Recurring Decimals and Geometric Sequences

Recurring decimals are decimals numbers whose digits repeat at regular intervals. For instance, 0.666… and 1.2341341341… are recurring decimals. They are denoted by  and  respectively.

All recurring decimals can be written as fractions. Below we give two methods by which the recurring decimal  can be converted into a fraction.

Let .

Then .

Subtracting the second equation from the first, we get .

Then .

Hence .