## Repeating Decimal

### A repeating decimal is a decimal number in which one or more digits repeat infinitely

A repeating decimal is a decimal number in which one or more digits repeat infinitely. It occurs when the division of two integers results in a non-terminating decimal. The repeating portion of the decimal usually appears in parentheses.

For example, consider the fraction 1/3. When you divide 1 by 3, you get the decimal 0.333…, where the digit 3 repeats infinitely. To represent this as a repeating decimal, we write it as 0.3(3), or simply 0.(3).

Another example is the fraction 5/6. When you divide 5 by 6, you get the decimal 0.833…, where the digit 8 repeats infinitely. To represent this as a repeating decimal, we write it as 0.8(3), or simply 0.(8).

There are two types of repeating decimals: pure repeating decimals and mixed repeating decimals.

1. Pure Repeating Decimal: In a pure repeating decimal, all the digits after the decimal point repeat infinitely. For example, 0.(3) and 0.(8) mentioned above are pure repeating decimals.

2. Mixed Repeating Decimal: In a mixed repeating decimal, there are some non-repeating digits before the repeating portion. For example, consider the fraction 7/12. When you divide 7 by 12, you get the decimal 0.58333…, where the digits 5 and 8 do not repeat, but the digit 3 repeats infinitely. To represent this as a repeating decimal, we write it as 0.58(3), where 58 are non-repeating digits and 3 repeats.

To convert a fraction to a repeating decimal, you can use long division or other methods such as converting fractions to equations and solving for the decimal form.

Repeating decimals are often represented with a bar over the repeating part, for example, 0.3̅ for the decimal 0.333… or 0.8̅ for the decimal 0.833…

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