  
  [1X17 [33X[0;0YRational Numbers[133X[101X
  
  [33X[0;0YThe  [13Xrationals[113X  form  a  very  important  field.  On  the one hand it is the
  quotient field of the integers (see chapter [14X14[114X). On the other hand it is the
  prime field of the fields of characteristic zero (see chapter [14X60[114X).[133X
  
  [33X[0;0YThe  former comment suggests the representation actually used. A rational is
  represented  as  a  pair  of  integers,  called  [13Xnumerator[113X  and [13Xdenominator[113X.
  Numerator  and  denominator are [13Xreduced[113X, 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.[133X
  
  [33X[0;0YBecause  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.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X2/3;[127X[104X
    [4X[28X2/3[128X[104X
    [4X[25Xgap>[125X [27X66/123;  # numerator and denominator are made relatively prime[127X[104X
    [4X[28X22/41[128X[104X
    [4X[25Xgap>[125X [27X17/-13;  # the numerator carries the sign;[127X[104X
    [4X[28X-17/13[128X[104X
    [4X[25Xgap>[125X [27X121/11;  # rationals with denominator 1 (when canceled) are integers[127X[104X
    [4X[28X11[128X[104X
  [4X[32X[104X
  
  
  [1X17.1 [33X[0;0YRationals: Global Variables[133X[101X
  
  [1X17.1-1 Rationals[101X
  
  [33X[1;0Y[29X[2XRationals[102X [32X global variable[133X
  [33X[1;0Y[29X[2XIsRationals[102X( [3Xobj[103X ) [32X filter[133X
  
  [33X[0;0Y[2XRationals[102X  is  the  field  [22Xℚ[122X  of  rational  integers, as a set of cyclotomic
  numbers,  see  Chapter [14X18[114X  for  basic  operations,  Functions  for the field
  [2XRationals[102X can be found in the chapters [14X58[114X and [14X60[114X.[133X
  
  [33X[0;0Y[2XIsRationals[102X  returns  [9Xtrue[109X  for  a  prime  field that consists of cyclotomic
  numbers  –for  example the [5XGAP[105X object [2XRationals[102X– and [9Xfalse[109X for all other [5XGAP[105X
  objects.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSize( Rationals ); 2/3 in Rationals;[127X[104X
    [4X[28Xinfinity[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X17.2 [33X[0;0YElementary Operations for Rationals[133X[101X
  
  [1X17.2-1 IsRat[101X
  
  [33X[1;0Y[29X[2XIsRat[102X( [3Xobj[103X ) [32X Category[133X
  
  [33X[0;0YEvery  rational number lies in the category [2XIsRat[102X, which is a subcategory of
  [2XIsCyc[102X ([14X18.1-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsRat( 2/3 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsRat( 17/-13 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsRat( 11 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsRat( IsRat );  # `IsRat' is a function, not a rational[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X17.2-2 IsPosRat[101X
  
  [33X[1;0Y[29X[2XIsPosRat[102X( [3Xobj[103X ) [32X Category[133X
  
  [33X[0;0YEvery positive rational number lies in the category [2XIsPosRat[102X.[133X
  
  [1X17.2-3 IsNegRat[101X
  
  [33X[1;0Y[29X[2XIsNegRat[102X( [3Xobj[103X ) [32X Category[133X
  
  [33X[0;0YEvery negative rational number lies in the category [2XIsNegRat[102X.[133X
  
  [1X17.2-4 NumeratorRat[101X
  
  [33X[1;0Y[29X[2XNumeratorRat[102X( [3Xrat[103X ) [32X function[133X
  
  [33X[0;0Y[2XNumeratorRat[102X  returns  the  numerator  of  the  rational  [3Xrat[103X.  Because  the
  numerator holds the sign of the rational it may be any integer. Integers are
  rationals with denominator [22X1[122X, thus [2XNumeratorRat[102X is the identity function for
  integers.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNumeratorRat( 2/3 );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27X# numerator and denominator are made relatively prime:[127X[104X
    [4X[25Xgap>[125X [27XNumeratorRat( 66/123 );[127X[104X
    [4X[28X22[128X[104X
    [4X[25Xgap>[125X [27XNumeratorRat( 17/-13 );  # numerator holds the sign of the rational[127X[104X
    [4X[28X-17[128X[104X
    [4X[25Xgap>[125X [27XNumeratorRat( 11 );      # integers are rationals with denominator 1[127X[104X
    [4X[28X11[128X[104X
  [4X[32X[104X
  
  [1X17.2-5 DenominatorRat[101X
  
  [33X[1;0Y[29X[2XDenominatorRat[102X( [3Xrat[103X ) [32X function[133X
  
  [33X[0;0Y[2XDenominatorRat[102X  returns  the  denominator  of  the rational [3Xrat[103X. Because the
  numerator  holds  the  sign  of  the  rational  the  denominator is always a
  positive  integer.  Integers  are  rationals  with  the  denominator 1, thus
  [2XDenominatorRat[102X returns 1 for integers.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDenominatorRat( 2/3 );[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27X# numerator and denominator are made relatively prime:[127X[104X
    [4X[25Xgap>[125X [27XDenominatorRat( 66/123 );[127X[104X
    [4X[28X41[128X[104X
    [4X[25Xgap>[125X [27X# the denominator holds the sign of the rational:[127X[104X
    [4X[25Xgap>[125X [27XDenominatorRat( 17/-13 );[127X[104X
    [4X[28X13[128X[104X
    [4X[25Xgap>[125X [27XDenominatorRat( 11 ); # integers are rationals with denominator 1[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [1X17.2-6 Rat[101X
  
  [33X[1;0Y[29X[2XRat[102X( [3Xelm[103X ) [32X attribute[133X
  
  [33X[0;0Y[2XRat[102X returns a rational number [3Xrat[103X whose meaning depends on the type of [3Xelm[103X.[133X
  
  [33X[0;0YIf  [3Xelm[103X  is a string consisting of digits [10X'0'[110X, [10X'1'[110X, [22X...[122X, [10X'9'[110X and [10X'-'[110X (at the
  first  position),  [10X'/'[110X  and  the  decimal  dot  [10X'.'[110X then [3Xrat[103X is the rational
  described by this string. If [3Xelm[103X is a rational number, then [10XRat[110X returns [3Xelm[103X.
  The  operation  [2XString[102X ([14X27.7-6[114X) can be used to compute a string for rational
  numbers, in fact for all cyclotomics.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRat( "1/2" );  Rat( "35/14" );  Rat( "35/-27" );  Rat( "3.14159" );[127X[104X
    [4X[28X1/2[128X[104X
    [4X[28X5/2[128X[104X
    [4X[28X-35/27[128X[104X
    [4X[28X314159/100000[128X[104X
  [4X[32X[104X
  
  [1X17.2-7 Random[101X
  
  [33X[1;0Y[29X[2XRandom[102X( [3XRationals[103X ) [32X operation[133X
  
  [33X[0;0Y[2XRandom[102X  for rationals returns pseudo random rationals which are the quotient
  of two random integers. See the description of [2XRandom[102X ([14X14.2-13[114X) for details.
  (Also see [2XRandom[102X ([14X30.7-1[114X).)[133X
  
