Rational and irrational numbers | Year 9

01 / LEARN

The idea, step by step

What makes a number rational?

A rational number can be written exactly as a fraction of two integers, with a nonzero denominator. Integers belong in this group: seven is seven divided by one. A terminating decimal is rational because it can be written using a power of ten as its denominator.

A recurring decimal is rational too. The repeating block continues forever. An irrational number cannot be expressed as an integer fraction: its decimal expansion neither ends nor settles into a repeating block. A few calculator digits alone cannot tell you whether later digits repeat.

a / b, a, b \in ℤ, b \neq 0

Square roots and pi

The square root of a perfect-square whole number is an integer. The square root of a positive whole number that is not a perfect square is irrational. The exact square root of two is irrational even though a calculator displays only a few digits.

Pi is irrational. A decimal such as 3.14 or 3.14159 is rational because it ends. It can approximate pi without being pi itself. Do not classify a number just by whether it contains a root sign: simplify any perfect square first.

A common trap

Every value written with a square root is irrational.

Instead: Simplify perfect-square roots first. The value, rather than its appearance, determines the classification.

02 / WORKED EXAMPLES

See the thinking

Example 1

A root sign does not decide the answer

Classify $\sqrt{49}$ and $\sqrt{50}$ .

$49 = 7^{2}$ , so $\sqrt{49} = 7 = 7 / 1$ .
$50$ is not a perfect square, so $\sqrt{50}$ is irrational.

\sqrt{49}

is rational;

\sqrt{50}

is irrational.

Example 2

An exact recurring decimal

Write $0. \bar{18}$ as a fraction and classify it.

Let $x = 0.181818 \dots$ . Then $100 x = 18.181818 \dots$ .
Subtract: $99 x = 18$ , so $x = 18 / 99 = 2 / 11$ .

Rational:

2 / 11

Check your understanding

Which is irrational: $\sqrt{81}$ , $0.125$ , $π$ or $- 4 / 7$ ?

Show answer

$π$ .

The other values are integer fractions: $9$ , $1 / 8$ and $- 4 / 7$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Classify $- 6$ and support your answer with a fraction.
2 marks
Classify $0.375$ and write it as a simplified fraction.
2 marks
Classify $\sqrt{121}$ and explain.
2 marks
Classify $\sqrt{13}$ and explain.
2 marks
Write $0. \bar{27}$ as a simplified fraction.
3 marks
A student says $3.142$ is irrational because it is close to $π$ . Explain the error.
3 marks

Read the worked solutions

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

Classify $- 6$ and support your answer with a fraction.
2 marks
$- 6 = - 6 / 1$ .
Rational.
Classify $0.375$ and write it as a simplified fraction.
2 marks
$0.375 = 375 / 1000 = 3 / 8$ .
Rational; $3 / 8$ .
Classify $\sqrt{121}$ and explain.
2 marks
$121 = 11^{2}$ .
Rational; $11$ .
Classify $\sqrt{13}$ and explain.
2 marks
$13$ is a positive integer that is not a perfect square.
Irrational.
Write $0. \bar{27}$ as a simplified fraction.
3 marks
$100 x - x = 27$ , so $x = 27 / 99 = 3 / 11$ .
$3 / 11$ .
A student says $3.142$ is irrational because it is close to $π$ . Explain the error.
3 marks
The decimal ends, so $3.142 = 3142 / 1000 = 1571 / 500$ .
An approximation to an irrational value need not be irrational.
$3.142 = 1571 / 500$ is rational.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Year 9 real-number classification and applications; root bounds and sensible digital approximation. Full surd manipulation is developed in Year 10.

Official curriculum source ↗

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