repeating decimal

Some divisions never finish cleanly — instead they fall into a loop, spitting out the same digits over and over. Divide 1 by 3 and you get 0.3333… with the 3 repeating forever; divide 1 by 7 and you get 0.142857142857… with the whole block 142857 cycling endlessly. These are repeating decimals.

A repeating decimal is one in which, from some point on, a fixed block of digits repeats without end. We mark the repeating block with a bar over it, so 0.3333… is written 0.3 with a bar, and 0.142857142857… is 0.142857 with a bar over those six digits. The repeating block can be one digit or many.

Crucially, every repeating decimal is a rational number, and there is a clean trick to recover the fraction. For 0.3333…, let x = 0.3333…, so 10x = 3.3333…; subtracting gives 9x = 3, hence x = 1/3. In fact a real number is rational exactly when its decimal either terminates or repeats — irrational numbers are precisely the ones whose decimals never repeat.