A recurring decimal goes on forever, repeating the same block of digits. It looks endless, yet it is really a tidy fraction in disguise. The trick is to write the decimal twice — shifted so the repeating tails line up — and then subtract to make the endless part cancel.

Take x = 0.\dot{3} = 0.333\ldots. Multiply by 10 to slide the point one place. Because the tail repeats, the two decimals have exactly the same digits after the point:

Subtract the second line from the first. Everything after the decimal point cancels, leaving a whole number:

So 0.\dot{3} = \tfrac{1}{3}. A single recurring digit always lands over 9.

When a two-digit block repeats, one shift isn't enough — multiply by 100 so the whole repeating block jumps past the point. Take x = 0.\dot{2}\dot{7} = 0.272727\ldots:

The pattern is clear: a single recurring digit gives a denominator of 9, and a two-digit block gives 99. The number of 9s matches the length of the repeating block.