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 is not 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 is written as 0.123456789101112131415... and so on, with every whole number appearing once. This number will not end, because there are infinite whole numbers, but it also will not repeat a pattern, because every whole number is unique. Therefore, this number is irrational.

Related pages

• 0.999...
• Rational number