A fraction is really a division: to write a fraction as a decimal you divide the top by the bottom. When you do, one of two things happens — and only ever these two.

Sometimes the division stops. The remainder runs out, and you are left with a tidy decimal that ends. This is a terminating decimal:

Other times the division never stops — the same remainders keep coming back, so the same digits repeat forever. This is a recurring decimal. A third is the classic example:

We do not write the threes out for ever — we put a dot over the repeating digit. So 0.\dot{3} means "3 repeating". Sometimes a whole block of digits repeats; we dot the first and last digit of the block. A seventh repeats a block of six:

Which fractions terminate?

There is a clean rule, and it comes straight from the number ten. Our decimal places are tenths, hundredths, thousandths — all powers of 10. And 10 = 2 \times 5, so a decimal that terminates is secretly a fraction whose denominator is built only from 2s and 5s.

So put the fraction in its lowest terms, then look at the prime factors of the denominator:

• Only 2s and 5s → terminates. For example \tfrac{1}{8} = \tfrac{1}{2^3} = 0.125, and \tfrac{1}{20} = \tfrac{1}{2^2 \times 5} = 0.05.
• Any other prime factor — a 3, a 7, an 11, … → recurs. For example \tfrac{1}{6} = \tfrac{1}{2 \times 3} has a stubborn 3, so \tfrac{1}{6} = 0.1\dot{6}.