Exact and Inexact Numbers

<BODY LANG="EN"> <HR><center> <table border=1> <td><A NAME="tex2html60" HREF="node3.html"><img height=36 width=36 alt="PREVIOUS" src="http://www.cs.utah.edu/icons/latex2html/previous.gif"></A></td> <td><A NAME="tex2html66" HREF="node2.html"><img height=36 width=36 alt="UP" src="http://www.cs.utah.edu/icons/latex2html/up.gif"></A></td> <td><A NAME="tex2html68" HREF="node5.html"><img height=36 width=36 alt="NEXT" src="http://www.cs.utah.edu/icons/latex2html/next.gif"></A></td> <td><a href="contents-node4.html"><img height=36 width=36 alt="CONTENTS" src="http://www.cs.utah.edu/icons/latex2html/toc.gif"></a></td> <td><a href="reset.hamlet"><img height=36 width=36 alt="RESET" src="http://www.cs.utah.edu/icons/latex2html/stop.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="http://www.cs.utah.edu/icons/latex2html/copyright.gif"></a></td> <td><a href="mailto:hamlet@cs.utah.edu"><img height=36 width=36 alt="FEEDBACK" src="http://www.cs.utah.edu/icons/latex2html/feedback.gif"></a></td> <td><a href="http://www.cs.utah.edu/%7Ehamlet/icons.html"><img height=36 width=36 alt="ABOUT" src="http://www.cs.utah.edu/icons/latex2html/about.gif"></a></td> </table> </center> <HR><H2><A NAME="SECTION00022000000000000000">Exact and Inexact Numbers</A></H2> <P> <A NAME="exact"> </A> Maple provides two kinds of numbers: <EM>exact</EM> and <EM>inexact</EM>. If you do arithmetic with exact numbers (and they aren't too large) the answer that you get back is exactly right. If you do arithmetic with inexact numbers (and they aren't too large or too small) the answer may have a small amount of roundoff error. Although we'll treat this distinction at length in future lessons, it is convenient to take a preliminary look right now. <P> In Maple, <EM>rational numbers</EM> (integers and fractions) are exact. If you do arithmetic with rational numbers, your answer will be correct to the last digit. Here are some examples. <P> <UL> <LI> We divide two integers, and end up with an exact fraction, expressed in lowest terms: <P> <table><td valign=top><a href="send5.hamlet"><img height=24 width=24 alt="EXECUTE" src="http://www.cs.utah.edu/icons/latex2html/execute.gif"></a></td><td valign=top><code>2468/1324;</code></td></table> <LI> We add two fractions, and end up with another fraction containing a lot of digits: <P> <table><td valign=top><a href="send6.hamlet"><img height=24 width=24 alt="EXECUTE" src="http://www.cs.utah.edu/icons/latex2html/execute.gif"></a></td><td valign=top><code>23848/837443 + 87533/9283932;</code></td></table> <LI> We do exponentiation and wind up with an astonishing number of digits: <P> <table><td valign=top><a href="send7.hamlet"><img height=24 width=24 alt="EXECUTE" src="http://www.cs.utah.edu/icons/latex2html/execute.gif"></a></td><td valign=top><code>444^555;</code></td></table> <P> </UL> <P> Maple also provides inexact <EM>floating-point numbers</EM>. Maple floating-point numbers look different from integers because they either contain a decimal point or are written using scientific notation. If you do arithmetic with floating-point numbers, your answer will be rounded off to ten digits. We will repeat the same three examples from above, using floating-point numbers instead of integers. <P> <UL> <LI> We add two floating-point numbers (note the decimal points), and wind up with a decimal approximation to the exact answer: <P> <table><td valign=top><a href="send8.hamlet"><img height=24 width=24 alt="EXECUTE" src="http://www.cs.utah.edu/icons/latex2html/execute.gif"></a></td><td valign=top><code>2468./1324.;</code></td></table> <LI> We add two ratios of floating-point numbers, and wind up with a floating-point answer. Note that we have written the numbers in scientific notation. The portion before the ``e'' is the mantissa, and the portion following the ``e'' is the exponent: <P> <table><td valign=top><a href="send9.hamlet"><img height=24 width=24 alt="EXECUTE" src="http://www.cs.utah.edu/icons/latex2html/execute.gif"></a></td><td valign=top><code>2.3848e4 / 8.37443e5 + 8.7533e4 / 9.283932e6;</code></td></table> <LI> We do an exponentiation. Even though one of the operands is an integer and the other is a floating-point number, the answer is a floating-point number. When even one inexact number is used in a computation, the result is always inexact: <P> <table><td valign=top><a href="send10.hamlet"><img height=24 width=24 alt="EXECUTE" src="http://www.cs.utah.edu/icons/latex2html/execute.gif"></a></td><td valign=top><code>444.^555;</code></td></table> <P> </UL> <P> Notice that each of the last three answers contains a ten-digit mantissa scaled, if necessary, by a power of ten. <P> Why, if Maple can do exact arithmetic to the very last digit, do we need to bother with floating-point numbers? Can you think of some reasons why? <P> <P><A NAME="tex2html5" HREF="answer63.html">Click here for the answer</A><P><HR><center> <table border=1> <td><A NAME="tex2html60" HREF="node3.html"><img height=36 width=36 alt="PREVIOUS" src="http://www.cs.utah.edu/icons/latex2html/previous.gif"></A></td> <td><A NAME="tex2html66" HREF="node2.html"><img height=36 width=36 alt="UP" src="http://www.cs.utah.edu/icons/latex2html/up.gif"></A></td> <td><A NAME="tex2html68" HREF="node5.html"><img height=36 width=36 alt="NEXT" src="http://www.cs.utah.edu/icons/latex2html/next.gif"></A></td> <td><a href="contents-node4.html"><img height=36 width=36 alt="CONTENTS" src="http://www.cs.utah.edu/icons/latex2html/toc.gif"></a></td> <td><a href="reset.hamlet"><img height=36 width=36 alt="RESET" src="http://www.cs.utah.edu/icons/latex2html/stop.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="http://www.cs.utah.edu/icons/latex2html/copyright.gif"></a></td> <td><a href="mailto:hamlet@cs.utah.edu"><img height=36 width=36 alt="FEEDBACK" src="http://www.cs.utah.edu/icons/latex2html/feedback.gif"></a></td> <td><a href="http://www.cs.utah.edu/%7Ehamlet/icons.html"><img height=36 width=36 alt="ABOUT" src="http://www.cs.utah.edu/icons/latex2html/about.gif"></a></td> </table> </center> <HR><P><ADDRESS> <I><A NAME="tex2html1" HREF="http://www.cs.utah.edu/~zachary">Joseph L. Zachary</A> <BR> <A NAME="tex2html2" HREF="http://www.cs.utah.edu/~hamlet">Hamlet Project</A> <BR> <A NAME="tex2html3" HREF="http://www.cs.utah.edu/">Department of Computer Science</A> <BR> <A NAME="tex2html4" HREF="http://www.utah.edu/ HTML_Docs/UofUHome.html">University of Utah</A></I> </ADDRESS> </BODY>