GAP (ref) - Chapter 17: Rational Numbers

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17 Rational Numbers
17.1 Rationals: Global Variables
17.1-1 Rationals
17.2 Elementary Operations for Rationals
17.2-1 IsRat
17.2-2 IsPosRat
17.2-3 IsNegRat
17.2-4 NumeratorRat
17.2-5 DenominatorRat
17.2-6 Rat
17.2-7 Random

### 17 Rational Numbers

The rationals form a very important field. On the one hand it is the quotient field of the integers (see chapter 14). On the other hand it is the prime field of the fields of characteristic zero (see chapter 60).

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 (when canceled) are integers
    11

#### 17.1 Rationals: Global Variables

##### 17.1-1 Rationals

`‣ Rationals` | ( global variable )

`‣ IsRationals`( obj ) | ( filter )

`Rationals` is the field \(ℚ\) of rational integers, as a set of cyclotomic numbers, see Chapter 18 for basic operations, Functions for the field `Rationals` can be found in the chapters 58 and 60.

`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

#### 17.2 Elementary Operations for Rationals

##### 17.2-1 IsRat

`‣ IsRat`( obj ) | ( category )

Every rational number lies in the category `IsRat`, which is a subcategory of `IsCyc` (18.1-3).

    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

##### 17.2-2 IsPosRat

`‣ IsPosRat`( obj ) | ( category )

Every positive rational number lies in the category `IsPosRat`.

##### 17.2-3 IsNegRat

`‣ IsNegRat`( obj ) | ( category )

Every negative rational number lies in the category `IsNegRat`.

##### 17.2-4 NumeratorRat

`‣ NumeratorRat`( rat ) | ( function )

`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> # numerator and denominator are made relatively prime:
    gap> NumeratorRat( 66/123 );
    22
    gap> NumeratorRat( 17/-13 );  # numerator holds the sign of the rational
    -17
    gap> NumeratorRat( 11 );      # integers are rationals with denominator 1
    11

##### 17.2-5 DenominatorRat

`‣ DenominatorRat`( rat ) | ( function )

`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> # numerator and denominator are made relatively prime:
    gap> DenominatorRat( 66/123 );
    41
    gap> # the denominator holds the sign of the rational:
    gap> DenominatorRat( 17/-13 );
    13
    gap> DenominatorRat( 11 ); # integers are rationals with denominator 1
    1

##### 17.2-6 Rat

`‣ Rat`( elm ) | ( attribute )

`Rat` returns a rational number rat whose meaning depends on the type of elm.

If elm is a string consisting of digits `'0'`, `'1'`, \(\ldots\), `'9'` and `'-'` (at the first position), `'/'` and the decimal dot `'.'` then rat is the rational described by this string. If elm is a rational number, then `Rat` returns elm. The operation `String` (27.7-6) 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

##### 17.2-7 Random

`‣ Random`( Rationals ) | ( operation )

`Random` for rationals returns pseudo random rationals which are the quotient of two random integers. See the description of `Random` (14.2-12) for details. (Also see `Random` (30.7-1).)

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