[ref] 16 Rational Numbers

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# 16 Rational Numbers

### Sections

1. Elementary Operations for Rationals

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;
    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

- `Rationals V`

      gap> Size( Rationals ); 2/3 in Rationals;
      infinity
      true

- `IsRationals(` obj`) C`

  `Rationals` is a subset of `Cyclotomics`, so see chapter Cyclotomic Numbers for arithmetic operations and comparison of rationals. Functions for the field `Rationals` can be found in chapters fields and division rings and Abelian Number Fields.

  ## 16.1 Elementary Operations for Rationals

- `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 );
      false    # `IsRat' is a function, not a rational

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

- `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 );
      41    # numerator and denominator are made relatively prime
      gap> DenominatorRat( 17/-13 );
      13    # the denominator holds the sign of the rational
      gap> DenominatorRat( 11 );
      1    # integers are rationals with denominator 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'`, ¼, `'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
  May 2002