Repeating Decimals

Divide 1 by 3. 10 divided by 3 is 3, remainder 1. Bring down a zero: 10 divided by 3 is 3, remainder 1, again. The same remainder keeps coming back, so the same digit does too, forever: $\frac{1}{3}$ is 0.333... This is a repeating decimal.

Instead of dots, a bar is drawn over the digits that repeat: $0.\overline{3}$ means the 3 goes on forever, and $0.\overline{09}$ means the pair 09 repeats. In $0.1\overline{6}$, which is $\frac{1}{6}$, only the 6 repeats, so the bar sits over the 6 alone.

Which fractions repeat? Our decimals count in tens, and 10 is $2\cdot 5$. A fraction in lowest terms ends when its bottom has no prime factors other than 2 and 5, like 4, 8, 20 or 25. Any other prime factor, such as 3, 7 or 11, makes the decimal repeat.