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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.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Convert a fraction to a decimal, and recognise terminating and recurring decimals.
  2. 2Write recurring decimals with dot notation.
  3. 3Convert a recurring decimal to a fraction (Higher tier).
  4. 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.

  • Terminating

    \(\dfrac{3}{8} = 0.375\) stops, because \(8 = 2^3\).

  • Recurring

    \(\dfrac{1}{3} = 0.3333\ldots\) never stops, because the bottom number has a factor of 3.

  • 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 \(0.\dot{2}\dot{7}\).

  • 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. 1 Divide \(5 \div 6 = 0.8333\ldots\).
  2. 2 Spot the repeat Only the 3 repeats.
  3. 3 Dot notation Put a dot over the 3.
  4. 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.

  • One repeating digit

    For \(x = 0.\dot{4}\), \(10x = 4.\dot{4}\). Subtract: \(9x = 4\), so \(x = \dfrac{4}{9}\).

  • Two repeating digits

    For \(x = 0.\dot{2}\dot{7}\), \(100x = 27.\dot{2}\dot{7}\). Subtract: \(99x = 27\), so \(x = \dfrac{27}{99} = \dfrac{3}{11}\).

  • Why it works

    Subtracting removes the infinite repeating tail.

  • 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. 1 Set up \(x = 0.454545\ldots\).
  2. 2 Multiply by 100 \(100x = 45.4545\ldots\).
  3. 3 Subtract \(100x - x = 45\), so \(99x = 45\).
  4. 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.

  • Rational

    Integers, fractions, terminating decimals and recurring decimals, such as \(0.\dot{3} = \dfrac{1}{3}\).

  • Irrational

    Numbers that cannot be written as a fraction, such as \(\pi\) and \(\sqrt{2}\). Their decimals never end and never repeat.

  • Roots

    \(\sqrt{9} = 3\) is rational, but \(\sqrt{5}\) is irrational.

  • Combining

    A rational number plus an irrational number is irrational.

Test yourself

  1. 1

    Is \(\dfrac{3}{8}\) a terminating or recurring decimal?

    Show answerHide answer

    Terminating, 0.375.

  2. 2

    What is \(\dfrac{1}{3}\) as a decimal?

    Show answerHide answer

    \(0.\dot{3}\).

  3. 3

    What are the dots in \(0.\dot{2}\dot{7}\) for?

    Show answerHide answer

    They show the repeating block, 27.

  4. 4

    Is \(\pi\) rational or irrational?

    Show answerHide answer

    Irrational.

  5. 5

    Is \(\sqrt{9}\) rational or irrational?

    Show answerHide answer

    Rational, because it equals 3.

Exam technique: recurring decimals

Show the subtraction.

  • Write the equation clearly

    Start with \(x = \ldots\).

  • Line up the repeats

    Multiply by 10, 100 or 1000 so the repeats match.

  • Subtract

    Then solve for \(x\).

  • 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.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Write 2 marks Easier

(a) Write \(\dfrac{2}{5}\) as a decimal. [1 mark] (b) Write \(\dfrac{4}{9}\) as a recurring decimal, using dot notation. [1 mark]

Mark scheme — 2 marks available

  • (a) 0.4 — B1
  • (b) \(0.\dot{4}\) — B1

Model answer

(a) \(2 \div 5 = 0.4\). (b) \(0.\dot{4}\).

2. Exam question Write 2 marks Core

Write \(\dfrac{7}{12}\) as a recurring decimal, using dot notation. [2 marks]

Mark scheme — 2 marks available

  • 0.5833 seen — M1
  • \(0.58\dot{3}\) — A1

Model answer

\(7 \div 12 = 0.58333\ldots = 0.58\dot{3}\).

3. Exam question Show that 3 marks Stretch

Write \(0.\dot{1}\dot{5}\) as a fraction in its simplest form. [3 marks]

Mark scheme — 3 marks available

  • \(100x = 15.1515\ldots\) — M1
  • \(99x = 15\) — M1
  • \(\dfrac{5}{33}\) — A1

Model answer

Let \(x = 0.1515\ldots\). Then \(100x = 15.1515\ldots\), so \(99x = 15\) and \(x = \dfrac{15}{99} = \dfrac{5}{33}\).

4. Exam question Show that 3 marks Stretch

Write \(0.0\dot{3}\) as a fraction in its simplest form. [3 marks]

Mark scheme — 3 marks available

  • \(10x\) and \(100x\) seen — M1
  • \(90x = 3\) — M1
  • \(\dfrac{1}{30}\) — A1

Model answer

Let \(x = 0.0333\ldots\). Then \(10x = 0.333\ldots\) and \(100x = 3.333\ldots\). Subtracting, \(90x = 3\), so \(x = \dfrac{3}{90} = \dfrac{1}{30}\).

5. Exam question Explain 2 marks Core

Is \(\sqrt{49}\) rational or irrational? Give a reason. [2 marks]

Mark scheme — 2 marks available

  • Rational — B1
  • \(\sqrt{49} = 7\) or written as a fraction — B1

Model answer

Rational. \(\sqrt{49} = 7\), which can be written as the fraction \(\dfrac{7}{1}\).

6. Exam question Show that 3 marks Stretch

Show that \(0.\dot{3}\dot{9} = \dfrac{13}{33}\). [3 marks]

Mark scheme — 3 marks available

  • \(100x = 39.3939\ldots\) — M1
  • \(99x = 39\) — M1
  • \(\dfrac{39}{99} = \dfrac{13}{33}\) — A1

Model answer

Let \(x = 0.3939\ldots\). Then \(100x = 39.3939\ldots\), so \(99x = 39\) and \(x = \dfrac{39}{99} = \dfrac{13}{33}\).

7. Multiple choice 1 mark Easier

What is \(\dfrac{3}{8}\) as a decimal?

  1. A 0.38
  2. B 0.375 Correct
  3. C 0.83
  4. D 0.\dot{3}

Why: \(3 \div 8 = 0.375\), which stops.

8. Multiple choice 1 mark Easier

What is \(\dfrac{1}{3}\) as a recurring decimal?

  1. A \(0.\dot{3}\) Correct
  2. B 0.3
  3. C 0.33
  4. D \(0.\dot{1}\dot{3}\)

Why: The 3 repeats for ever.

9. Multiple choice 1 mark Core

What is \(\dfrac{3}{11}\) as a recurring decimal?

  1. A \(0.\dot{3}\)
  2. B \(0.\dot{2}\dot{3}\)
  3. C \(0.2\dot{7}\)
  4. D \(0.\dot{2}\dot{7}\) Correct

Why: \(3 \div 11 = 0.272727\ldots\), where 27 repeats.

10. Multiple choice 1 mark Core

Which fraction gives a terminating decimal?

  1. A \(\dfrac{1}{3}\)
  2. B \(\dfrac{2}{7}\)
  3. C \(\dfrac{7}{20}\) Correct
  4. D \(\dfrac{5}{6}\)

Why: \(20 = 2^2 \times 5\), so the decimal 0.35 stops.

11. Multiple choice 1 mark Easier

Is \(\pi\) rational or irrational?

  1. A Rational
  2. B Irrational Correct
  3. C Both
  4. D Neither

Why: \(\pi\) cannot be written as a fraction.

12. Multiple choice 1 mark Core

Is \(\sqrt{16}\) rational or irrational?

  1. A Rational, because it equals 4 Correct
  2. B Irrational, because it is a root
  3. C Irrational, because it is not a fraction
  4. D Neither

Why: \(\sqrt{16} = 4 = \dfrac{4}{1}\).

13. Multiple choice 1 mark Core

Write \(0.\dot{4}\) as a fraction.

  1. A \(\dfrac{4}{10}\)
  2. B \(\dfrac{4}{99}\)
  3. C \(\dfrac{1}{4}\)
  4. D \(\dfrac{4}{9}\) Correct

Why: \(x = 0.\dot{4}\), \(10x = 4.\dot{4}\), so \(9x = 4\).

14. Multiple choice 1 mark Stretch

Write \(0.\dot{4}\dot{5}\) as a fraction in its simplest form.

  1. A \(\dfrac{45}{100}\)
  2. B \(\dfrac{9}{20}\)
  3. C \(\dfrac{5}{11}\) Correct
  4. D \(\dfrac{4}{9}\)

Why: \(100x - x = 45\), so \(x = \dfrac{45}{99} = \dfrac{5}{11}\).

15. Multiple choice 1 mark Stretch

Write \(0.1\dot{6}\) as a fraction in its simplest form.

  1. A \(\dfrac{16}{99}\)
  2. B \(\dfrac{1}{6}\) Correct
  3. C \(\dfrac{16}{100}\)
  4. D \(\dfrac{1}{16}\)

Why: \(100x - 10x = 15\), so \(x = \dfrac{15}{90} = \dfrac{1}{6}\).