Helpful Base Conversion Property

I decided to write this question since I haven't yet seen other number base questions dealing with decimal conversions. We may all know how to convert whole numbers, but the process is a little bit different when dealing with fractions. To start: $\frac{1}{3}=\frac{3}{10^{1}}+\frac{3}{10^{2}}+\frac{3}{10^{3}}+...$ . We represent a decimal in base-10 by fractions of powers of 10. So naturally, a decimal represented in base-8 (octal) should be represented by fractions of powers of 8: $\frac{1}{3}=\frac{x_{1}}{8^{1}}+\frac{x_{2}}{8^{2}}+\frac{x_{3}}{8^{3}}+...$ . To solve we first multiply $\frac{1}{3}$ by 8: $\frac{8}{3}$ . We then change this improper fraction to a mixed number, giving us our first $x_{n}$ : $\frac{8}{3}=2\frac{2}{3};\hspace{1mm}x_{1}=2$ . We simply repeat this process to get closer: $\frac{2}{3}\cdot{8}=5\frac{1}{3}$ . If you look at the result of the second iteration of this algorithm, we get $\frac{1}{3}$ again, so we can be certain that this pattern of 252525 will continue on forever.