals

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"`.

## 12.1 IsRat

`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 ```

## 12.2 Numerator

`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`.

## 12.3 Denominator

`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`.

## 12.4 Floor

`Floor(r)`

This function returns the largest integer smaller or equal to r.

```    gap> Floor(-2/3);
-1
gap> Floor(2/3);
0```

## 12.5 Mod1

`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 ]```

## 12.6 Comparisons of Rationals

`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 ```

## 12.7 Operations for Rationals

`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
24 Jun 2022