recurring decimal
Definition
Noun: A recurring decimal is a decimal fraction in which a digit, or a group of digits, repeats infinitely without end. It is also commonly called a repeating decimal.
Usage
The term is used in mathematics to describe the decimal representation of a rational number that, when divided, results in a remainder that cycles, causing a pattern of digits to repeat forever.
Examples
- The fraction 1/3 is equal to the recurring decimal 0.333...
- When you convert 2/7 to a decimal, you get the recurring decimal 0.285714285714...
- A single repeating digit is shown as 0.5̅, which is a notation for a recurring decimal.
Advanced Usage
- Notation: In mathematical notation, a recurring decimal is often indicated by placing a dot (or a bar) over the repeating digit(s). For example, 0.1̅6̅ means the digits "16" repeat infinitely (0.161616...).
- Terminology Distinction: While all recurring decimals are repeating, some mathematicians distinguish a as one where the repetition starts immediately after the decimal point, whereas a may have non-repeating digits before the repeating block.
Variants and Related Words
- Repeating Decimal (noun): The most common synonym for .
- Circulating Decimal (noun): An older, less common term with the same meaning.
- Terminating Decimal (noun): The opposite concept; a decimal that ends (e.g., 0.5, 0.25).
Synonyms
- Repeating decimal
- Periodic decimal
- Circulating decimal
Antonyms
- Terminating decimal
- Non-repeating decimal (which is characteristic of irrational numbers)
Related Phrases/Concepts
- Rational Number: A number that can be expressed as a fraction of two integers. All rational numbers have decimal expansions that are either terminating or recurring.
- Repetend: The sequence of digits that repeats in a recurring decimal.
- To recur: (verb) To occur again periodically or repeatedly. This is the verbal root of the term.
Noun
-
a decimal with a sequence of digits that repeats itself indefinitely
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