Sets of Numbers

Lessons

Senior 1 · Mathematics

Sets of Numbers

5 lessons

Lesson 2 of 5

Integers and their properties

25 min

Numbers below zero

In a building with underground parking, the lift numbers the ground floor 0, the floors above it 1, 2, 3, … and the basement floors −1, −2, … A basement floor is below the ground in the same way that a debt is below having no money, or a temperature of −5 °C is below freezing point. The natural numbers cannot describe these situations, so we extend them.

Definition

Integers: the whole numbers together with their negatives.

\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}

Notation: $\mathbb{Z}$ , from the German word Zahlen, "numbers". The positive integers are $\mathbb{Z}^+ = \{1, 2, 3, \ldots\} = \mathbb{N}$ and the negative integers are $\mathbb{Z}^- = \{-1, -2, -3, \ldots\}$ . So $\mathbb{Z} = \mathbb{Z}^- \cup \{0\} \cup \mathbb{Z}^+$ , and 0 is neither positive nor negative.

Example: $-7 \in \mathbb{Z}$ , $0 \in \mathbb{Z}$ , but $2.5 \notin \mathbb{Z}$ . Every natural number is an integer: $\mathbb{N} \subset W \subset \mathbb{Z}$ .

$- 4$ and $4$ are opposites: they are the same distance from 0, on opposite sides.

Definition

Opposite (additive inverse) of an integer $a$ : the integer $- a$ , the same distance from 0 on the other side, so that $a + (- a) = 0$ .

Example: the opposite of 4 is $- 4$ ; the opposite of $- 9$ is $- (- 9) = 9$ .

Definition

Absolute value of an integer $a$ : its distance from 0 on the number line, which is never negative.

Notation: $∣ a ∣$ . Example: $∣ - 4 ∣ = 4$ , $∣ 4 ∣ = 4$ and $∣ 0 ∣ = 0$ .

Comparing integers

On a number line drawn from left to right, a number is greater than every number to its left. So $2 \gt -5$ , $0 \gt -1$ , and $-2 \gt -7$ , because $- 2$ lies to the right of $- 7$ .

Warning

A bigger digit does not mean a bigger negative number. Although $7 \gt 2$ , we have $-7 \lt -2$ : owing 7,000 Frw is worse than owing 2,000 Frw, and $-7\,^\circ\mathrm{C}$ is colder than $-2\,^\circ\mathrm{C}$ . The number further from 0 on the negative side is the smaller one.

Adding and subtracting integers

Think of adding a positive number as moving right along the number line, and adding a negative number as moving left. Subtracting a number is the same as adding its opposite:

a - b = a + (- b) .

$- 3 + 5 = 2$ : start at $- 3$ (orange) and move 5 steps right.

$2 - 6 = 2 + (- 6) = - 4$ : start at 2 and move 6 steps left.

To add two integers with …	Rule	Example
the same sign	add the absolute values; keep the common sign	$- 8 + (- 5) = - 13$
different signs	subtract the smaller absolute value from the larger; take the sign of the number with the larger absolute value	$- 8 + 5 = - 3$ ; $8 + (- 5) = 3$

Worked example 1: differences in height and temperature

(a) The summit of Mount Karisimbi, Rwanda's highest point, is 4,507 m above sea level. The shore of the Dead Sea is about 430 m below sea level, which we write as $- 430$ m. How much higher is the summit?
(b) Milk in a freezer is at $-18\,^\circ\mathrm{C}$ and the kitchen is at $22\,^\circ\mathrm{C}$ . What is the difference in temperature?

Solution A difference is "larger minus smaller".

(a) $4507 - (- 430) = 4507 + 430 = 4937$ . The summit is about 4,937 m higher.

(b) $22 - (- 18) = 22 + 18 = 40$ . The difference is $40\,^\circ\mathrm{C}$ .

Multiplying and dividing integers

Signs	Product or quotient	Examples
$(+)$ and $(+)$	positive	$6 \times 3 = 18$ , $12 \div 4 = 3$
$(+)$ and $(-)$	negative	$6 \times (-3) = -18$ , $12 \div (-4) = -3$
$(-)$ and $(+)$	negative	$(-6) \times 3 = -18$ , $(-12) \div 4 = -3$
$(-)$ and $(-)$	positive	$(-6) \times (-3) = 18$ , $(-12) \div (-4) = 3$

Same signs give a positive answer; different signs give a negative answer.

Important

Why a negative times a negative is positive. Follow the pattern as the second factor goes down by 1 each time:

Product	Value
$(-3) \times 3$	$- 9$
$(-3) \times 2$	$- 6$
$(-3) \times 1$	$- 3$
$(-3) \times 0$	$0$
$(-3) \times (-1)$	$3$
$(-3) \times (-2)$	$6$

Each step adds 3 to the product. For the pattern to continue past 0, $(-3) \times (-1)$ must be 3. The distributive property gives the same answer: $0 = (-3) \times (1 + (-1)) = -3 + (-3) \times (-1)$ , so $(-3) \times (-1) = 3$ .

Worked example 2: mixed operations

Calculate $(-4) \times (-6) \div (-3) + (-2)^3$ .

Solution Powers first, then multiplication and division from left to right, then addition.

(-2)^3 = (-2) \times (-2) \times (-2) = -8

(-4) \times (-6) = 24, \qquad 24 \div (-3) = -8

- 8 + (- 8) = - 16

Properties of the operations in $\mathbb{Z}$

Addition and multiplication of integers are commutative and associative, and multiplication distributes over addition, exactly as in $\mathbb{N}$ . The identity for addition, 0, and the identity for multiplication, 1, are both integers. Two things are new:

every integer $a$ has an additive inverse $- a$ in $\mathbb{Z}$ , with $a + (- a) = 0$ ;
$\mathbb{Z}$ is closed under subtraction, because $a - b = a + (- b)$ is always an integer.

Operation	Closed in $\mathbb{N}$ ?	Closed in $\mathbb{Z}$ ?	Counterexample where it fails
addition	yes	yes	—
subtraction	no	yes	$3 - 5 = -2 \notin \mathbb{N}$
multiplication	yes	yes	—
division	no	no	$7 \div 2 = 3.5 \notin \mathbb{Z}$

That is why $\mathbb{Z}$ was built: it repairs subtraction. The next lesson repairs division.

Important

Division by zero is not defined. Suppose $5 \div 0 = q$ . Then $q \times 0$ would have to equal 5, but $q \times 0 = 0$ for every number $q$ . No such $q$ exists, so $5 \div 0$ has no value. (But $0 \div 5 = 0$ , because $0 \times 5 = 0$ .)

Tip

For a long product, count the negative factors: an even number of negative factors gives a positive answer, an odd number gives a negative answer. So $(-1)^{10} = 1$ and $(-1)^{7} = -1$ .

Khan Academy: Why a negative times a negative is a positive (6 min). Watch how Sal uses the distributive property to show that the rule is forced, not just a convention to memorise.

✦

Practice

Work without a calculator. The last two are harder.

Arrange in increasing order: $3, - 5, 0, - 1, 7, - 8$ .
Find $∣ - 12 ∣$ , $∣ 7 ∣$ and $- ∣ - 3 ∣$ .
Calculate (a) $- 8 + 13$ (b) $- 6 - 9$ (c) $4 - (- 11)$ (d) $- 15 + (- 7)$
Calculate (a) $(-7) \times 6$ (b) $(-48) \div (-8)$ (c) $(- 2)^{4}$ (d) $(- 3)^{3}$
At 5 a.m. the temperature at the top of a mountain was $-6\,^\circ\mathrm{C}$ ; at noon it was $9\,^\circ\mathrm{C}$ . By how many degrees did it rise?
A trader starts the day owing 3,000 Frw, which we write as $- 3000$ Frw. She earns 5,500 Frw and then pays 1,200 Frw for transport. What is her balance at the end of the day?
Is $\mathbb{Z}^-$ , the set of negative integers, closed under addition? Under multiplication? Give a reason or a counterexample for each.

Show answersHide answers

$-8 \lt -5 \lt -1 \lt 0 \lt 3 \lt 7$
$12$ , $7$ and $- 3$
(a) $5$ (b) $- 15$ (c) $15$ (d) $- 22$
(a) $- 42$ (b) $6$ (c) $16$ (d) $- 27$
$9 - (- 6) = 15$ degrees
$- 3000 + 5500 - 1200 = 1300$ : she has 1,300 Frw.
Addition: yes, the sum of two negative integers is negative, e.g. $- 2 + (- 3) = - 5$ . Multiplication: no, $(-2) \times (-3) = 6 \notin \mathbb{Z}^-$ .

Sources and credits

3.2: The Integers, from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
7.5: Properties of Identity, Inverses, and Zero, from Prealgebra by Lynn Marecek and MaryAnne Anthony-Smith, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
Photo: Elevator Negative Floor Numbers in Ireland (16785350923).jpg, Tony Webster, Wikimedia Commons, licensed under CC BY 2.0.
Video: Why a negative times a negative is a positive, Khan Academy, embedded from YouTube.

The Rwandan worked examples, the diagrams, the practice items and the quiz questions were written by Ganzaa.

1Numbers below zero

2Comparing integers

3Adding and subtracting integers

4Multiplying and dividing integers

5Properties of the operations in Z \mathbb{Z} Z

Numbers below zero

Comparing integers

Adding and subtracting integers

Multiplying and dividing integers

Properties of the operations in $\mathbb{Z}$