Rational and irrational numbers; repeating decimals

About this lesson

Grade 8 › Pythagorean theorem and irrational numbers

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Rational and irrational numbers; repeating decimals

Warm-up

Three quick questions from earlier lessons.

Discover

13=0.333…\tfrac{1}{3} = 0.333\ldots goes on forever. Is it rational?

  1. Yes
  2. No: it never ends

Key idea

Rational numbers are fractions; their decimals end or repeat.

A rational number can be written ab\tfrac{a}{b} with integers aa, b≠0b \neq 0; its decimal expansion terminates or repeats. An irrational number cannot: its decimal never ends and never repeats. n\sqrt{n} is irrational unless nn is a perfect square; π\pi is irrational.

Worked example

Write 0.27‾0.\overline{27} as a fraction.

100x=27.2727…100x = 27.2727\ldots
Let x = 0.2727…
99x=2799x = 27
Subtract x.
x=2799=311x = \tfrac{27}{99} = \tfrac{3}{11}
Simplify.

Your turn, guided

Your turn: write 0.5‾0.\overline{5} as a fraction.

59\tfrac{5}{9}
10x − x = 5.

Practice ★

  1. Which number is irrational?
    1. 2\sqrt{ 2 }
    2. 4\sqrt{ 4 }
    3. 0.1‾0.\overline{ 1 }
  2. Which number is irrational?
    1. 10\sqrt{ 10 }
    2. 16\sqrt{ 16 }
    3. 0.3‾0.\overline{ 3 }
  1. Write 0.1‾0.\overline{ 1 } (the digit 1 repeating forever) as a fraction.

Watch out!

Practice ★★

  1. Write 0.48‾0.\overline{ 48 } (the digits 48 repeating forever) as a fraction.
  2. Write 0.70‾0.\overline{ 70 } (the digits 70 repeating forever) as a fraction.

Exit ticket

Three questions to check you have got it.

  1. Write 0.1‾0.\overline{ 1 } (the digit 1 repeating forever) as a fraction.
  2. Write 0.2‾0.\overline{ 2 } (the digit 2 repeating forever) as a fraction.
  3. Write 0.8‾0.\overline{ 8 } (the digit 8 repeating forever) as a fraction.

Summary card

Rational and irrational

Ends or repeats: rational. Neither: irrational.

Watch out: a repeating decimal is a fraction over 9, 99, …

End of the lesson

Next lesson: The Pythagorean theorem Back to the chapter

Answers: Rational and irrational numbers; repeating decimals

A

  1. 2\sqrt{ 2 }
  2. 10\sqrt{ 10 }

B

  1. 19\frac{1}{9}

C

  1. 1633\frac{16}{33}
  2. 7099\frac{70}{99}

D

  1. 19\frac{1}{9}
  2. 29\frac{2}{9}
  3. 89\frac{8}{9}

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