%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A rational.tex GAP documentation Martin Schoenert %% %A @(#)$Id: rational.tex,v 3.6 1993/02/19 10:48:42 gap Rel $ %% %Y Copyright 1990-1992, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %% This file describes the rational datatype, its operations and functions. %% %H $Log: rational.tex,v $ %H Revision 3.6 1993/02/19 10:48:42 gap %H adjustments in line length and spelling %H %H Revision 3.5 1992/04/06 16:40:34 martin %H fixed some more typos %H %H Revision 3.4 1992/04/02 21:06:23 martin %H changed *domain functions* to *set theoretic functions* %H %H Revision 3.3 1992/03/19 15:57:34 martin %H renamed chapter "Number Fields" to "Subfields of Cyclotomic Fields" %H %H Revision 3.2 1991/12/27 16:07:04 martin %H revised everything for GAP 3.1 manual %H %H Revision 3.1 1991/07/25 16:16:59 martin %H fixed some minor typos %H %H Revision 3.0 1991/04/11 11:33:12 martin %H Initial revision under RCS %H %% \Chapter{Rationals}% \index{type!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 {\GAP} 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\"'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsRat}% \index{test!for a rational} 'IsRat( )' 'IsRat' returns 'true' if , 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 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 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Numerator}% \index{numerator!of a rational} 'Numerator( )' 'Numerator' returns the numerator of the rational . 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'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Denominator}% \index{denominator!of a rational} 'Denominator( )' 'Denominator' returns the denominator of the rational . 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'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Comparisons of Rationals}% \index{comparisons!of rationals} ' = ' \\ ' \<> ' The equality operator '=' evaluates to 'true' if the two rationals and are equal and to 'false' otherwise. The inequality operator '\<>' evaluates to 'true' if the two rationals and are not equal and to 'false' otherwise. | gap> 2/3 = -4/-6; true gap> 66/123 <> 22/41; false gap> 17/13 = 11; false | ' \<\ ' \\ ' \<= ' \\ ' > ' \\ ' >= ' The operators '\<', '\<=', '>', and '=>' evaluate to 'true' if the rational is less than, less than or equal to, greater than, and greater than or equal to the rational and to 'false' otherwise. One rational $q_1 = n_1/d_1$ is less than another $q_2 = n_2/d_2$ if and only if $n_1 d_2 \< n_2 d_2$. 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 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Operations for Rationals}% \index{operations!for rationals} ' + ' \\ ' - ' \\ ' \*\ ' \\ ' / ' The operators '+', '-', '\*' and '/' evaluate to the sum, difference, product, and quotient of the two rationals and . For the quotient '/' must of course be nonzero, otherwise an error is signalled. Either operand may also be an integer , 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 | ' \^\ ' The powering operator '\^' returns the -th power of the rational . must be an integer. If the exponent is zero, '\^' is defined as 1; if is positive, '\^' is defined as the -fold product '\*\*..\*'; finally, if is negative, '\^' is defined as '(1/)\^-'. In this case 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 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Set Functions for Rationals}% As was already mentioned in the introduction of this chapter the {\GAP} 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Field Functions for Rationals}% As was already mentioned in the introduction of this chapter the {\GAP} 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, )' returns '[ ]', 'Norm( Rationals, )' and 'Trace( Rationals, )' return , and 'CharPol( Rationals, )' and 'MinPol( Rationals, )' both return '[ -, 1 ]'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E Emacs . . . . . . . . . . . . . . . . . . . . . local Emacs variables %% %% Local Variables: %% mode: outline %% outline-regexp: "\\\\Chapter\\|\\\\Section" %% fill-column: 73 %% eval: (hide-body) %% End: %%