document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=645485>Question 645485</A>: <I><SPAN STYLE=\"word-break: break-all;\"> HI tutors!  \r<BR>\n" );
document.write( "\n" );
document.write( "I have this problem in my college math book:<BR>\n" );
document.write( "The decimal expansion of the irrational number pi symbol begins 3.141592653589793 . . . . Use your calculator to determine which of the following rational numbers is the best approximation for the irrational number pi symbol:<BR>\n" );
document.write( "22/7 , 355/113 , 103,993/33,102 , 2,508,429,787/798,458,000.<BR>\n" );
document.write( "Your calculator may tell you that some of these numbers are equal to pi symbol, but that just indicates that the number agrees with pi symbol for as many decimal places as your calculator can handle (usually 10-14).  No rational number is exactly equal to pi symbol.\r<BR>\n" );
document.write( "\n" );
document.write( "Here is what I get from left to right for these fractions:<BR>\n" );
document.write( "3.142857143 , 3.14159292 , 3.141592653 , and 3.141592654.\r<BR>\n" );
document.write( "\n" );
document.write( "The book's answer is the last fraction.  \r<BR>\n" );
document.write( "\n" );
document.write( "Why is the last fraction the right answer? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #405496 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=405496\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The fractions below\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "22/7, 355/113, 103993/33102, 2508429787/798458000\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "convert to the following numbers in decimal form\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3.14285714285714, 3.14159292035398, 3.1415926530119, 3.141592653589794\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Now compare this with the given value of pi, which is 3.141592653589793...\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First Rational Approximation (22/7)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3.141592653589793<BR>\n" );
document.write( "3.14285714285714\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: first number is the given value of pi and second number is the rational approximation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice how 22/7 only matches up to 3.14 and then it's off for the rest of the expansion.\r<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Second Rational Approximation (103993/33102)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3.141592653589793<BR>\n" );
document.write( "3.14159292035398\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: first number is the given value of pi and second number is the rational approximation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We got a better approximation this time since it matches 3.141592\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So far, the second approximation is better\r<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Third Rational Approximation (355/113)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3.141592653589793<BR>\n" );
document.write( "3.1415926530119\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: first number is the given value of pi and second number is the rational approximation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We got a better approximation this time since it matches 3.141592653\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So far, the third approximation is best\r<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Fourth Rational Approximation (2508429787/798458000)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3.141592653589793<BR>\n" );
document.write( "3.141592653589794\r<BR>\n" );
document.write( "\n" );
document.write( "Note: first number is the given value of pi and second number is the rational approximation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We get the most accurate match this time since it matches 3.14159265358979\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The only difference between this rational approximation and the given decimal expansion is that the '3' at the end doesn't match with the 4.\r<BR>\n" );
document.write( "\n" );
document.write( "==========================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the fraction 2508429787/798458000 gives the best approximation of pi\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<font color=\"red\">--------------------------------------------------------------------------------------------------------------</font><BR>\n" );
document.write( "If you need more help, email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com?Subject=I%20Need%20Algebra%20Help\">jim_thompson5910@hotmail.com</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Also, please consider visiting my website: <a  rel=nofollow HREF=\"http://www.freewebs.com/jimthompson5910/home.html\">http://www.freewebs.com/jimthompson5910/home.html</a> and making a donation. Thank you\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Jim<BR>\n" );
document.write( "<font color=\"red\">--------------------------------------------------------------------------------------------------------------</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );