## Monday, June 18, 2012

### Cool Decimal Representations of Fractions

I believe that God created the world in such a way that we can discover it, by creating things with pattern and order. Over the years, I've learned a lot of cool patterns -- and quite a few of them in the realm of fractions. I wanted to share just a few of them with you.

The first are the 'sevenths', shown below:
$\begin{matrix} \tfrac{1}{7} = 0.{\color{Blue} 142857}14285714... \\ \tfrac{2}{7} = 0.2857{\color{Blue} 142857}1428... \\ \tfrac{3}{7} = 0.42857{\color{Blue} 142857}142... \\ \tfrac{4}{7} = 0.57{\color{Blue} 142857}142857... \\ \tfrac{5}{7} = 0.7{\color{Blue} 142857}1428571... \\ \tfrac{6}{7} = 0.857{\color{Blue} 142857}14285... \\ \end{matrix}$

I remember first being amazed when showed me that 1/7 was what's called a repeating decimal -- that is, the digits 142857 repeat over and over again in that order forever, which is often written as $\inline 0.\overline{142857}$.  I remember trying hard to remember that sequence of six digits. As amazing as that was, I was blown away when I looked at the other seventh's fractions, and notice that same string of digits show up in each of their representations!  The only difference between them is where in the sequence they start!

Here's a few other sequences that are easy to remember:
$\inline \begin{matrix} \tfrac{1}{9}= 0.111111111... \\ \tfrac{2}{9}= 0.222222222... \\ \tfrac{3}{9}= 0.333333333... \\ \tfrac{4}{9}= 0.444444444... \\ \tfrac{5}{9}= 0.555555555.. \\ \tfrac{6}{9}= 0.666666666... \\ \tfrac{7}{9}= 0.777777777... \\ \tfrac{8}{9}= 0.888888888... \\ \end{matrix}$
Each of the 'ninths' is simply the numerator repeated forever. In fact, this is just the simplest case of a much broader pattern -- that of any denominator of all 9's.  Any fraction with all 9's in the denominator will have a representation that simply repeats the numerator over and over again. (A small catch: you may have to add a few zeroes to pad the numerator so it has as many digits as the number of nines in the denominator).  Here's a few examples:
$\frac{{\color{Blue} 237}}{999} = 0.{\color{Blue} 237}237237237237... \\$
$\frac{{\color{Blue} 8514}}{99999} = 0.{\color{Blue} 08514}08514085$

If I haven't bored you enough, I'll show a few others that I've learned
$\begin{matrix} \tfrac{1}{11} = 0.09090909090909... \\ \tfrac{2}{11} = 0.18181818181818... \\ \tfrac{3}{11} = 0.27272727272727... \\ \tfrac{4}{11} = 0.36363636363636... \\ \tfrac{5}{11} = 0.45454545454545... \\ \tfrac{6}{11} = 0.54545454545454... \\ \tfrac{7}{11} = 0.63636363636363... \\ \tfrac{8}{11} = 0.72727272727272... \\ \tfrac{9}{11} = 0.81818181818181... \\ \tfrac{10}{11} = 0.90909090909090... \\ \end{matrix}$
The pattern here may not be quite as obvious, but 'elevenths' have a nice trick to them too. Take the numerator, and multiply it by nine, and that's the number that gets repeated.

Finally, a pair of fractions that might have been twins separated at birth:
$\begin{matrix} \frac{1}{{\color{Red} 27}} = 0.{\color{Blue} 037}037037037037037... \\ \\ \frac{1}{{\color{Blue} 37}} = 0.{\color{Red} 027}027027027027027... \\ \end{matrix}$

It's ok if you just said "Awesome" or something similar. As I said above - I think God designed these patterns to help us learn and understand math better, and this lost pair is not the only pair that exists, just the first one I was ever shown. I won't list anymore, but I will point out that 27*37 = 999, and that is perhaps enough of a hint to get you on a path to discovering your own. If you find some others, feel free to post them in the comments below!