Post Closed as "too localized" by Bruce Westbury, Yemon Choi, Daniel Litt, Igor Rivin, fedja
3 empiric test with a c program

We know $\frac{1}{81}$ gives us $0.\overline{0123456790}$

How do we create a recurrent decimal with the property of repeating:

$0.\overline{0123456789}$

a) Is there a method to construct such number?

b) Is there a solution?

c) Is the solution in $\mathbb{Q}$?

Thanks!

Beco

Edited:

According with wikipedia page: http://en.wikipedia.org/wiki/Decimal One could get this number by applying this series. Supppose:

$M=123456789$, $x=10^{10}$, then $0.\overline{0123456789}= \frac{M}{x}\cdot$ $\sum$ ${(10^{-9})}^k$ $=\frac{M}{x}\cdot\frac{1}{1-10^{-9}}$ $=\frac{M}{9999999990}$

Unless my calculator is crazy, this is giving me $0.012345679$, not the expected number. Although the example of wikipedia works fine with $0.\overline{123}$.

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.

2nd Edition:

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:

#include <stdio.h>
int main(void)
{
long double b;
b=123456789.0/9999999990.0;
printf("%.40Lf\n", b);
}


Result: $0.0123456789123456787266031042804570461158$

2 some equations I tired

We know $\frac{1}{81}$ gives us $0.\overline{0123456790}$

How do we create a recurrent decimal with the property of repeating:

$0.\overline{0123456789}$

a) Is there a method to construct such number?

b) Is there a solution?

c) Is the solution in $\mathbb{Q}$?

Thanks!

Beco

Edited:

According with wikipedia page: http://en.wikipedia.org/wiki/Decimal One could get this number by applying this series. Supppose:

$M=123456789$, $x=10^{10}$, then $0.\overline{0123456789}= \frac{M}{x}\cdot$ $\sum$ ${(10^{-9})}^k$ $=\frac{M}{x}\cdot\frac{1}{1-10^{-9}}$ $=\frac{M}{9999999990}$

Unless my calculator is crazy, this is giving me $0.012345679$, not the expected number. Although the example of wikipedia works fine with $0.\overline{123}$.

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.

1