Maths · Standard Form and Accuracy
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Recurring Decimals and Rational Numbers
Converting fractions to recurring decimals, writing recurring decimals as fractions, and telling rational from irrational numbers.
Learning Objectives
- 1Convert a fraction to a decimal, and recognise terminating and recurring decimals.
- 2Write recurring decimals with dot notation.
- 3Convert a recurring decimal to a fraction (Higher tier).
- 4Say whether a number is rational or irrational.
Decimals that never end
Some fractions give decimals that stop, such as \(\dfrac{3}{8} = 0.375\), and others give decimals that repeat for ever, such as \(\dfrac{1}{3} = 0.333\ldots\). A decimal that repeats is called a recurring decimal and is written with dots over the digits that repeat. Writing a fraction as a decimal is Foundation content, while turning a recurring decimal back into a fraction is Higher tier and uses a short algebra trick. The last part of the lesson uses these ideas to separate rational numbers, which can be written as fractions, from irrational numbers, which cannot.
Terminating and recurring decimals
A fraction in its simplest form gives a terminating decimal only if the bottom number has no prime factors other than 2 and 5.
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Terminating
\(\dfrac{3}{8} = 0.375\) stops, because \(8 = 2^3\).
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Recurring
\(\dfrac{1}{3} = 0.3333\ldots\) never stops, because the bottom number has a factor of 3.
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Dot notation
One dot over a single repeating digit, as in \(0.\dot{3}\). For a block, a dot over the first and last digits, as in \(0.\dot{2}\dot{7}\).
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Dividing
\(\dfrac{3}{11}\) gives \(0.272727\ldots = 0.\dot{2}\dot{7}\), found by dividing 3 by 11.
Writing a fraction as a recurring decimal
Write \(\dfrac{5}{6}\) as a decimal, using dot notation.
Show the solutionHide the solution
- 1 Divide \(5 \div 6 = 0.8333\ldots\).
- 2 Spot the repeat Only the 3 repeats.
- 3 Dot notation Put a dot over the 3.
- 4 Write it \(0.8\dot{3}\).
Answer\(0.8\dot{3}\)
Recurring decimals to fractions (Higher tier)
Multiply by a power of 10 so that the repeating part lines up, then subtract.
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One repeating digit
For \(x = 0.\dot{4}\), \(10x = 4.\dot{4}\). Subtract: \(9x = 4\), so \(x = \dfrac{4}{9}\).
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Two repeating digits
For \(x = 0.\dot{2}\dot{7}\), \(100x = 27.\dot{2}\dot{7}\). Subtract: \(99x = 27\), so \(x = \dfrac{27}{99} = \dfrac{3}{11}\).
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Why it works
Subtracting removes the infinite repeating tail.
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Part not repeating
For \(0.1\dot{6}\), \(10x = 1.\dot{6}\) and \(100x = 16.\dot{6}\). Subtract: \(90x = 15\), so \(x = \dfrac{1}{6}\).
Converting a recurring decimal (Higher tier)
Write \(0.\dot{4}\dot{5}\) as a fraction in its simplest form.
Show the solutionHide the solution
- 1 Set up \(x = 0.454545\ldots\).
- 2 Multiply by 100 \(100x = 45.4545\ldots\).
- 3 Subtract \(100x - x = 45\), so \(99x = 45\).
- 4 Solve and simplify \(x = \dfrac{45}{99} = \dfrac{5}{11}\).
Answer\(\dfrac{5}{11}\)
Rational and irrational numbers
A rational number can be written as a fraction of two whole numbers.
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Rational
Integers, fractions, terminating decimals and recurring decimals, such as \(0.\dot{3} = \dfrac{1}{3}\).
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Irrational
Numbers that cannot be written as a fraction, such as \(\pi\) and \(\sqrt{2}\). Their decimals never end and never repeat.
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Roots
\(\sqrt{9} = 3\) is rational, but \(\sqrt{5}\) is irrational.
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Combining
A rational number plus an irrational number is irrational.
Test yourself
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1
Is \(\dfrac{3}{8}\) a terminating or recurring decimal?
Show answerHide answer
Terminating, 0.375.
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2
What is \(\dfrac{1}{3}\) as a decimal?
Show answerHide answer
\(0.\dot{3}\).
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3
What are the dots in \(0.\dot{2}\dot{7}\) for?
Show answerHide answer
They show the repeating block, 27.
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4
Is \(\pi\) rational or irrational?
Show answerHide answer
Irrational.
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5
Is \(\sqrt{9}\) rational or irrational?
Show answerHide answer
Rational, because it equals 3.
Exam technique: recurring decimals
Show the subtraction.
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Write the equation clearly
Start with \(x = \ldots\).
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Line up the repeats
Multiply by 10, 100 or 1000 so the repeats match.
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Subtract
Then solve for \(x\).
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Simplify
Cancel the fraction to its simplest form.
Summary and exam focus
- A fraction gives a terminating or a recurring decimal, and recurring decimals use dot notation.
- To change a recurring decimal to a fraction, multiply to line up the repeats, then subtract (Higher tier).
- Rational numbers can be written as fractions, and irrational numbers cannot.
- \(\pi\) and surds like \(\sqrt{2}\) are irrational.
Exam focus
Write \(0.\dot{3}\dot{6}\) as a fraction in its simplest form. (2 marks) (2 marks)
\(x = 0.3636\ldots\), so \(100x = 36.3636\ldots\) and \(99x = 36\). Then \(x = \dfrac{36}{99} = \dfrac{4}{11}\). Write \(100x - x\) to show the method for the first mark.
Key terms
The words this lesson expects you to use. Each one is linked from the first place it appears above.
- Terminating decimal
- A decimal that stops after a certain number of digits.
- Recurring decimal
- A decimal in which one or more digits repeat for ever.
- Dot notation
- Dots over repeating digits to show a recurring decimal.
- Rational number
- A number that can be written as a fraction of two integers.
- Irrational number
- A number that cannot be written as a fraction.
- Surd
- An irrational root, such as \(\sqrt{2}\).
- Integer
- A whole number, positive, negative or zero.
- Simplest form
- A fraction that cannot be cancelled further.
- Repeating block
- The digits that repeat in a recurring decimal.
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