Terminators vs. Repeaters

Here is the test: put the fraction in lowest terms and look at the denominator's prime factors. If they are only 2s and 5s, the decimal terminates. Any other prime, and it repeats.

• 1/8 ends (8 = 2 cubed)
• 1/20 ends (20 = 2 squared times 5)
• 1/3 repeats (3 is neither a 2 nor a 5)

The outcome is fixed: check the denominator and you already know the decimal's fate.

A repeating decimal is written with a bar over the block that repeats. For 1/6 = 0.1666..., only the 6 repeats, so the bar sits over a single 6 (the 1 is written normally). This notation is compact and exact. Every rational number (a fraction of integers) is either a terminating or a repeating decimal — no other behavior is possible.