Repeating Decimals in Other Bases - Convert to Base 10

To convert a repeating decimal in another base to a base-10 fraction, you can use the infinite geometric series formula or apply the same logic as base-10 repeating decimals, but with the new base.

For $.xyxy..._b$, the base-10 fraction is $\frac{(xy)_b}{b^2-1}$, where $(xy)_b$ is first converted to base 10.

For $.xyyy..._b$, the fraction is $\frac{(xy)_b - x}{b(b-1)}$.

Example: $.3535..._8$. The numerator is $35_8 = 3 \times 8 + 5 = 29_{10}$. The denominator is $8^2-1=63$. The fraction is $\frac{29}{63}$.