When fractions are expressed in different bases, they can be terminating or repeating. For example, when $\frac{1}{5}$ is expressed in base $3,$ the result is

Question

0.\overline{0121}_3 = 0.01210121 \dots,
which is repeating.



When \frac{1}{288} is expressed in base 25, is it terminating or repeating?

Answer 1

To determine if $\frac{1}{288}$ is terminating or repeating in base 25, we need to find the decimal representation of this fraction in base 25.

We start by converting 288 into base 25. We have:

$288 = 11 \cdot 25^1 + 13 \cdot 25^0 = B_{25}$ in base 25.

Therefore, $\frac{1}{288}$ in base 25 is:

$\frac{1}{B_{25}} = \frac{1}{11 \cdot 25^1 + 13 \cdot 25^0} = \frac{1}{11B_{25} + 13}.$

To express this fraction in decimal form, we perform long division:

\[
\begin{array}{c|c l}
0.004_25 = & 0.01323330 \cdots \\
\cline{2-2}
& 0.011\overline{01323330} \\
\cline{2-2}
& \cdots
\end{array}
\]

Thus, in base 25, $\frac{1}{288}$ is repeating.