Integers (Grade 8) — Mathematics Grades 8–11

Ordering integers and calculating with positive and negative numbers.

Gr 8 Term 1 — IntegersTerm 1

Practise Integers in the app →

What gets asked

Add or subtract a mix of positive and negative numbers.
Multiply or divide integers and work out the sign of the answer.
Order a list of integers from smallest to largest.
Solve a multi-step problem using several operations on integers.

You must be able to

Subtract a negative integer by adding its opposite.
Work out the sign of a product or quotient of two integers.
Order integers correctly, remembering a more negative number is smaller.
Use brackets and the order of operations correctly with negative numbers.

Traps that cost marks

Thinking a negative number is bigger than a positive one because it has more digits, e.g. $- 620 > 4$ . On the number line, every negative number is smaller than every positive number.
Treating $5 - (- 6)$ as $5 - 6$ . Subtracting a negative is the same as adding. $5 - (- 6) = 5 + 6 = 11$ .
Using the addition sign rules for multiplication, e.g. saying $(- 4) \times (- 4) = - 16$ . Two negatives multiplied give a positive: $(- 4) \times (- 4) = 16$ .
Saying the square of a negative integer is negative. Squaring a negative number always gives a positive result: $(- 6)^{2} = 36$ .

Worked example

Calculate:

- 8 + 3 \times (- 2)

Do the multiplication first: $3 \times (- 2) = - 6$ .
Now add: $- 8 + (- 6)$ .
$- 8 + (- 6) = - 14$ .

Integers on the number line

Numbers below zero: how to count them, how to put them in order, and how to add them.

integer: A whole number together with its negative, and zero. So $- 3$ , $0$ and $7$ are integers.
negative number: A number less than $0$ , written with a minus sign in front, like $- 5$ .
number line: A line with the integers marked in order, growing as you move to the right.
additive inverse: The number you add to get $0$ . For $6$ it is $- 6$ , and for $- 4$ it is $4$ .

Integers carry on past zero into the negatives. Counting back from $2$ goes $2$ , $1$ , $0$ , $- 1$ , $- 2$ . Zero is not the end of the line. The size of your step does not change when you cross it.

On the number line, numbers get bigger as you move right. So $- 2$ is bigger than $- 7$ , even though $7$ is the bigger numeral. Every negative number is smaller than every positive one.

Use $<$ for is less than, and $>$ for is more than. The open end always faces the bigger number. So $- 7 < - 2$ is true, and so is $4 > - 620$ .

Adding a positive moves you right along the number line. Adding a negative moves you left. So $- 8 + 3 = - 5$ : start at $- 8$ and take three steps to the right.

Subtracting a negative is the same as adding. Taking a debt away leaves you better off. So $7 - (- 4) = 7 + 4 = 11$ . You are adding the additive inverse of $- 4$ , which is $4$ .

Rules to remember

$a + (- a) = 0$
$a - (- b) = a + b$

Examples

A freezer reads

- 4

degrees. It drops

3

degrees every hour. What does it read after three hours?

Worked answer

After one hour: $- 4 - 3 = - 7$ .
After two hours: $- 7 - 3 = - 10$ .
After three hours: $- 10 - 3 = - 13$ .

Answer: $- 13$ degrees

The step of $3$ stays the same the whole way down, and crossing zero would not change it.

Put these bank balances in order, smallest first:

- 450

120

- 425

0

Worked answer

Every negative balance comes before $0$ .
Of the two negatives, $- 450$ sits further left, so it is the smaller.
The order is $- 450$ , then $- 425$ , then $0$ , then $120$ .

Answer: $- 450$ , $- 425$ , $0$ , $120$

Once you are past zero, the bigger numeral makes the smaller number.

A Karoo town drops to

- 6

degrees at night and reaches

19

degrees by midday. How much did it rise?

Worked answer

A rise is the top value minus the bottom value.
So work out $19 - (- 6)$ .
Subtracting a negative adds: $19 + 6 = 25$ .

Answer: $25$ degrees

Counting the steps from $- 6$ up to $19$ on the number line also gives $25$ .

Traps

Counting backwards past zero as if the numbers start again at $1$ . After $0$ the count carries straight on: $- 1$ , then $- 2$ , then $- 3$ .
Thinking a negative number is bigger than a positive one because it has more digits, e.g. $- 620 > 4$ . On the number line, every negative number is smaller than every positive number.
Ordering negatives by their numeral, so $- 450$ is placed above $- 425$ . The bigger the numeral, the further left a negative sits. So $- 450$ is the smaller one.
Treating $5 - (- 6)$ as $5 - 6$ . Subtracting a negative is the same as adding. $5 - (- 6) = 5 + 6 = 11$ .

Learn and practise “Integers on the number line” in the app →

Multiply, divide and powers

What the signs do when you multiply or divide, and what happens when you square or cube.

exponent: For a whole-number exponent, it says how many times to use the number as a factor.
square: To square a number is to multiply it by itself: $5$ squared is $5 \times 5 = 25$ .
cube: To cube a number is to use it as a factor three times: $2 \times 2 \times 2 = 8$ .
square root: The number that multiplies by itself to give it. So $36 = 6$ .
cube root: The number used as a factor three times to give it. So $327 = 3$ .

When you multiply two integers, the signs decide the sign of the answer. Two signs the same give a positive. Two signs different give a negative. So $(- 4) \times (- 4) = 16$ , while $(- 4) \times 4 = - 16$ .

The rules for multiplying are not the rules for adding. $(- 5) + (- 5) = - 10$ , but $(- 5) \times (- 5) = 25$ . Look at the sign in the middle before you decide anything.

Division follows the same sign rule as multiplication. $- 20 \div (- 4) = 5$ , and $- 20 \div 4 = - 5$ . Losing the minus sign in a division is the commonest slip of all.

In a longer calculation, brackets come first. Then multiplying and dividing, and only then adding and subtracting. So in $- 8 + 3 \times (- 2) = - 14$ , the multiplying is done first.

An exponent of $2$ means square and $3$ means cube. Squaring a negative number always gives a positive: $(- 6)^{2} = 36$ . Cubing keeps the sign: $(- 3)^{3} = - 27$ . Nothing squares to a negative, so $- 9$ does not exist.

Rules to remember

$(- a) \times (- b) = ab$
$(- a) \times b = - ab$
$(- a)^{2} = a^{2}$

Examples

A stokvel member misses

4

payments of R

150

each. Write the effect on her account as an integer.

Worked answer

A missed payment is a negative amount: $- 150$ .
There are four of them: $4 \times (- 150) = - 600$ .
So the effect is $- 600$ , meaning she is R $600$ short.

Answer: $- 600$

Signs that differ give a negative answer, which matches money going missing.

Calculate:

(- 5) \times (- 4) + 3 \times (- 6)

Worked answer

Do both multiplications before the addition.
$(- 5) \times (- 4) = 20$ .
$3 \times (- 6) = - 18$ .
Now add: $20 + (- 18) = 2$ .

Answer: $2$

Two negatives multiplied give a positive, so the first product is $20$ and not $- 20$ .

Calculate:

(- 9)^{2} - 3 - 27

Worked answer

$(- 9)^{2} = 81$ , because squaring a negative gives a positive.
$3 - 27 = - 3$ , because $(- 3)^{3} = - 27$ .
So the sum is $81 - (- 3)$ .
$81 + 3 = 84$ .

Answer: $84$

A cube root of a negative number exists, but a square root of one does not.

Traps

Using the addition sign rules for multiplication, e.g. saying $(- 4) \times (- 4) = - 16$ . Two negatives multiplied give a positive: $(- 4) \times (- 4) = 16$ .
Dropping the minus sign in a division, so $- 25 \div 5$ is given as $5$ . Signs that differ give a negative: $- 25 \div 5 = - 5$ .
Working left to right, so the adding in $- 8 + 3 \times (- 2)$ is done first. Multiplying comes first. $3 \times (- 2) = - 6$ , and then $- 8 + (- 6) = - 14$ .
Saying the square of a negative integer is negative. Squaring a negative number always gives a positive result: $(- 6)^{2} = 36$ .

Learn and practise “Multiply, divide and powers” in the app →

Properties and problems

Rearranging a calculation to make it easier, and reading real problems that go below zero.

commutative: You may swap the order of two integers when you add or multiply them.
associative: You may choose which pair to work out first when you add or multiply.
distributive: Multiplying a bracket means multiplying every integer inside it.
multiplicative inverse: The number you multiply by to get $1$ . Only $1$ and $- 1$ have one that is an integer.
inverse operation: The operation that undoes another. Dividing undoes multiplying.

Adding and multiplying integers obey the same properties as whole numbers. The commutative property lets you swap two integers. The associative property lets you choose which pair to do first. Every sign must travel with its own number.

Subtracting is not commutative. $8 - (- 3) = 11$ , but $(- 3) - 8 = - 11$ . Those are different answers, so keep the order the question gives you. Dividing is not commutative either.

The distributive property works over a bracket of integers too. Take $- 4 \times (5 - 2) = - 20 + 8$ . Both sides come to $- 12$ , and the second term kept its own sign.

Every integer has an additive inverse, the number that brings it back to $0$ . For $- 7$ it is $7$ , because $- 7 + 7 = 0$ . The multiplicative inverse gives $1$ instead, so for $5$ it is $\frac{1}{5}$ .

Check an integer calculation with the inverse operation, never by repeating it. If $- 6 \times 7 = - 42$ , then $- 42 \div 7$ must bring you back to $- 6$ . In a word problem, decide first which amounts are negative.

Rules to remember

$a \times (b + c) = ab + a c$
$(a + b) + c = a + (b + c)$
Subtracting and dividing are not commutative.

Examples

Calculate

(- 25) + 63 + 25

the easy way.

Worked answer

Move the numbers so the pair that cancels meets.
$(- 25) + 25 = 0$ .
That leaves $0 + 63 = 63$ .

Answer: $63$

$- 25$ and $25$ are additive inverses, so pairing them clears them both away.

Use the distributive property:

- 6 \times (30 - 2)

Worked answer

Multiply each number in the bracket by $- 6$ .
$- 6 \times 30 = - 180$ .
$- 6 \times (- 2) = 12$ .
Add the parts: $- 180 + 12 = - 168$ .

Answer: $- 168$

The second term keeps its own sign, so that product is $12$ and not $- 12$ .

Nomsa has R

220

in her account. A debit order of R

450

goes off, then she deposits R

100

. What is her balance?

Worked answer

Money going out is negative, so the debit order is $- 450$ .
$220 + (- 450) = - 230$ .
The deposit adds: $- 230 + 100 = - 130$ .
So the balance is $- 130$ , which is R $130$ overdrawn.

Answer: $- 130$

Treating every amount as positive would give R $770$ , which is nowhere near right.

Traps

Treating $8 - (- 3)$ and $(- 3) - 8$ as the same calculation. Order matters when you subtract. $8 - (- 3) = 11$ , and $(- 3) - 8 = - 11$ .
Losing a minus sign while regrouping, so $- 25$ turns into $25$ . Move each sign with its own number. The sign belongs to the number, not the place.
Giving the additive inverse of $- 9$ as $- 9$ again. The two must add to $0$ , so the partner of $- 9$ is $9$ .
Treating every value in a word problem as positive. Decide first which way each amount points. Money going out is negative.

Learn and practise “Properties and problems” in the app →

Grade 8 Integers

What gets asked

You must be able to

Traps that cost marks

Worked example

Integers on the number line

Rules to remember

Examples

Traps

Multiply, divide and powers

Rules to remember

Examples

Traps

Properties and problems

Rules to remember

Examples

Traps

About this material