Recurring Decimals (Cambridge International IGCSE Mathematics)

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

  • $\begin{gathered} x = 0.35353535... \end{gathered}$

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

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

    • 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

  • $\begin{gathered} 100 x - x & = 35.353535... - 0.35353535... \end{gathered}$
  • $\begin{gathered} 99 x & = 35 \end{gathered}$

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

  • $x=\frac{35}{99}$

Worked Example Write $0. \dot{3} 0 \dot{7}$ as a fraction in its lowest terms.

Answer:

Write as $x=...$ to show the pattern

$x=0.307307307307...$

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

$\begin{gathered} 10 x & = 3.07307307307... \\ 100 x & = 30.730730730... \\ 1000 x & = 307.307307307... \end{gathered}$

Notice that $x$ and $1000x$ have matching recurring decimal parts

Subtract one from the other

$\begin{gathered} 1000 x - x & = 307.307307307... - 0.307307307... \\ 999 x & = 307 \end{gathered}$

Divide both sides by 999

$x=\frac{307}{999}$

This cannot be simplified, so this is the final answer

$\frac{307}{999}$