Irrational numbers - Year 8 Mathematics

Start here: a strange length

Imagine a perfect square that is $1$ m along each side. Its corners are nice whole numbers. How long is the diagonal?

A unit square has diagonal of length √2 - a number that can never be written exactly as a fraction.

Using Pythagoras, the diagonal satisfies $d^2 = 1^2 + 1^2 = 2$ , so $d = \sqrt{2}$ . If you grab a ruler and measure carefully, it is just over $1.41$ m.

Now try to write $\sqrt{2}$ as a fraction $\dfrac{a}{b}$ where $a$ and $b$ are whole numbers.

$\dfrac{14}{10} = 1.4$ ? Too small. $\dfrac{141}{100} = 1.41$ ? Still too small. $\dfrac{1414}{1000} = 1.414$ ? Better, but $1.414^2 = 1.999\,396$ , not exactly $2$ .

You can get closer and closer — but no fraction ever hits $\sqrt{2}$ exactly. This is what “irrational” means: it can’t be written as a ratio of integers.

What you will learn

distinguish rational from irrational numbers,
recognise $\pi$ and square roots of non-perfect squares as irrational,
estimate an irrational number and locate it between two whole numbers or tenths,
see where irrationals show up in real contexts.

Worked example E Very easy: rational or irrational?

Is $0.75$ rational or irrational?

$0.75 = \dfrac{3}{4}$ , a ratio of two integers. Its decimal terminates. So $0.75$ is rational.

Worked example 0 Real-world example: when rounding matters

You need to cut a piece of timber for the diagonal brace of a $1$ m $\times$ $1$ m frame. Your calculator says $\sqrt{2} = 1.41421356\ldots$

Round to $1$ decimal place: $1.4$ m — the brace is $14$ mm too short. It won’t fit.
Round to $2$ decimal places: $1.41$ m — still $1.4$ mm short.
Round to $3$ decimal places: $1.414$ m — error is $0.2$ mm, within workshop tolerance.

Key idea: $\sqrt{2}$ is irrational — its decimal never ends. Every time you round, you introduce error. The more precision you need, the more decimal places you keep. That is why maths uses $\sqrt{2}$ (exact) and only rounds at the very last step.

1. Rational vs irrational

A rational number can be written as a ratio $\dfrac{a}{b}$ of integers (with $b \neq 0$ ). Every rational has a decimal expansion that either terminates (ends) or recurs (has a repeating block).

$\dfrac{3}{8} = 0.375$ — terminates.
$\dfrac{1}{3} = 0.333\ldots$ — recurs.

An irrational number cannot be written as such a ratio. Its decimal expansion goes on forever without ever settling into a repeating pattern.

Two famous irrationals:

$\pi \approx 3.14159\ldots$ (circumference ÷ diameter of any circle).
$\sqrt{2} \approx 1.41421\ldots$ (the square’s diagonal from the opening).

2. Roots of perfect squares vs non-perfect squares

Worked example E Very easy: a perfect square

Is $\sqrt{9}$ rational or irrational?

$\sqrt{9} = 3$ , a whole number. Whole numbers are rational (you can write them as a ratio, e.g. $\dfrac{3}{1}$ ). So $\sqrt{9}$ is rational.

The square root of a perfect square (like $9, 16, 25, 49, 100$ ) is always a whole number, hence rational.

The square root of a number that is not a perfect square is always irrational:

\sqrt{2},\ \sqrt{3},\ \sqrt{5},\ \sqrt{6},\ \sqrt{7},\ \sqrt{10}, \ldots

Worked example 1 Which numbers are irrational?

Classify each: $\;\dfrac{5}{7},\ \sqrt{16},\ \pi,\ 0.75,\ \sqrt{3}$ .

$\dfrac{5}{7}$ : ratio of integers — rational.
$\sqrt{16} = 4$ : perfect square root — rational.
$\pi$ : decimal never terminates or recurs — irrational.
$0.75 = \dfrac{3}{4}$ : terminating decimal — rational.
$\sqrt{3}$ : not a perfect square — irrational.

3. Placing an irrational on the number line

You can’t write an irrational exactly, but you can squeeze it between two rationals — as close as you want.

Worked example F Very easy: between which whole numbers?

Between which two consecutive whole numbers does $\sqrt{20}$ lie?

$4^2 = 16$ and $5^2 = 25$ . Since $16 < 20 < 25$ , we have $4 < \sqrt{20} < 5$ .

Worked example 2 Pin it down to two decimal places

Estimate $\sqrt{43}$ more precisely.

Start: $6^2 = 36$ , $7^2 = 49$ . So $6 < \sqrt{43} < 7$ .
Narrow: $6.5^2 = 42.25$ , $6.6^2 = 43.56$ . So $6.5 < \sqrt{43} < 6.6$ .
Narrow again: $6.55^2 = 42.9025$ . So $6.55 < \sqrt{43} < 6.6$ .

So $\sqrt{43} \approx 6.56$ to two decimal places.

4. Where irrationals show up in real contexts

Circles. The circumference $C = \pi d$ of any circle involves $\pi$ , so circle measurements are usually irrational.
Diagonals. The diagonal of a $1 \times 1$ square is $\sqrt{2}$ . The diagonal of a $1 \times 2$ rectangle is $\sqrt{5}$ .
Right-angled triangles. Pythagoras often produces irrational side lengths: $3, 4, 5$ is rational, but $1, 1, \sqrt{2}$ and $1, 2, \sqrt{5}$ are not.

Practice: Year 8 core

Fluency

Classify and identify

Try mentally first for each: can it be written as a fraction of whole numbers?

Classify: $7$ .
Classify: $0.5$ .
Classify: $\sqrt{9}$ .
Classify as rational or irrational: $\dfrac{4}{9}$ .
Classify: $\sqrt{25}$ .
Classify: $\sqrt{7}$ .
Classify: $0.\overline{6}$ .
Classify: $\pi$ .
Classify: $-3$ .
Classify: $\sqrt{0.25}$ .
Classify: $1.41421356\ldots$ (the digits do not repeat).

Fluency

Estimate a root

Between which two consecutive whole numbers does $\sqrt{20}$ lie?
Between which two consecutive whole numbers does $\sqrt{90}$ lie?
Between which two tenths does $\sqrt{20}$ lie? Use trial with $4.4^2$ and $4.5^2$ .
Use trial to estimate $\sqrt{75}$ to one decimal place.
Use trial to estimate $\sqrt{150}$ to one decimal place.

Reasoning

Explain and spot the mistake

Kim says ” $0.9999\ldots$ is irrational because it has infinite digits”. Is Kim correct? Explain.
Lee writes $\pi = \dfrac{22}{7}$ . Is this correct, or an approximation? Explain the difference.
Is the product of two irrational numbers always irrational? Give an example that supports your answer.
Aisha claims $\sqrt{49} + \sqrt{2}$ is rational because $\sqrt{49}$ is rational. Is she correct? Explain.

Problem solving

Real contexts

A square has side $1$ m. Find the length of its diagonal. Give the exact value and then an estimate to 2 decimal places.
A pizza has diameter $30$ cm. Find its circumference exactly (in terms of $\pi$ ) and to the nearest cm.
Explain why the area of any circle cannot be exactly a rational number (when the radius is rational).
A right-angled triangle has legs $5$ cm and $12$ cm. Find the exact length of the hypotenuse, then classify it as rational or irrational.

Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Year 8 core - answers

Fluency

Classify and identify

Rational ( $7 = \dfrac{7}{1}$ , an integer).
Rational ( $0.5 = \dfrac{1}{2}$ , a terminating decimal).
Rational ( $\sqrt{9} = 3$ ).
Rational (already a fraction of integers).
Rational ( $\sqrt{25} = 5$ ).
Irrational ( $7$ is not a perfect square, so $\sqrt{7}$ is non-terminating and non-repeating).
Rational ( $0.\overline{6} = \dfrac{2}{3}$ ; repeating decimals are rational).
Irrational ( $\pi$ is non-terminating and non-repeating).
Rational ( $-3 = \dfrac{-3}{1}$ ).
Rational ( $\sqrt{0.25} = 0.5 = \dfrac{1}{2}$ ).
Irrational (the digits do not terminate and do not repeat; in fact this is $\sqrt{2}$ ).

Fluency

Estimate a root

Between $4$ and $5$ . Method: $4^2 = 16$ , $5^2 = 25$ .
Between $9$ and $10$ .
Between $4.4$ and $4.5$ . Method: $4.4^2 = 19.36$ , $4.5^2 = 20.25$ .
$\sqrt{75} \approx 8.7$ . Method: $8.6^2 = 73.96$ , $8.7^2 = 75.69$ .
$\sqrt{150} \approx 12.2$ . Method: $12.2^2 = 148.84$ , $12.3^2 = 151.29$ .

Reasoning

Explain and spot the mistake

Kim is wrong. $0.\overline{9} = 1$ exactly, and $1$ is rational. Infinite digits do not make a number irrational - only non-repeating infinite digits do.
Not correct. $\dfrac{22}{7}$ is a rational approximation to $\pi$ . The true $\pi$ never terminates or repeats; $\dfrac{22}{7}$ does repeat ( $3.\overline{142857}$ ) and so it cannot equal $\pi$ .
Not always irrational. Example: $\sqrt{2} \times \sqrt{2} = 2$ , which is rational. So the product of two irrationals can be rational.
Wrong. $\sqrt{49} + \sqrt{2} = 7 + \sqrt{2}$ , and adding a rational to an irrational gives an irrational.

Problem solving

Real contexts

Exact: $\sqrt{2}$ m. Approximate: $1.41$ m.
Exact: $30\pi$ cm. Approximate: $94$ cm.
If $r$ is rational, $r^2$ is rational; area $= \pi r^2$ is rational times irrational ( $\pi$ ), which is irrational.
Exact: $13$ cm. Method: $5^2 + 12^2 = 25 + 144 = 169 = 13^2$ . Classification: rational (a Pythagorean triple - $13$ is a whole number).

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