Repeating Decimals

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Introduction

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, $rac{1}{3}$ as a decimal is $0.33333333...$

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 $0.overline{3}$

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

Terms

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.

Lesson

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

$rac{1}{9} = 0.overline{1}$

$rac{2}{9} = 0.overline{2}$

$rac{3}{9} = rac{1}{3} = 0.overline{3}$

$rac{4}{9} = 0.overline{4}$

$rac{5}{9} = 0.overline{5}$

$rac{6}{9} = rac{2}{3} = 0.overline{6}$

$rac{7}{9} = 0.overline{7}$

$rac{8}{9} = 0.overline{8}$

$rac{9}{9} = 1$

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

Examples