Changing Bases: Miscellaneous Topics

This section explores problems that combine base conversions with other mathematical tricks.

One example is converting a repeating decimal in a different base to a base-10 fraction. This requires using the formula for the sum of an infinite geometric series. For example, $.333..._7 = \frac{3}{7} + \frac{3}{49} + \frac{3}{343} + \dots$. This is a geometric series with $a = \frac{3}{7}$ and $r = \frac{1}{7}$, so the sum is $\frac{3/7}{1 - 1/7} = \frac{1}{2}$.

Another example involves finding the remainder when a base-n number is divided by n-1. The remainder is simply the sum of the digits of the base-n number, divided by n-1.