How do I create a recurrent decimal with this property: - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T20:58:40Z http://mathoverflow.net/feeds/question/59912 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59912/how-do-i-create-a-recurrent-decimal-with-this-property How do I create a recurrent decimal with this property: Beco 2011-03-29T01:18:17Z 2011-03-29T02:23:39Z <p>We know $\frac{1}{81}$ gives us $0.\overline{0123456790}$</p> <p>How do we create a recurrent decimal with the property of repeating:</p> <p>$0.\overline{0123456789}$</p> <p>a) Is there a method to construct such number?</p> <p>b) Is there a solution?</p> <p>c) Is the solution in $\mathbb{Q}$?</p> <p>Thanks!</p> <p>Beco</p> <p>Edited:</p> <p>According with wikipedia page: <a href="http://en.wikipedia.org/wiki/Decimal" rel="nofollow">http://en.wikipedia.org/wiki/Decimal</a> One could get this number by applying this series. Supppose:</p> <p>$M=123456789$, $x=10^{10}$, then $0.\overline{0123456789}= \frac{M}{x}\cdot$ $\sum$ ${(10^{-9})}^k$ $=\frac{M}{x}\cdot\frac{1}{1-10^{-9}}$ $=\frac{M}{9999999990}$</p> <p>Unless my calculator is crazy, this is giving me $0.012345679$, not the expected number. Although the example of wikipedia works fine with $0.\overline{123}$.</p> <p>The answer that was there for some time and then deleted gave me another equation: $\frac{M}{1-10^{-10}}$. Well, that does not work either.</p> <p>2nd Edition:</p> <p>Just to get rid of the gnome calculator, Running a simple program written in C with very large precision (long double) I get this result:</p> <pre><code>#include &lt;stdio.h&gt; int main(void) { long double b; b=123456789.0/9999999990.0; printf("%.40Lf\n", b); } </code></pre> <p>Result: $0.0123456789123456787266031042804570461158$</p>