Rational and irrational numbers | Year 8

01 / LEARN

Understand the idea.

A fraction is the test

A rational number can be written as $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ . Negative numbers and integers qualify: $- 5 = \frac{- 5}{1}$ . Terminating decimals have a power-of-ten denominator; repeating decimals also have fraction forms.

An irrational number is a real number with no such integer ratio. Its decimal continues without an eventually repeating pattern. A calculator showing many digits does not, by itself, establish that a number is irrational.

Roots and approximations

Simplify a square root before classifying it. The square root of a nonnegative perfect-square integer is rational, such as $\sqrt{49} = 7$ . The square root of a positive integer that is not a perfect square is irrational.

Keep an exact number separate from its approximation. The number $π$ is irrational, but $3.14 = \frac{314}{100}$ and $\frac{22}{7}$ are rational. Being close to $π$ does not give an approximation the same classification.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Simplify before deciding

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

Worked solution

1
Simplify the perfect-square root.
$\sqrt{36} = 6 = \frac{6}{1}$
Six squared is thirty-six, and six is a ratio of integers.
2
Locate seven among perfect squares.
$2^{2} = 4 < 7 < 9 = 3^{2}$
Seven is a positive integer that is not a perfect square.
3
Classify the second root.
$\sqrt{7} is irrational$
The square root of a positive nonsquare integer cannot be expressed as a ratio of integers.

Answer

$\sqrt{36}$ rational; $\sqrt{7}$ irrational.

EXAMPLE 02

A repeating decimal

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

Worked solution

1
Name the repeating value.
$x = 0.181818 \dots$
Naming the value lets us compare it with a shifted copy of the same decimal.
2
Shift by one whole repeating block.
$100 x = 18.181818 \dots$
Multiplying by one hundred shifts the two-digit block two places.
3
Subtract the repeating tails.
$\begin{aligned} 100 x - x & = 18 \\ 99 x & = 18 \end{aligned}$
The identical infinite decimal tails cancel exactly, rather than being rounded.
4
Write a reduced fraction.
$x = \frac{18}{99} = \frac{2}{11}$
An exact ratio of integers with nonzero denominator establishes that the number is rational.

Answer

Rational, because it equals $\frac{2}{11}$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Rational.

The number $- \frac{13}{4}$ already has integer numerator $- 13$ and nonzero integer denominator $4$ , so it meets the definition of rational.

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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.
$0. \bar{3} = \frac{1}{3}$ .

Checking saved progress…

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 ↗

In this lesson

Understand the idea.

A fraction is the test

Roots and approximations

See the method in action.

Simplify before deciding

Simplify the perfect-square root.

Locate seven among perfect squares.

Classify the second root.

A repeating decimal

Name the repeating value.

Shift by one whole repeating block.

Subtract the repeating tails.

Write a reduced fraction.

Try it yourself.

Check your understanding

Keep practising