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Repeating Decimal: Definition and Examples | EDU.COM

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Repeating Decimal: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Repeating DecimalRepeating Decimal: Definition and ExamplesTable of ContentsRepeating Decimals: Definition, Types, and Conversion to Fractions Definition of Repeating Decimals

Repeating decimals are decimals in which a digit or a group of digits after the decimal point repeats indefinitely and at regular intervals such that the decimal representation becomes periodic. These are also known as "recurring decimals." Examples of repeating decimals include 0.3333...0.3333...0.3333... (where 333 repeats forever), 1.454545...1.454545...1.454545... (where 454545 repeats), and 0.626262...0.626262...0.626262... (where 626262 repeats).

Decimals can be classified into two main categories: terminating and non-terminating decimals. Terminating decimals have a finite number of digits after the decimal point, while non-terminating decimals have an infinite number of digits. Non-terminating decimals are further classified as repeating decimals (where digits or groups of digits repeat themselves) and non-repeating decimals (where digits don't follow any repeating pattern). All repeating decimals are rational numbers, meaning they can be expressed as a fraction pq\frac{p}{q}qp​ where ppp and qqq are integers and q≠0q ≠ 0q=0.

Examples of Converting Repeating Decimals Example 1: Identifying a Pure Repeating Decimal Problem:

Which of the following is a pure repeating decimal?

0.123123…0.123123…0.123123… 0.123333…0.123333…0.123333… 0.123456…0.123456…0.123456… Step-by-step solution:

Step 1, Examine the decimal tails.

Step 2, 0.123123…0.123123…0.123123… repeats the block 123123123 without any non-repeating prefix.

Step 3, Therefore, a) 0.123123…0.123123…0.123123… is a pure repeating decimal.

Example 2: Converting a Pure Repeating Decimal to a Fraction Problem:

Convert 0.3‾0.\overline{3}0.3 (where 333 repeats) to a fraction.

Step-by-step solution: Step 1: Let x=0.333333...x = 0.333333...x=0.333333... Step 2: Multiply both sides by 10: 10x=3.333333...10x = 3.333333...10x=3.333333... Step 3: Subtract the original equation: 10x−x=3.333333...−0.333333...10x - x = 3.333333... - 0.333333...10x−x=3.333333...−0.333333... Step 4: Simplify: 9x=39x = 39x=3 Step 5: Solve for xxx: x=39=13x = \frac{3}{9} = \frac{1}{3}x=93​=31​ Example 3: Classifying a Mixed Repeating Decimal Problem:

Identify the type of repeating decimal: 0.58333…0.58333…0.58333….

Step-by-step solution:

Step 1, Note the digits after the decimal.

Step 2, The digits 585858 do not repeat, while 333 repeats endlessly.

Step 3, Because it contains both non-repeating and repeating parts, 0.58333…0.58333…0.58333… is a mixed repeating decimal.

Comments(1)NNatureLover95September 16, 2025I used the examples on this page to help my kid understand repeating decimals, and it really clicked for him! The step-by-step method for converting them to fractions was super helpful.

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