Rational and irrational numbers | Year 8

01 / LEARN

The idea, step by step

A fraction is the test

A rational number can be written as a ratio of integers with a nonzero denominator. Every integer qualifies: $- 5 = - \frac{5}{1}$ . Terminating decimals can be written over a power of ten, and recurring decimals also have fraction forms. An irrational real number cannot be written as such a ratio; its decimal neither terminates nor repeats.

Roots and approximations

A square root of a nonnegative perfect-square integer is rational: $\sqrt{49} = 7$ . The square root of a positive integer that is not a perfect square is irrational. The number $π$ is irrational, while 3.14 and $\frac{22}{7}$ are rational approximations. Classifying the approximation is not the same as classifying the original number.

A common trap

A square-root symbol always means an irrational number.

Instead: Check the value under the root: $\sqrt{81} = 9$ is an integer and therefore rational.

02 / WORKED EXAMPLES

See the thinking

Example 1

Simplify before deciding

Classify $\sqrt{36}$ and $\sqrt{7}$ .

$\sqrt{36} = 6 = \frac{6}{1}$ , so it is rational.
Seven is not a perfect square, so its square root is irrational.

\sqrt{36}

rational;

\sqrt{7}

irrational.

Example 2

A repeating decimal

Is $0. \bar{18}$ rational?

The two-digit repeating block gives the fraction $18 / 99$ .
Reduce: $\frac{18}{99} = \frac{2}{11}$ .

Rational, because it equals

\frac{2}{11}

Check your understanding

Classify $- \frac{13}{4}$ .

Show answer

Rational.

It is already written as a ratio of integers, with nonzero denominator 4.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Explain why -12 is rational.
2 marks
Classify $\sqrt{81}$ and justify.
2 marks
Classify $\sqrt{11}$ and justify.
2 marks
Classify $0. \bar{4}$ and give a fraction.
2 marks
Is $\frac{22}{7}$ irrational because it approximates $π$ ? Explain.
2 marks
Leila says every decimal that goes on forever is irrational. Give a counterexample.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Explain why -12 is rational.
2 marks
Both -12 and 1 are integers and the denominator is not zero.
$- 12 = - \frac{12}{1}$ .
Classify $\sqrt{81}$ and justify.
2 marks
$\sqrt{81} = 9 = \frac{9}{1}$ .
Rational.
Classify $\sqrt{11}$ and justify.
2 marks
Eleven is a positive integer that is not a perfect square.
Irrational.
Classify $0. \bar{4}$ and give a fraction.
2 marks
The repeating decimal 0.444… equals four ninths.
Rational; $\frac{4}{9}$ .
Is $\frac{22}{7}$ irrational because it approximates $π$ ? Explain.
2 marks
It is a ratio of integers.
Being close to an irrational value does not make a fraction irrational.
No. It is rational.
Leila says every decimal that goes on forever is irrational. Give a counterexample.
3 marks
This decimal does not end but repeats.
Its fraction form proves it is rational.
$0. \bar{3} = \frac{1}{3}$ .

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Estimating square roots and recognising pi

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Four operations with integers; signed fraction/decimal calculations; order of operations; terminating and recurring decimals; rational and irrational numbers; estimating square roots and recognising pi; choosing efficient exact/decimal representations.

Official curriculum source ↗

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