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<!Converted with LaTeX2HTML 0.6.5 (Tue Nov 15 1994) by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds >
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<TITLE>12.4. Arithmetic Operations</TITLE>
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<meta name="description" value=" Arithmetic Operations">
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<b>Common Lisp the Language, 2nd Edition</b>
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<BR> <HR><A NAME=tex2html3090 HREF="node126.html"><IMG ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME=tex2html3088 HREF="node121.html"><IMG ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME=tex2html3082 HREF="node124.html"><IMG ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <A NAME=tex2html3092 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="contents" SRC="icons/contents_motif.gif"></A> <A NAME=tex2html3093 HREF="index.html"><IMG ALIGN=BOTTOM ALT="index" SRC="icons/index_motif.gif"></A> <BR>
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<B> Next:</B> <A NAME=tex2html3091 HREF="node126.html"> Irrational and Transcendental </A>
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<B>Up:</B> <A NAME=tex2html3089 HREF="node121.html"> Numbers</A>
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<B> Previous:</B> <A NAME=tex2html3083 HREF="node124.html"> Comparisons on Numbers</A>
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<HR> <P>
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<H1><A NAME=SECTION001640000000000000000>12.4. Arithmetic Operations</A></H1>
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<P>
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Each of the functions in this section requires that its arguments all be
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numbers; to call one with a non-number is an error. Unless otherwise
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specified, each works on all types of numbers, automatically performing
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any required coercions when arguments are of different types.
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<P>
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<BR><b>[Function]</b><BR>
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<tt>+ &rest <i>numbers</i></tt><P>This returns the sum of the arguments. If there are no arguments, the result
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is <tt>0</tt>, which is an identity for this operation.
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<P>
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<hr>
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<b>Compatibility note:</b> While <tt>+</tt> is compatible with its use in Lisp Machine Lisp,
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it is incompatible with MacLisp, which uses <tt>+</tt> for fixnum-only
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addition.
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<hr>
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<P>
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<BR><b>[Function]</b><BR>
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<tt>- <i>number</i> &rest <i>more-numbers</i></tt><P>The function <tt>-</tt>, when given one argument, returns the negative
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of that argument.
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<P>
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The function <tt>-</tt>, when given more than one argument, successively subtracts
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from the first argument all the others, and returns the result.
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For example, <tt>(- 3 4 5)</tt> => <tt>-6</tt>.
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<P>
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<hr>
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<b>Compatibility note:</b> While <tt>-</tt> is compatible with its use in Lisp Machine Lisp,
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it is incompatible with MacLisp, which uses <tt>-</tt> for fixnum-only
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subtraction.
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Also, <tt>-</tt> differs from <tt>difference</tt> as used in most Lisp
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systems in the case of one argument.
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<hr>
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<P>
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<BR><b>[Function]</b><BR>
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<tt>* &rest <i>numbers</i></tt><P>This returns the product of the arguments.
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If there are no arguments, the result
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is <tt>1</tt>, which is an identity for this operation.
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<P>
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<hr>
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<b>Compatibility note:</b> While <tt>*</tt> is compatible with its use in Lisp Machine Lisp,
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it is incompatible with MacLisp, which uses <tt>*</tt> for fixnum-only
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multiplication.
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<hr>
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<P>
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<BR><b>[Function]</b><BR>
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<tt>/ <i>number</i> &rest <i>more-numbers</i></tt><P>The function <tt>/</tt>, when given more than one argument, successively divides
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the first argument by all the others and returns the result.
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<P>
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<img align=bottom alt="change_begin" src="gif/change_begin.gif"><br>
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It is generally accepted that it is an error for any argument other than the
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first to be zero.
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<br><img align=bottom alt="change_end" src="gif/change_end.gif">
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<P>
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With one argument, <tt>/</tt> reciprocates the argument.
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<P>
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<img align=bottom alt="change_begin" src="gif/change_begin.gif"><br>
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It is generally accepted that it is an error in this case for the argument
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to be zero.
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<br><img align=bottom alt="change_end" src="gif/change_end.gif">
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<P>
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<tt>/</tt> will produce a ratio if the mathematical quotient of two integers
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is not an exact integer. For example:
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<P><pre>
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(/ 12 4) => 3
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(/ 13 4) => 13/4
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(/ -8) => -1/8
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(/ 3 4 5) => 3/20
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</pre><P>
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To divide one integer by another producing an integer result,
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use one of the functions <tt>floor</tt>, <tt>ceiling</tt>, <tt>truncate</tt>,
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or <tt>round</tt>.
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<P>
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If any argument is a floating-point number,
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then the rules of floating-point contagion apply.
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<P>
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<hr>
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<b>Compatibility note:</b> What <tt>/</tt> does is totally unlike what the usual
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<tt>//</tt> or <tt>quotient</tt> operator does. In most Lisp systems,
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<tt>quotient</tt> behaves like <tt>/</tt> except when dividing integers,
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in which case it behaves like <tt>truncate</tt> of two arguments;
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this behavior is mathematically intractable, leading to such
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anomalies as
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<P><pre>
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(quotient 1.0 2.0) => 0.5 but (quotient 1 2) => 0
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</pre><P>
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In contrast, the Common Lisp function <tt>/</tt> produces these results:
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<P><pre>
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(/ 1.0 2.0) => 0.5 and (/ 1 2) => 1/2
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</pre><P>
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In practice <tt>quotient</tt> is used only when one is sure that both arguments
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are integers, <i>or</i> when one is sure that at least one argument
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is a floating-point number. <tt>/</tt> is tractable for its purpose
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and works for <i>any</i> numbers.
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<hr>
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<P>
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<BR><b>[Function]</b><BR>
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<tt>1+ <i>number</i> <BR></tt><tt>1- <i>number</i></tt><P><tt>(1+ <i>x</i>)</tt> is the same as <tt>(+ <i>x</i> 1)</tt>.
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<P>
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<tt>(1- <i>x</i>)</tt> is the same as <tt>(- <i>x</i> 1)</tt>.
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Note that the short name may be confusing: <tt>(1- <i>x</i>)</tt> does <i>not</i> mean
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1-<i>x</i>; rather, it means <i>x</i>-1.
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<P>
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<hr>
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<b>Rationale:</b> These are included primarily for compatibility with MacLisp
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and Lisp Machine Lisp. Some programmers prefer always to write <tt>(+ x 1)</tt> and
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<tt>(- x 1)</tt> instead of <tt>(1+ x)</tt> and <tt>(1- x)</tt>.
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<hr>
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<b>Implementation note:</b> Compiler writers are very strongly encouraged to ensure
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that <tt>(1+ x)</tt> and <tt>(+ x 1)</tt> compile into identical code, and
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similarly for <tt>(1- x)</tt> and <tt>(- x 1)</tt>, to avoid pressure on a Lisp
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programmer to write possibly less clear code for the sake of efficiency.
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This can easily be done as a source-language transformation.
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<hr>
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<P>
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<BR><b>[Macro]</b><BR>
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<tt>incf <i>place</i> [<i>delta</i>] <BR></tt><tt>decf <i>place</i> [<i>delta</i>]</tt><P>The number produced by the form <i>delta</i>
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is added to (<tt>incf</tt>) or subtracted from (<tt>decf</tt>)
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the number in the generalized variable named by <i>place</i>,
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and the sum is stored back into <i>place</i> and returned.
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The form <i>place</i> may be any form acceptable
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as a generalized variable to <tt>setf</tt>.
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If <i>delta</i> is not supplied, then the number in <i>place</i> is changed
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by <tt>1</tt>.
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For example:
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<P><pre>
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(setq n 0)
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(incf n) => 1 and now n => 1
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(decf n 3) => -2 and now n => -2
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(decf n -5) => 3 and now n => 3
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(decf n) => 2 and now n => 2
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</pre><P>
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The effect of <tt>(incf <i>place</i> <i>delta</i>)</tt>
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is roughly equivalent to
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<P><pre>
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(setf <i>place</i> (+ <i>place</i> <i>delta</i>))
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</pre><P>
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except that the latter would evaluate any subforms of <i>place</i>
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twice, whereas <tt>incf</tt> takes care to evaluate them only once.
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Moreover, for certain <i>place</i> forms <tt>incf</tt> may be
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significantly more efficient than the <tt>setf</tt> version.
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<p>
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<img align=bottom alt="change_begin" src="gif/change_begin.gif"><br>
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X3J13 voted in March 1988 (PUSH-EVALUATION-ORDER) <A NAME=11688> </A>
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to clarify order of evaluation (see section <A HREF="node80.html#SETFSECTION">7.2</A>).
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<br><img align=bottom alt="change_end" src="gif/change_end.gif">
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<P>
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<BR><b>[Function]</b><BR>
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<tt>conjugate <i>number</i></tt><P>This returns the complex conjugate of <i>number</i>. The conjugate
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of a non-complex number is itself. For a complex number <tt>z</tt>,
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<P><pre>
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(conjugate z) == (complex (realpart z) (- (imagpart z)))
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</pre><P>
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For example:
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<P><pre>
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(conjugate #C(3/5 4/5)) => #C(3/5 -4/5)
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(conjugate #C(0.0D0 -1.0D0)) => #C(0.0D0 1.0D0)
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(conjugate 3.7) => 3.7
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</pre><P>
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<P>
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<BR><b>[Function]</b><BR>
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<tt>gcd &rest <i>integers</i></tt><P>This returns the greatest common divisor of all the arguments,
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which must be integers. The result of <tt>gcd</tt> is always a non-negative
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integer.
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If one argument is given, its absolute value is returned.
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If no arguments are given, <tt>gcd</tt> returns <tt>0</tt>,
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which is an identity for this operation.
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For three or more arguments,
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<P><pre>
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(gcd <i>a</i> <i>b</i> <i>c</i> ... <i>z</i>) == (gcd (gcd <i>a</i> <i>b</i>) <i>c</i> ... <i>z</i>)
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</pre><P>
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<P>
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Here are some examples of the use of <tt>gcd</tt>:
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<P><pre>
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(gcd 91 -49) => 7
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(gcd 63 -42 35) => 7
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(gcd 5) => 5
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(gcd -4) => 4
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(gcd) => 0
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</pre><P>
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<P>
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<BR><b>[Function]</b><BR>
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<tt>lcm <i>integer</i> &rest <i>more-integers</i></tt><P>This returns the least common multiple of its arguments,
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which must be integers.
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The result of <tt>lcm</tt> is always a non-negative integer.
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For two arguments that are not both zero,
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<P><pre>
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(lcm <i>a</i> <i>b</i>) == (/ (abs (* <i>a</i> <i>b</i>)) (gcd <i>a</i> <i>b</i>))
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</pre><P>
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If one or both arguments are zero,
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<P><pre>
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(lcm <i>a</i> 0) == (lcm 0 <i>a</i>) == 0
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</pre><P>
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<P>
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For one argument, <tt>lcm</tt> returns the absolute value of that argument.
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For three or more arguments,
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<P><pre>
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(lcm <i>a</i> <i>b</i> <i>c</i> ... <i>z</i>) == (lcm (lcm <i>a</i> <i>b</i>) <i>c</i> ... <i>z</i>)
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</pre><P>
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<P>
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Some examples:
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<P><pre>
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(lcm 14 35) => 70
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(lcm 0 5) => 0
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(lcm 1 2 3 4 5 6) => 60
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</pre><P>
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<P>
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Mathematically, <tt>(lcm)</tt> should return infinity. Because Common Lisp
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does not have a representation for infinity, <tt>lcm</tt>, unlike <tt>gcd</tt>,
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always requires at least one argument.
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<P>
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<img align=bottom alt="change_begin" src="gif/change_begin.gif"><br>
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X3J13 voted in January 1989
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(LCM-NO-ARGUMENTS) <A NAME=11749> </A>
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to specify that <tt>(lcm) => 1</tt>.
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<P>
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This is one of my biggest boners. The identity for <tt>lcm</tt> is of course 1,
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not infinity, and so <tt>(lcm)</tt> ought to have been defined to return 1.
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Sorry about that, though in point of fact very few users have complained
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to me that this mistake in the first edition has cramped their programming style.
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<br><img align=bottom alt="change_end" src="gif/change_end.gif">
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<P>
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<BR> <HR><A NAME=tex2html3090 HREF="node126.html"><IMG ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME=tex2html3088 HREF="node121.html"><IMG ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME=tex2html3082 HREF="node124.html"><IMG ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <A NAME=tex2html3092 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="contents" SRC="icons/contents_motif.gif"></A> <A NAME=tex2html3093 HREF="index.html"><IMG ALIGN=BOTTOM ALT="index" SRC="icons/index_motif.gif"></A> <BR>
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<B> Next:</B> <A NAME=tex2html3091 HREF="node126.html"> Irrational and Transcendental </A>
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<B>Up:</B> <A NAME=tex2html3089 HREF="node121.html"> Numbers</A>
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<B> Previous:</B> <A NAME=tex2html3083 HREF="node124.html"> Comparisons on Numbers</A>
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<HR> <P>
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<HR>
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<P><ADDRESS>
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AI.Repository@cs.cmu.edu
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