+2653 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?
+1926 First, let's convert 1/288 into base 13.
Since 1 is the smae in every base, and we have
\((192)_{13} = (1 × 13^2) + (9 × 13^1) + (2 × 13^0) = (288)_{10} \), we have the equation
\(\frac{1}{192}_{13} = \frac{1}{288}_{10}\)
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! :)
