Recurring Decimals (Cambridge (CIE) IGCSE Maths: Extended): Revision Note

Recurring decimals

What are recurring decimals?

• When writing a rational number as a decimal, it will either be:

  • A decimal that stops, called a "terminating" decimal

    • $\frac{1}{4}=0.25$

  • Or a decimal that repeats with a pattern, called a "recurring" decimal

    • $\frac{32}{99}=0.32323232...$

• The recurring part can be written with a dot above the digit that repeats

• If multiple digits repeat, dots are used on the first and last digits that repeat

  • $0.3333...=0.\dot{3}$

  • $0.121212...=0.\dot{1}\dot{2}$

  • $0.325632563256...=0.\dot{3}25\dot{6}$

• Recurring decimals are rational numbers, they are not irrational

  • They can be written as fractions

How do I write recurring decimals as fractions?

Write out the first few decimal places to show the recurring pattern and then:

• STEP 1
  Write the recurring decimal as $x=...$

  • $x=0.35353535...$

• STEP 2
  Multiply both sides by 10 repeatedly until two lines have the same recurring decimal part

  • $\begin{array}{rcl}x& =& 0.35353535...\end{array}$

  • $\begin{array}{rcl}10x& =& 3.5353535...\end{array}$

  • $\begin{array}{rcl}100x& =& 35.353535...\end{array}$

    • Note that x and 100x have 35 repeating after the decimal point, the repeating pattern after 10x is 53 repeating

• STEP 3
  Subtract the two lines which have matching recurring decimal parts

  • $\begin{array}{rcl}100x-x& =& 35.353535...-0.35353535...\end{array}$

  • $\begin{array}{rcl}99x& =& 35\end{array}$

• STEP 4
  Divide both sides to get $x=...$
  Cancel if necessary to get fraction in its lowest terms

  • $x=\frac{35}{99}$