2.3

Multiplication, Division and Combined Operations of Integers

AMultiplication

We have learnt that multiplication of whole numbers can be regarded as repeated addition. For example, 3 × 4 = 4 + 4 + 4 = 12.

Let us use algebra discs to help us make sense of the multiplication of integers to find their product.

Activity 4

Objective: To perform multiplication of integers using algebra discs.

Multiplying a positive integer by a negative integer

Example: Evaluate 2 × (−5).

Take 2 groups of −5:

−1−1−1−1−1

→ ungroup →

−1−1−1−1−1 −1−1−1−1−1

∴ 2 × (−5) = −10

① Evaluate the following.

(a) 5 × (−2) = (b) 4 × (−3) =

Multiplying a negative integer by a positive integer

Example (i): Evaluate (−1) × 3.

(−1) × 3 is the same as −(1 × 3). We use algebra discs to show −(1 × 3), that is, the negative of 1 group of 3, by flipping the discs.

111

→ flip →

−1−1−1

∴ (−1) × 3 = −3

Example (ii): Evaluate (−3) × 4.

(−3) × 4 = −(3 × 4). Flip the discs to show the negative of 3 groups of 4.

1111

→ flip →

−1−1−1−1 −1−1−1−1 −1−1−1−1

∴ (−3) × 4 = −12

② Evaluate the following.

(a) (−5) × 2 = (b) (−2) × 3 =

Multiplying two negative integers

Example: Evaluate (−2) × (−3).

(−2) × (−3) = −[2 × (−3)]. Flip the discs to show the negative of 2 groups of (−3).

−1−1−1

→ flip →

111 111

∴ (−2) × (−3) = 6

③ Evaluate the following.

(a) (−3) × (−2) = (b) (−4) × (−3) =

④ Using the results from ①, ② and ③, decide if the following is positive or negative.

(a) Multiplying two positive integers → (b) Positive × Negative →

⑤ Evaluate the following. What do you observe?

(a) 2 × 5 = 5 × 2 =

(b) 2 × (−5) = (−5) × 2 =

观察一下： a × b 和 b × a 的结果有什么关系？

自评：

From Activity 4, we observe the following regarding the multiplication of two integers.

Positive × Positive = Positive

e.g., 2 × 3 = 6

Positive × Negative = Negative

e.g., 2 × (−3) = −6

Negative × Positive = Negative

e.g., (−2) × 3 = −6

Negative × Negative = Positive

e.g., (−2) × (−3) = 6

Multiplication of integers is commutative: 5 × 2 = 2 × 5, 2 × (−5) = (−5) × 2, (−3) × (−6) = (−6) × (−3).

📖 WORKED EXAMPLE 8

Find the values of the following without using a calculator or algebra discs.

(a) (−6) × 9 (b) (−4) × (−4) (c) (−2) × (−2) × (−2) (d) 8 × (−4) × (−7)

Solution

(a) (−6) × 9 = −(6 × 9) = −54
(−) × (+) → (−)

(b) (−4) × (−4) = 4 × 4 = 16
(−) × (−) → (+)

(d) 8 × (−4) × (−7) = (−32) × (−7) = 32 × 7 = 224
(−) × (−) → (+)

✏️ TRY IT YOURSELF 8

Find the values of the following without using a calculator or algebra discs.

(a) (−12) × 8 = (b) (−66) × 0 = (c) (−7) × (−7) =

(d) (−3) × (−3) × (−3) = (e) (−4)³ = (f) (−5) × 6 × (−4) =

📖 WORKED EXAMPLE 9

Without using a calculator, find

(a) the square roots of 36, (b) the cube root of −1000.

Analysis. In Worked Example 8(b), we observed that 4 and −4 are both square roots of 16. If we can find the positive square root, ▢, of 36, then −▢ is also a square root of 36.

Solution

(a) 36 = 2 × 2 × 3 × 3 = (2 × 3) × (2 × 3) = (2 × 3)²
Square roots of 36 = ±√36 = ±√(2 × 3)² = ±(2 × 3) = ±6
Note: '±' is read as 'plus minus'.

(b) 1000 = 2 × 2 × 2 × 5 × 5 × 5 = (2 × 5) × (2 × 5) × (2 × 5) = (2 × 5)³
−1000 = −(2 × 5)³
Cube root of −1000 = ∛(−1000) = ∛(−(2 × 5)³) = −(2 × 5) = −10

THINK²

1. Is it possible to get an integer that is a square root of −16? Explain your answer.

自评：

2. (a) Do both positive and negative numbers have cube roots? (b) How many cube roots does each number have? Explain.

自评：

✏️ TRY IT YOURSELF 9

Without using a calculator, find

(a) the square roots of 64 →

(b) the cube root of −64 →

(d) −∛(−125) =

BDivision

Previously, we learnt that 12 ÷ 3 can be written as 12 × ⅓ to give the same result, 4.

12 ÷ 3 = 12 × ⅓ = 4

We say that ⅓ is the reciprocal of 3 and vice versa. That is, 3 is the reciprocal of ⅓. Hence the rules for division of integers can be derived from the rules for multiplication of integers.

Examples:

Multiplication	Equivalent Division	Result
4 × 3 = 12	12 ÷ 3 = 12/3 = 12 × ⅓	= 4
4 × (−3) = −12	(−12) ÷ (−3) = −12/−3 = (−12) × 1/(−3)	= 4
(−4) × 3 = −12	(−12) ÷ 3 = −12/3 = (−12) × ⅓	= −4
(−4) × (−3) = 12	12 ÷ (−3) = 12/−3 = 12 × 1/(−3)	= −4

The sign rules for division match those for multiplication:
(+) ÷ (+) = (+), (+) ÷ (−) = (−), (−) ÷ (+) = (−), (−) ÷ (−) = (+).

📖 WORKED EXAMPLE 10

Find the values of the following.

(a) 20 ÷ 5 (b) (−42) ÷ (−7) (c) 24 ÷ (−6) (d) (−36) ÷ 3

Solution

(a) 20 ÷ 5 = 20/5 = 4

(b) (−42) ÷ (−7) = −42/−7 = 6

(d) (−36) ÷ 3 = −(36/3) = −12

✏️ TRY IT YOURSELF 10

Find the values of the following.

(a) 16 ÷ 2 = (b) (−75) ÷ (−5) = (c) 63 ÷ (−7) = (d) (−54) ÷ 6 =

CCombined Operations on Integers

In this section, we shall extend the order of operations on whole numbers to all integers as follows:

First, evaluate the expressions in brackets, starting with the innermost pair of brackets if there are more than one pair of brackets. (Brackets)
Next, evaluate the powers and roots. (Powers and Roots)
Then, multiply and divide from left to right. (Multiplication and Division)
Finally, add and subtract from left to right. (Addition and Subtraction)

📖 WORKED EXAMPLE 11

Find the values of the following without using a calculator.

(a) 23 − 56 ÷ (−8) + (−5)
(b) (−2)³ × 7 − [(−5) + (−1)]² ÷ (−4)
(c) (−11) − (−6)³ ÷ [(−4) + (−8)] + (−5)²

Solution

(a)

= 23 − [56 ÷ (−8)] + (−5)Do the division first.
= 23 − (−7) + (−5)Add and subtract from left to right.
= 23 + 7 − 5
= 25

(b)

= (−2)³ × 7 − (−6)² ÷ (−4)Do the sum within the square brackets first.
= (−8) × 7 − 36 ÷ (−4)Evaluate the powers.
= −56 − (−9)Do the multiplication and division.
= −56 + 9Do the addition.
= −47

(c)

= (−11) − (−6)³ ÷ (−12) + (−5)²Do the sum within the square brackets first.
= (−11) − (−216) ÷ (−12) + 25Evaluate the powers.
= (−11) − 18 + 25Do the division.
= −11 − 18 + 25Add and subtract from left to right.
= −4

✏️ TRY IT YOURSELF 11

Find the values of the following without using a calculator.

(a) 15 − 5 × (−2) − [30 − (1 − 3²)] =

(b) (−6)² ÷ (−12) + [(−8) − (−3)]² × (−2) =

📝 PRACTICE EXERCISE 2.3

Do not use a calculator for this exercise.

BASIC MASTERY

Evaluate the following.

(a) 8 × (−9) = (b) (−5) × (−4) = (c) (−6) × 7 = (d) −3 × 0 =

(e) (−12)² = (f) −(−4)³ = (g) (−3)² × (−5) = (h) (−2)³ × (−9) =
Evaluate the following.

(a) (−38) ÷ (−2) = (b) 132 ÷ (−11) = (c) 65 ÷ 5 = (d) (−57) ÷ 3 =

(e) 0 ÷ (−7) = (f) (−144) ÷ (−9) = (g) (−6)³ = (h) 162 ÷ (−3)³ =
Evaluate the following.

(a) √169 = (b) Square root of 361 = (c) −∛27 = (d) ∛(−343) =

INTERMEDIATE

Evaluate the following.

(a) (−6) × 3 × (−1) = (b) (−84) ÷ 7 × 5 = (c) (−37) × 0 − (−8) =

(d) 63 ÷ (−9) + (−2) × (−10) = (e) (−47) − 33 ÷ (−3) − 3 × (−7) = (f) 196 ÷ [(−8) + (−6)] × (−2) =

(g) [(−23) + 14] × (−2)² = (h) (−45) ÷ 6 × (−3)³ = (i) (−7) × (−8) × 0 × 34 − 4 × (−5)² =

(j) (−6)³ ÷ 3² − [−9 − (−8)]³ =
After joining a fitness programme, Megan's mass decreases by 2 kg every month for 3 months.

(a) Represent the decrease in Megan's mass per month by a negative number. kg

(b) If Megan's original mass was 65 kg, find her mass after 3 months. kg

ADVANCED

The volume of water in a tank is decreasing at the rate of 3 litres per minute. At 9.00 a.m., the volume of water is 120 litres. Find the volume of water at the following times.

(a) 9.30 a.m. → L

(b) 8.50 a.m. → L
Tom borrowed some money from his father. He started repaying the money 7 months ago. He returned $400 per month to his father. Currently, he still owes his father $2500.

(a) How much money did he borrow from his father? $

(b) How much will he owe his father 3 months from now? $
Jamilah jogs along a road. Starting from her house, she jogs at 4 m/s due North for 20 minutes, then at 5 m/s due South for 15 minutes and finally at 3 m/s due South for 10 minutes. Find her final position from her house.

N ↑

+

S ↓

−

Distance from house = m Direction =
The highest point of Bukit Timah Hill (BTH) is about 164 m above ground level. Jurong Rock Cavern (JRC) is 132 m below ground level.

(a) Find the vertical distance between the top of BTH and the base of JRC. m

(b) How high is an office building above ground level if its top is vertically midway between the top of BTH and the base of JRC? m
[OPEN] A quiz consists of 5 'true/false' questions. The marking scheme is as follows: 3 marks for every correct answer; −2 marks for every incorrect answer; and 0 marks for every unattempted question.

(a) What is the maximum score for the quiz? marks

(b) What is the minimum score for the quiz? marks

(c) Write about a situation where a student scores 3 marks for the quiz.

ENOne possibility: 1 correct + 0 wrong + 4 unattempted → 1×3 + 0×(−2) + 4×0 = 3 marks.
(Another: 3 correct + 3 wrong is impossible since 3+3 > 5; but 1 correct alone works.)

⏱ 20 秒后查看中文版

中文例如：对 1 题 + 错 0 题 + 不答 4 题 → 1×3 + 0×(−2) + 4×0 = 3 分。

自评：

(d) When would a student score −3 marks for the quiz?

ENOne possibility: 1 correct + 3 wrong + 1 unattempted → 1×3 + 3×(−2) + 1×0 = 3 − 6 = −3 marks.

⏱ 20 秒后查看中文版

中文例如：对 1 题 + 错 3 题 + 不答 1 题 → 1×3 + 3×(−2) + 1×0 = 3 − 6 = −3 分。

自评：

[OPEN] (a) Fill the same shape with the same integer in the following table. (b) Evaluate each expression and compare the results on the left side with those on the right side. What do you notice?

Left side	Right side
□ × ○	○ × □
(□ × ○) × ◇	□ × (○ × ◇)
□ ÷ ○	○ ÷ □
(□ ÷ ○) ÷ ◇	□ ÷ (○ ÷ ◇)

选择 □、○、◇ 为三个不同的非零整数，分别计算每一行左右两边的值并对比。

自评：

章末概念检查 · Concept Checkpoints

5 道封闭题，自动判分。本节关键：同号得正、异号得负；除法符号规则同乘法；运算顺序"括号→幂方→乘除→加减"。