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Repeating Decimals


Repeating decimals are decimal numbers that have either one digit or a group of digits that repeat over and over without ever ending.

For example, 13 rac{1}{3} as a decimal is 0.33333333...0.33333333...

This is a repeating decimal, sometimes known as '0.3 recurring'. To show that a decimal is recurring, you draw a horizontal line over the number or numbers that repeat, such as 0.30.overline{3}

All repeating decimals can be rewritten as fractions, meaning that they are rational numbers.


Decimal Point - The decimal point looks like a period and is always placed after the “ones” position of a number.
Decimal Notation - Decimal notation is useful for writing numbers which are not whole numbers.
Whole Number - A whole number is a number with nothing but zeroes after the decimal point.


All repeating decimals are rational numbers, and can therefore be turned into fractions. 9 as a denominator has a lot of repeating decimals:

19=0.1 rac{1}{9} = 0.overline{1}

29=0.2 rac{2}{9} = 0.overline{2}

39=13=0.3 rac{3}{9} = rac{1}{3} = 0.overline{3}

49=0.4 rac{4}{9} = 0.overline{4}

59=0.5 rac{5}{9} = 0.overline{5}

69=23=0.6 rac{6}{9} = rac{2}{3} = 0.overline{6}

79=0.7 rac{7}{9} = 0.overline{7}

89=0.8 rac{8}{9} = 0.overline{8}

99=1 rac{9}{9} = 1

All repeating decimals have a denominator that is a factor of a prime number other than 2 or 5 (3, 7, 11 etc.)


Repeating Decimals Worksheets (PDF)

Repeating Decimals in Bar Notation Worksheet 1

Repeating Decimals in Bar Notation Worksheet 2

Repeating Decimals in Bar Notation Worksheet 3