Repeating Decimals in Other Bases - Keeping Same Base

To convert a repeating decimal in base $b$ to a fraction in the same base, you can first convert to a base-10 fraction and then convert the numerator and denominator back to base $b$.

For example, to convert $0.444..._8$ to a base-8 fraction: First, convert to base 10: $\frac{4}{7}$. Then convert the numerator and denominator to base 8: $4_{10}=4_8$ and $7_{10}=7_8$. So the answer is $\frac{4_8}{7_8}$.