80 lines
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80 lines
4.2 KiB
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<!DOCTYPE HTML PUBLIC "-//W3O//DTD W3 HTML 2.0//EN">
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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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<HEAD>
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<TITLE>2.1.2. Ratios</TITLE>
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<BODY>
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<meta name="description" value=" Ratios">
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<meta name="keywords" value="clm">
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<meta name="resource-type" value="document">
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<meta name="distribution" value="global">
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<P>
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<b>Common Lisp the Language, 2nd Edition</b>
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<BR> <HR><A NAME=tex2html1733 HREF="node19.html"><IMG ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME=tex2html1731 HREF="node16.html"><IMG ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME=tex2html1725 HREF="node17.html"><IMG ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <A NAME=tex2html1735 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="contents" SRC="icons/contents_motif.gif"></A> <A NAME=tex2html1736 HREF="index.html"><IMG ALIGN=BOTTOM ALT="index" SRC="icons/index_motif.gif"></A> <BR>
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<B> Next:</B> <A NAME=tex2html1734 HREF="node19.html"> Floating-Point Numbers</A>
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<B>Up:</B> <A NAME=tex2html1732 HREF="node16.html"> Numbers</A>
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<B> Previous:</B> <A NAME=tex2html1726 HREF="node17.html"> Integers</A>
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<HR> <P>
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<H2><A NAME=SECTION00612000000000000000>2.1.2. Ratios</A></H2>
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<P>
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<A NAME=591>A</A>
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<A NAME=592><i>ratio</i></A>
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is a number representing the mathematical ratio
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of two integers. Integers and ratios collectively constitute
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the type <tt>rational</tt>.
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The canonical representation of a rational number is as an
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integer if its value is integral, and otherwise as the ratio of two
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integers, the <i>numerator</i> and <i>denominator</i>, whose greatest
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common divisor is 1, and of which the denominator is positive (and in
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fact greater than 1, or else the value would be integral).
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A ratio is notated with
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<tt>/</tt> as a separator, thus: <tt>3/5</tt>. It is possible to notate
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ratios in non-canonical (unreduced) forms, such as <tt>4/6</tt>, but the
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Lisp function <tt>prin1</tt> always prints the canonical form for a
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ratio.
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<P>
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If any computation produces a result that is a ratio of
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two integers such that the denominator evenly divides the
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numerator, then the result is immediately converted to the equivalent
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integer. This is called the rule of <i>rational canonicalization</i>.
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<P>
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Rational numbers may be written as the possibly signed quotient of
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decimal numerals: an optional sign followed by two non-empty sequences of
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digits separated by a <tt>/</tt>. This syntax may be described as
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follows:
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<PRE><TT> <i>ratio</i> ::= [<i>sign</i>] {<i>digit</i>}+ <tt>/</tt> {<i>digit</i>}+
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<P>
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</TT></PRE>
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The second sequence may not consist
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entirely of zeros.
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For example:
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<P><pre>
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2/3 ;This is in canonical form
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4/6 ;A non-canonical form for the same number
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-17/23 ;A not very interesting ratio
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-30517578125/32768 ;This is <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40209.gif">
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10/5 ;The canonical form for this is <tt>2</tt>
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</pre>
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<P>
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To notate rational numbers in radices other than ten,
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one uses the same radix specifiers
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(one of <tt>#<i>nn</i>R</tt>, <tt>#O</tt>, <tt>#B</tt>, or <tt>#X</tt>) as for integers.
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For example:
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<P><pre>
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#o-101/75 ;Octal notation for <tt>-65/61</tt>
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#3r120/21 ;Ternary notation for <tt>15/7</tt>
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#Xbc/ad ;Hexadecimal notation for <tt>188/173</tt>
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#xFADED/FACADE ;Hexadecimal notation for <tt>1027565/16435934</tt>
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</pre>
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<P>
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<BR> <HR><A NAME=tex2html1733 HREF="node19.html"><IMG ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME=tex2html1731 HREF="node16.html"><IMG ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME=tex2html1725 HREF="node17.html"><IMG ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <A NAME=tex2html1735 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="contents" SRC="icons/contents_motif.gif"></A> <A NAME=tex2html1736 HREF="index.html"><IMG ALIGN=BOTTOM ALT="index" SRC="icons/index_motif.gif"></A> <BR>
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<B> Next:</B> <A NAME=tex2html1734 HREF="node19.html"> Floating-Point Numbers</A>
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<B>Up:</B> <A NAME=tex2html1732 HREF="node16.html"> Numbers</A>
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<B> Previous:</B> <A NAME=tex2html1726 HREF="node17.html"> Integers</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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</ADDRESS>
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