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

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

which is repeating.

When $\frac{1}{288}$ is expressed in base $13,$ is it terminating or repeating?

BRAlNBOLT Jul 18, 2024

#1**+1 **

First, let's convert 1/288 into base 13.

Since 1 is the smae in every base, and we have

\((192)_{13} = (1 × 13^2) + (9 × 13^1) + (2 × 13^0) = (288)_{10}\)

This means that

\(\frac{1}{192}_{13} = \frac{1}{288}_{10}\)

Now, we complete base division. The answer is approximately

\(0.0078229A44BC67118933...\)

I don't think the number is repeating, although I am not sure.

If I had to geuss with the first 20 digits, it isn't repeating, but maybe it is.

Thanks! :)

NotThatSmart Jul 18, 2024