Sets of Numbers

Lessons

Senior 1 · Mathematics

Sets of Numbers

5 lessons

Lesson 5 of 5

Irrational numbers and the real numbers

25 min

A length that is not a fraction

A classroom floor in Huye is covered with square tiles 1 m long. Four tiles make a 2 m by 2 m square. Join the midpoints of its sides and you get a tilted square whose sides are the diagonals of the tiles. How long is one diagonal, $s$ ?

The tilted square has area 2, so its side $s$ , the diagonal of a 1 m tile, satisfies $s \times s = 2$ .

So $s^{2} = 2$ . Try some decimals:

$s$	1.4	1.41	1.414	1.415	1.42
$s^{2}$	1.96	1.9881	1.999396	2.002225	2.0164

The squares get closer and closer to 2 but never equal it. In fact no fraction at all has a square of exactly 2.

Definition

Square root of a number $a \ge 0$ : the number that is not negative and whose square is $a$ .

Notation: $\sqrt{a}$ . Example: $\sqrt{9} = 3$ , because $3^{2} = 9$ and $3 \ge 0$ ; the diagonal above is $s = \sqrt{2} \approx 1.414$ .

Definition

Irrational number: a number on the number line that cannot be written as $\frac{a}{b}$ with $a$ , $b$ integers and $b \neq 0$ . Its decimal never terminates and never repeats.

Examples: $\sqrt{2} = 1.41421356\ldots$ , $\sqrt{3}$ , $\sqrt{5}$ , $\pi = 3.14159265\ldots$ (the ratio of any circle's circumference to its diameter) and $0.1010010001\ldots$ , where the blocks of 0s keep getting longer.

Important

Why $\sqrt{2}$ cannot be a fraction. Suppose it could: $\sqrt{2} = \frac{a}{b}$ , a fraction in simplest form, so $a$ and $b$ have no common factor.

Squaring, $2 = \frac{a^2}{b^2}$ , so $a^{2} = 2 b^{2}$ . Then $a^{2}$ is even, so $a$ is even (the square of an odd number is odd). Write $a = 2 k$ .
Then $(2 k)^{2} = 2 b^{2}$ , so $4 k^{2} = 2 b^{2}$ and $b^{2} = 2 k^{2}$ . So $b^{2}$ is even, and $b$ is even too.
Now $a$ and $b$ are both even: they share the factor 2. That contradicts "simplest form".

The only way out is that the supposition was false: $\sqrt{2}$ is irrational. This style of argument is called proof by contradiction.

Irrational numbers on the number line

Even though its decimal never ends, $\sqrt{2}$ has an exact place on the number line, and you can find it with a ruler and a pair of compasses.

Stand a unit square on the segment from 0 to 1. Its diagonal has length $\sqrt{2}$ . With the compass point at 0, swing the diagonal down onto the line: it lands at $\sqrt{2} \approx 1.414$ .

Definition

Real numbers: all the rational and irrational numbers together. Every point on the number line is a real number, and every real number is a point on the line.

Notation: $\mathbb{R}$ . Example: $- 3$ , $0.75$ , $\frac{1}{3}$ , $\sqrt{2}$ and $π \pi$ are all real numbers.

How the sets fit together

\mathbb{N} \subset \mathbb{Z} \subset \mathbb{D} \subset \mathbb{Q} \subset \mathbb{R}

Each set contains the one before it, and every new set was built to repair something the old one could not do: $\mathbb{Z}$ allows subtraction, $\mathbb{Q}$ allows division, and $\mathbb{R}$ fills the gaps where numbers like $\sqrt{2}$ and $π \pi$ belong. The irrational numbers are the set $\mathbb{R} - \mathbb{Q}$ .

$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{D} \subset \mathbb{Q} \subset \mathbb{R}$ . Each number is written in the smallest set it belongs to; the irrational numbers lie in $\mathbb{R}$ but outside $\mathbb{Q}$ .

The same sets on the number line. $\mathbb{N}$ and $\mathbb{Z}$ are separate dots; $\mathbb{Q}$ crowds the line with dots but leaves gaps such as $\sqrt{2}$ ; $\mathbb{R}$ is the whole line, with no gaps.

Worked example 1: classifying numbers

Tick every set each number belongs to: $7$ , $- 3$ , $0.75$ , $\frac{2}{3}$ , $\sqrt{9}$ , $\sqrt{7}$ , $π \pi$ , $-\frac{8}{2}$ .

Solution Simplify first: $\sqrt{9} = 3$ and $-\frac{8}{2} = -4$ . Then use the chain $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{D} \subset \mathbb{Q} \subset \mathbb{R}$ : a number in one set is in every set to its right.

Number	$\mathbb{N}$	$\mathbb{Z}$	$\mathbb{D}$	$\mathbb{Q}$	$\mathbb{R}$	Irrational
$7$	✓	✓	✓	✓	✓	—
$- 3$	—	✓	✓	✓	✓	—
$0.75$	—	—	✓	✓	✓	—
$\frac{2}{3} = 0.\dot{6}$	—	—	—	✓	✓	—
$\sqrt{9} = 3$	✓	✓	✓	✓	✓	—
$\sqrt{7}$	—	—	—	—	✓	✓
$π \pi$	—	—	—	—	✓	✓
$-\frac{8}{2} = -4$	—	✓	✓	✓	✓	—

A square root is not automatically irrational: $\sqrt{9}$ is a natural number. Only the square root of a natural number that is not a perfect square is irrational.

Worked example 2: numbers between 1.4 and 1.5

Find a rational number and an irrational number between 1.4 and 1.5.

Solution

Rational: $1.45 = \frac{145}{100} = \frac{29}{20}$ .
Irrational: $\sqrt{2}$ . Check by squaring the ends: $1.4^2 = 1.96 \lt 2 \lt 2.25 = 1.5^2$ , so $1.4 \lt \sqrt{2} \lt 1.5$ . Another answer is $1.41010010001\ldots$ .

Properties of the four operations on $\mathbb{R}$

For all real numbers $a$ , $b$ and $c$ :

Property	Addition	Multiplication
Closure	$a + b \in \mathbb{R}$	$ab \in \mathbb{R}$
Commutative	$a + b = b + a$	$a b = b a$
Associative	$(a + b) + c = a + (b + c)$	$(a b) c = a (b c)$
Identity element	$a + 0 = a$	$a \times 1 = a$
Inverse	$a + (- a) = 0$	$a \times \frac{1}{a} = 1$ , for $a \neq 0$
Distributive	$a (b + c) = a b + a c$

Subtraction and division are neither commutative nor associative, in $\mathbb{R}$ as in $\mathbb{N}$ : $5 - 2 \neq 2 - 5$ and $(8 \div 4) \div 2 \neq 8 \div (4 \div 2)$ .

The table below collects where each operation stays inside each set (division by non-zero numbers only).

Set	$+$	$-$	$\times$	$\div$	A failure
$\mathbb{N}$	✓	—	✓	—	$2 - 5 = - 3$
$\mathbb{Z}$	✓	✓	✓	—	$1 \div 2 = 0.5$
$\mathbb{D}$	✓	✓	✓	—	$1 \div 3 = 0.\dot{3}$
$\mathbb{Q}$	✓	✓	✓	✓	—
$\mathbb{R}$	✓	✓	✓	✓	—
irrational numbers	—	—	—	—	$\sqrt{2} \times \sqrt{2} = 2$

✓ = closed, — = not closed.

Worked example 3: the irrational numbers are not closed

Show that the sum and the product of two irrational numbers can be rational.

Solution One example of each is enough.

\sqrt{2} + (-\sqrt{2}) = 0, \qquad \sqrt{2} \times \sqrt{2} = 2.

Both answers are rational, although every number used was irrational. (By contrast, a rational number plus an irrational number is always irrational: if $r + x = q$ with $r$ , $q$ rational, then $x = q - r$ would be rational.)

Warning

$π \pi$ is not $\frac{22}{7}$ , and not 3.14. Both are rational approximations: $\frac{22}{7} = 3.\dot{1}4285\dot{7} = 3.142857\ldots$ , while $\pi = 3.141592\ldots$ . Use them to estimate, but write $π \pi$ when you need the exact value.

Tip

To place a square root between two integers, square the integers around it: $25 \lt 30 \lt 36$ , so $5 \lt \sqrt{30} \lt 6$ . And $\sqrt{n}$ is rational only when $n$ is a perfect square (1, 4, 9, 16, 25, …).

Khan Academy: Introduction to rational and irrational numbers (6 min). Sal explains what makes a number rational and why some numbers are not. Compare his examples with the nested diagram in this lesson.

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Practice

Show your reasoning. The last two are harder.

List every set ( $\mathbb{N}$ , $\mathbb{Z}$ , $\mathbb{D}$ , $\mathbb{Q}$ , $\mathbb{R}$ ) that each number belongs to, and say which numbers are irrational: $- 5$ , $0$ , $\frac{7}{4}$ , $\sqrt{16}$ , $\sqrt{10}$ , $0.\dot{4}$ , $\pi + 1$ .
Between which two consecutive integers does each number lie? (a) $\sqrt{30}$ (b) $\sqrt{70}$
Find a rational number and an irrational number between 2 and 3.
True or false? Give a reason or an example. (a) Every natural number is an integer. (b) Every rational number is an integer. (c) The product of two irrational numbers is always irrational. (d) 0 is a rational number.
Name the property: (a) $\sqrt{2} + 3 = 3 + \sqrt{2}$ (b) $\pi \times 1 = \pi$ (c) $5(\sqrt{2} + 1) = 5\sqrt{2} + 5$ (d) $(- 7) + 7 = 0$
Use a property of the operations to calculate $3.7 \times 45 + 3.7 \times 55$ without a calculator.
Is $0.121121112\ldots$ (one more 1 in each block) rational or irrational? Explain.

Show answersHide answers

$- 5$ : $\mathbb{Z}, \mathbb{D}, \mathbb{Q}, \mathbb{R}$ . $0$ : $\mathbb{Z}, \mathbb{D}, \mathbb{Q}, \mathbb{R}$ (not $\mathbb{N}$ ). $\frac{7}{4} = 1.75$ : $\mathbb{D}, \mathbb{Q}, \mathbb{R}$ . $\sqrt{16} = 4$ : all five. $\sqrt{10}$ : $\mathbb{R}$ , irrational. $0.\dot{4} = \frac{4}{9}$ : $\mathbb{Q}, \mathbb{R}$ . $\pi + 1$ : $\mathbb{R}$ , irrational.
(a) $25 \lt 30 \lt 36$ , so between 5 and 6 (b) $64 \lt 70 \lt 81$ , so between 8 and 9
For example 2.5, and $\sqrt{5}$ (since $4 \lt 5 \lt 9$ ); $\sqrt{7}$ or $2.1010010001\ldots$ also work.
(a) True: $\mathbb{N} \subset \mathbb{Z}$ (b) False: $\frac{1}{2}$ is rational but not an integer (c) False: $\sqrt{2} \times \sqrt{2} = 2$ (d) True: $0 = \frac{0}{1}$
(a) commutative (addition) (b) identity element for multiplication (c) distributive (d) additive inverse
$3.7 \times (45 + 55) = 3.7 \times 100 = 370$ (distributive property)
Irrational. Its decimal never terminates, and it never repeats, because the blocks of 1s keep getting longer, so no block of digits can repeat for ever. A rational number's decimal must terminate or repeat.

1A length that is not a fraction

2Irrational numbers on the number line

3How the sets fit together

4Properties of the four operations on R \mathbb{R} R

A length that is not a fraction

Irrational numbers on the number line

How the sets fit together

Properties of the four operations on $\mathbb{R}$