Irrational numbers

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.