Terminating and Recurring Decimals

When you turn a fraction into a decimal, exactly one of two things happens. Either it terminates, meaning it stops, like three eighths which is 0.375 and finished after three decimal places. Or it recurs, meaning a block of digits repeats forever, like one third which is nought point three three three going on with no end. There is no third possibility — every fraction does one or the other. Two things are worth knowing straight away. A recurring decimal never behaves randomly; a fixed block of digits comes round on a cycle. And a recurring decimal is exact, so nought point three recurring genuinely is one third rather than an approximation of it. Your calculator hides this by showing 0.3333333 and stopping, but that is only the screen running out of room.

Why does one fraction stop and another go on forever? Watch the remainders in the long division. For 3 divided by 8, you get 30 divided by 8 is 3 remainder 6, then 60 divided by 8 is 7 remainder 4, then 40 divided by 8 is 5 remainder 0. Once the remainder hits zero there is nothing left to divide, so it terminates at 0.375. Now 1 divided by 3: 10 divided by 3 is 3 remainder 1, and you bring down a zero to get 10 again, remainder 1 again, forever. The loop can be longer. For 2 divided by 11 you get remainder 9, then remainder 2, then remainder 9 again, producing the repeating block one-eight. And since dividing by 11 leaves only ten possible remainders, one has to repeat eventually.

There is a proper way to write a recurring decimal. You write the repeating block once and put a dot over it. One third is nought point three with a dot on the 3. Five sixths is nought point eight three with the dot on the 3 only, because the 8 happens once and never comes back. Two elevenths is nought point one eight with dots on both digits, since that pair repeats together. One seventh has a six-digit block, so you dot the first digit and the last to show everything between is included. You may also see a bar drawn over the block, which means the same thing. The main error is careless dotting: putting the dot on the 8 of five sixths would claim the number is nought point eight three eight three, which is something else.

You can predict which kind of decimal you will get without doing any division. Write the fraction in simplest form, then break the denominator into prime factors. If the only primes are 2s and 5s, it terminates. If any other prime turns up — a 3, a 7, an 11 — it recurs. The reason is that 2 and 5 are the factors of 10, the base our place-value columns are built on. Three eighths has 8 equals two times two times two, so it terminates at 0.375. Seven twentieths has 20 equals two times two times five, so it terminates at 0.35. But five sixths has a 3 hiding in the 6, so it recurs. You must simplify first, though: six fifteenths looks like it should recur, but it is really two fifths, which is 0.4.

Because a recurring decimal never ends, you have to round it before it is usable. One third to two decimal places is 0.33 and to three places is 0.333. Two thirds is nought point six recurring, so at two places the next digit is a 6 and it rounds up to 0.67. Five sixths gives 0.83 and 0.833, and one seventh gives 0.14 and 0.143. The important rule is to round from the exact value, never from something you already rounded — starting at 0.66 would wrongly leave you with 0.66. And know when not to round at all. Keep the fraction when the question wants an exact answer or more calculation is coming, and use a decimal for comparing, for money, or when the question names a number of places.