Question

Express 3.4444...(in base 13) as a base 10 fraction in reduced form.

Answer 1

You can write the base \(13\) number as a sum of decimal numbers:
\[\begin{align*}
3.\bar4_{13}&=3\times13^0+4\times13^{-1}+4\times13^{-2}+\cdots\\[1ex]
&=3+4\sum_{i=1}^\infty\frac{1}{13^i}
\end{align*}\]Perhaps you're familiar with the sum in this form? It's geometric, of the form \(\displaystyle\sum_{k\ge1}r^k\), which means if \(|r|<1\), the sum converges to \(\dfrac{r}{1-r}\).