Significant Digits

<BODY LANG="EN"> <HR><center> <table border=1> <td><A NAME="tex2html144" HREF="node12.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html148" HREF="node12.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html150" HREF="node14.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node13.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="/hamlet-bin/qhandler?exit" target="results"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="/UCES/hamlet/"><img height=36 width=36 alt="HAMLET" src="/UCESIcons/tools/hamlet.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="/UCES/hamlet/usage.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </table> </center> <HR><H2><A NAME="SECTION00091000000000000000">Significant Digits</A></H2> <P> Let's begin by helping you get a feel for what it means to have ten-digit floating point numbers. Keep in mind that when we say a floating point number has ten digits, we mean that, when written in scientific notation, its mantissa has ten digits. <P> Suppose you are doing a calculation that depends upon knowing the surface area of the earth. Suppose further that your calculation must be accurate to ten significant digits. If your measure of the earth's surface area is not accurate to ten digits, then there's no way your calculation can be accurate to ten digits. In other words, you're calculations can be no more accurate than your data. <P> Is it reasonable to expect to be able to measure the surface area of the earth to ten digits of accuracy? Let's see exactly what that would entail. First of all, use Mathematica to calculate the surface area of the earth subject to the following two assumptions: <UL><LI> The radius of the earth is 6371 kilometers.<LI> The earth is a perfect sphere. </UL> Recall that the surface area of a sphere of radius <I>r</I> is <I>4 Pi r2</I>. <P> Thanks to Brigham Young, a city block in Salt Lake City is a square one- seventh of a mile on a side. What is the surface area of the earth expressed in city blocks? (In case you've forgotten, there are 5280 feet in a mile, 12 inches in a foot, 2.54 centimeters in an inch, and 100,000 centimeters in a kilometer.) <UL><LI> At a minimum, how many city blocks would you have to overlook in order to change the last of the ten digits that express your estimate of the surface area of the earth? </UL> Of course, the earth is not a perfect sphere and the radius that we have given is not entirely precise either. Furthermore, the surface of the earth is extremely wrinkled. All of this makes your estimate of the earth's surface area, with its ten digits of ``accuracy,'' rather suspect. Suppose that your estimate is accurate to one-tenth of one percent. <UL><LI> In that case, how many of the ten digits can you trust, and how many are just garbage? Explain your answer. </UL><HR><center> <table border=1> <td><A NAME="tex2html144" HREF="node12.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html148" HREF="node12.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html150" HREF="node14.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node13.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="/hamlet-bin/qhandler?exit" target="results"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="/UCES/hamlet/"><img height=36 width=36 alt="HAMLET" src="/UCESIcons/tools/hamlet.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="/UCES/hamlet/usage.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </table> </center> <HR><P><ADDRESS> <I>Joseph Zachary and Thomas L. Marchioro</I> </ADDRESS> </BODY>