Terminating and Repeating Decimals

A fraction is a division. 3/8 is 3 ÷ 8, which is 0.375.

Only two outcomes

The division either stops, or it falls into a repeating cycle. Nothing else can happen.

Why it must repeat

Dividing by 7 leaves a remainder from 1 to 6. With only six possibilities, one has to come round again — and once it does, the digits repeat.

Writing a repeat

1/3 = 0.333… is written with a dot or bar over the repeating part.

Every fraction gives one of two kinds

Divide the numerator by the denominator and the decimal either stops — 1/4 = 0.25 — or repeats forever — 1/3 = 0.333… There is no third possibility, because the remainders in the division must eventually repeat.

Predicting which you will get

A fraction in simplest form terminates exactly when its denominator has no prime factors besides 2 and 5. So 1/8 terminates and 1/6 does not. This works because our number system bundles in tens, and 10 = 2 × 5.

Writing a repeating decimal

A bar over the repeating digits, or a dot at each end of the repeating block, records the pattern exactly. Writing 0.333 without any marking is a rounded approximation, not the same number as 1/3.

What this tells you about rational numbers

A rational number is one that can be written as a fraction of integers, and every such number gives a terminating or repeating decimal. Numbers whose decimals never repeat — like π — are irrational, and cannot be written as any fraction at all.