Lesson 4: Repeating Decimals

Property

A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly. A bar is placed over the repeating digit or digits.

Examples

• To convert $\frac{2}{3}$ to a decimal, dividing 2 by 3 gives $0.666...$. The digit 6 repeats endlessly, so we write it as $0.\overline{6}$.

• For the fraction $\frac{5}{11}$, dividing 5 by 11 gives $0.454545...$. The block of digits 45 repeats, so we write it as $0.\overline{45}$.

Property

To convert a common fraction to a decimal, divide the numerator by the denominator. Some decimals terminate (end), while others have a repeating pattern of digits. We use a bar over the repeating digits, for example, $\frac{1}{3} = 0.333... = 0.\overline{3}$.

Examples

• To convert $\frac{3}{4}$ to a decimal, we calculate $3 \div 4$, which gives $0.75$. This is a terminating decimal.
• To convert $\frac{4}{9}$ to a decimal, we calculate $4 \div 9$, which gives $0.444...$. We write this repeating decimal as $0.\overline{4}$.
• The fraction $\frac{5}{12}$ is converted by calculating $5 \div 12 = 0.41666...$. We write this as $0.41\overline{6}$, with the bar only over the repeating digit.

Explanation

A fraction is just a division problem in disguise! When you perform the division, the answer is its decimal form. It either stops neatly (terminates) or repeats a pattern forever.

Property

To convert a repeating decimal to a fraction: Let $x$ equal the repeating decimal, multiply both sides by the appropriate power of 10 to shift the decimal point, subtract the original equation to eliminate the repeating part, then solve for $x$.

Examples