Repeating decimals

Type 1 : 4 into a calculator and 0.25 appears, job done. Type 1 : 3 and the display fills up with threes: 0.3333333. Both fractions look equally harmless, yet one behaves completely differently from the other. The nice part: you can tell from the denominator which case you are in, before you calculate at all.

The fraction bar is a division

Every fraction is a division: 1/4 means 1 divided by 4. If you work it out on paper, you bring in zeros. The 1 becomes 10 tenths, divided by 4 that is 2 tenths with remainder 2 tenths. Those 2 tenths are 20 hundredths, divided by 4 that is 5 hundredths with remainder 0. No remainder means the calculation is over, and what stands there is 0.25. With 1 : 3 that never happens. 10 tenths divided by 3 is 3 tenths, remainder 1 tenth. That remainder becomes 10 hundredths again, gives 3 again, leaves remainder 1 again. The same remainder comes back, so the same digit comes back, endlessly.

The bar over the period

Endless digit sequences are not written with dots but with a bar. The digits that repeat are called the period, and the bar sits exactly above them: 0.3333... becomes $0.\overline{3}$. For 1/7 = 0.142857142857... six digits repeat, so the bar goes over 142857. Sometimes the repetition starts later: 1/6 gives 0.1666..., where the 1 is not part of it and the bar covers only the 6, giving $0.1\overline{6}$. Digits before the period like that are called the pre-period.

Which denominators come out even?

A fraction gives a terminating decimal exactly when it can be rewritten with a power of ten as denominator, that is 10, 100, 1000 and so on. And those numbers are built only from twos and fives: 10 = 2 · 5, 100 = 2 · 2 · 5 · 5. That gives the rule: a fully simplified fraction terminates when its denominator contains only the prime factors 2 and 5. It works for 8 = 2 · 2 · 2, because 1/8 = 125/1000 = 0.125. It works for 20 = 2 · 2 · 5 as well, because 1/20 = 5/100 = 0.05. But if any other prime sits in the denominator, such as the 3 in 6 or the 7 in 14, the denominator can never be turned into a power of ten, and the number repeats. Simplifying first matters: 6/12 looks like a repeater because of the 12, but simplified it is 1/2 and therefore a clean 0.5.