Repeating decimal

A repeating decimal or recurring decimal is a rational number that repeats a pattern of numbers forever. For example, the fraction 1/15 is 0.066666666... with sixes repeating forever. You can also write these numbers by putting a line over the pattern; 1/15 is 0.06.

Repeating patterns can have more than one digit. For example, 1/11 is 0.09, a 2-digit pattern. A more complicated example is 10/53, which is 0.1886792452830, a 13-digit pattern.

We use the decimal or base ten system for writing numbers. This affects what fractions are repeating decimals. Specifically, dividing by a number that isn't a power of a factor of ten makes a repeating decimal. For example, 1/8 is 0.125 with no repeating pattern, because 8 is $2^{3}$ and 2 is a factor of 10. 1/7 is 0.142857 with the repeating pattern "142857", because 7 is not a factor of 10.

All rational numbers either end, or repeat a pattern forever. We can use this fact to prove a number is rational or irrational. For example, imagine a number that's written as 0.123456789101112131415... and so on, with every whole number appearing once. This number won't end, because there are infinite whole numbers, but it also won't repeat a pattern, because every whole number is unique. Therefore, this number is irrational.

Related pages

• 0.999...
• Rational number