What is a Recurring Decimal?
A decimal in which the same digit or group of digits repeats forever without stopping. For example, 0.333333... and 0.272727... are both recurring decimals.
When you divide 1 by 3 on a calculator, you get 0.333333... — the 3s go on forever! Your calculator can only show a limited number of digits, but the pattern never actually ends.
We use a special notation to write recurring decimals neatly:
- A dot (or bar) above a digit means that digit repeats:
- A bar over multiple digits means that whole block repeats:
A recurring decimal is also called a periodic or repeating decimal. The repeating block of digits is called the period of the decimal.
Think of a recurring decimal like a song stuck on repeat — the same sequence of notes (digits) plays over and over without ever stopping.
Terminating vs Recurring Decimals
Not all decimals go on forever. It helps to understand the two main types:
Terminating (Finite) Decimals
These decimals stop after a certain number of digits.
- Examples: , , ,
Recurring (Infinite, Periodic) Decimals
These decimals continue forever with a repeating pattern.
- Examples: , ,
A number that can be written as a ratio , where and are both integers and . Both terminating and recurring decimals are rational numbers.
When a fraction in its lowest terms has a denominator whose only prime factors are 2 and 5, it produces a terminating decimal. If the denominator has any other prime factor, the decimal will recur. For example, recurs because 11 is a prime factor other than 2 or 5.
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