A repeating decimal, also referred to as a recurring decimal, is a decimal number with a digit, or group of digits, that repeat on and on, without end; in other words, the digits are periodic. The repeating digits also cannot all be zero; 1.000000 is not a repeating decimal even though we can add an infinite number of 0s after the decimal point.
There are two commonly used methods for indicating a repeating decimal. One method is to write the repeating portion of the decimal, referred to as the repetend, followed by an ellipsis (...). The other method is to write a bar, referred to as a vinculum, over the repetend.
- expressed as a decimal is 0.3333..., or 0.3.
- expressed as a decimal is 0.1818..., or 0.18.
Repeating, non-terminating, and terminating decimals
These three types of decimals are often discussed together because they are closely related. A non-terminating decimal is a decimal that never ends. It has an infinite number of digits.
There are two types of non-terminating decimals, ones that repeat and ones that do not repeat. Non-terminating decimals that repeat are repeating decimals. As described above, repeating decimals have an infinite number of known digits, and the repetend is not 0. For non-terminating decimals that do not repeat, not all of the digits are known. No matter how many digits are known, there will always be a digit following it that needs to be determined.
A terminating decimal is one that has a finite number of digits. All of the digits in a terminating decimal are known.
- Repeating decimal: = 0.09090909...
- Terminating decimal: = 0.25
- Non-terminating decimal: = 0.3333333...
Note that ⅓ is both a non-terminating decimal as well as a repeating decimal. Understanding the differences between these types of decimals is important when trying to distinguish rational and irrational numbers. All repeating decimals are non-terminating decimals and rational numbers, but not all non-terminating decimals are rational numbers. Rational numbers can either be terminating decimals or repeating decimals. Irrational numbers on the other hand, must be both non-terminating and non-repeating decimals. Examples include π (3.14159...) and the square root of 2 (1.4142135...). Regardless of the number of digits we compute, neither π nor the square root of 2 will ever terminate or repeat.