Integers (Grade 9) — Mathematics Grades 8–11

Calculating with positive and negative numbers, and squares, cubes and roots of integers.

Gr 9 Term 1 — IntegersTerm 1

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What gets asked

Calculate an expression that mixes all four operations with integers.
Evaluate a square, cube, square root or cube root of an integer.
Write down the additive or multiplicative inverse of an integer.
Solve a word problem using more than one operation with integers.

You must be able to

Add, subtract, multiply and divide positive and negative numbers correctly.
Work out the square, cube, square root or cube root of a given integer.
Find the additive inverse (opposite) and multiplicative inverse (reciprocal) of a number.
Apply the correct order of operations when integers are mixed together.

Traps that cost marks

Treating the addition of two negative numbers like a multiplication. Adding two negatives makes a bigger negative: $- 3 + (- 4) = - 7$ .
Evaluating a negative number squared without brackets as positive. Without brackets, only the number is squared: $- 3^{2} = - 9$ . Only $(- 3)^{2} = 9$ .
Giving a square root for a negative number. No real number squares to give a negative answer, so a negative number has no square root.

Worked example

Calculate:

(- 6) - (- 9) + (- 2) \times 3

Subtracting a negative adds: $- 6 - (- 9) = - 6 + 9 = 3$ .
Multiply next: $(- 2) \times 3 = - 6$ .
Add the results: $3 + (- 6) = - 3$ .

Calculate with integers

Signs, brackets and the three rules that let you move numbers around safely.

integer: A counting number, its negative, or $0$ . So $- 7$ , $0$ and $12$ are all integers.
additive inverse: The number you add to bring it back to $0$ . The additive inverse of $7$ is $- 7$ .
multiplicative inverse: The number you multiply by to get $1$ . The multiplicative inverse of $5$ is $\frac{1}{5}$ .
commutative: The order may be swapped without changing the answer, as in $3 + 8 = 8 + 3$ .
associative: The grouping may be changed without changing the answer, as in $(2 + 5) + 4 = 2 + (5 + 4)$ .
distributive: A multiplication spreads over a bracket: $3 (4 + 6) = 3 \times 4 + 3 \times 6$ .

Adding a negative takes you further down: $- 3 + (- 4) = - 7$ . Subtracting a negative sends you back up: $5 - (- 3) = 8$ . Two minus signs standing next to each other turn into a plus.

When you multiply or divide, the same signs give a positive and different signs give a negative. That rule belongs to multiplying and dividing only. It says nothing about adding.

Brackets decide what the exponent sits on. $(- 3)^{2} = 9$ , because the minus is inside. $- 3^{2} = - 9$ , because only the $3$ is squared. A cube keeps the sign, so $3 - 8 = - 2$ .

Adding and multiplying are commutative and associative. Subtracting and dividing are neither: $8 - 3$ is not the same as $3 - 8$ . The distributive rule is the one that opens a bracket.

Every integer has an additive inverse. Every integer except $0$ also has a multiplicative inverse. The two are easy to mix up, so read which one the question wants.

Rules to remember

$- 3 + (- 4) = - 7$
$5 - (- 3) = 8$
$(- 3)^{2} = 9$ , but $- 3^{2} = - 9$

Examples

Calculate:

- 8 + (- 5) - (- 3)

Worked answer

Adding a negative takes you lower: $- 8 + (- 5) = - 13$ .
Subtracting a negative adds: $- 13 - (- 3) = - 13 + 3$ .
$- 13 + 3 = - 10$ .

Answer: $- 10$

The signs were handled one at a time, so no pair of them got treated as a multiplication.

Calculate:

- 4^{2} + (- 4)^{2} + 3 - 27

Worked answer

With no brackets only the $4$ is squared: $- 4^{2} = - 16$ .
With brackets the minus is squared too: $(- 4)^{2} = 16$ .
$3 - 27 = - 3$ , because $(- 3)^{3} = - 27$ .
Add them up: $- 16 + 16 + (- 3) = - 3$ .

Answer: $- 3$

The first two terms differ only by a pair of brackets, and they cancel each other out.

One winter night in Sutherland the temperature is

4

°C. It falls

9

°C by midnight, then rises

2

°C by sunrise. Find the sunrise temperature.

Worked answer

A fall of $9$ means adding $- 9$ .
$4 + (- 9) = - 5$ , so at midnight it is $- 5$ °C.
A rise of $2$ means adding $2$ .
$- 5 + 2 = - 3$ .

Answer: $- 3$ °C

Direction had to be written into the signs before any adding could be done.

Traps

Treating the addition of two negative numbers like a multiplication. Adding two negatives makes a bigger negative: $- 3 + (- 4) = - 7$ .
Evaluating a negative number squared without brackets as positive. Without brackets, only the number is squared: $- 3^{2} = - 9$ . Only $(- 3)^{2} = 9$ .
Giving a square root for a negative number. No real number squares to give a negative answer, so a negative number has no square root.
Answering $\frac{1}{6}$ when the additive inverse of $6$ was asked for. The additive inverse brings you to $0$ , so it is $- 6$ . The multiplicative inverse is $\frac{1}{6}$ .

Learn and practise “Calculate with integers” in the app →

Grade 9 Integers

What gets asked

You must be able to

Traps that cost marks

Worked example

Calculate with integers

Rules to remember

Examples

Traps

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