2.2

Addition and Subtraction of Integers

In the previous section, we learnt that positive and negative integers can be represented on a number line. These numbers can also be represented using algebra discs. Each disc has two sides as shown below. One side shows the number 1 and the other side shows the number −1.

front

−1

back

We flip a disc to obtain the negative of a number as shown below.

1 → flip → −1 and −1 → flip → 1

We write −(1) = −1 and −(−1) = 1.

We can use four 1 discs to represent the number 4.

1111

We write this as 1 + 1 + 1 + 1 = 4.

We use three −1 discs to represent the number −3.

−1−1−1

We write this as (−1) + (−1) + (−1) = −1 − 1 − 1 = −3.

AAddition

Suppose the temperature of a piece of frozen meat is −3 °C. During defrosting, its temperature rises by 4 °C. What is the temperature then?

The answer involves the addition of integers (−3) + 4.

We can illustrate the addition of integers using algebra discs. A key concept of the discs is that 1 and −1 form a zero pair.

1 −1 zero pair

1 + (−1) = 1 − 1 = 0

1 −1

3 zero pairs

3 + (−3) = 3 − 3 = 0

Let us see how we can make sense of addition of integers using algebra discs.

Activity 2

Objective: To perform addition of integers using algebra discs.

Adding two integers with the same sign

Example: Evaluate (−3) + (−2).

−1−1−1 Place 3 −1 discs to represent −3.

−1−1 Add 2 −1 discs to represent +(−2).

−5

∴ (−3) + (−2) = −5

In fact, we can write (−3) + (−2) = −3 − 2 = −5.

① Evaluate each of the following.

(a) 5 + 1 = (b) (−5) + (−5) = (c) (−2) + (−3) = (d) (−3) + (−5) =

Example: Evaluate (−5) + 2.

−1

−1−1−1 Place 5 −1 discs to represent −5.

Add 2 1 discs to represent +2. Remove the zero pairs.

Here we add a negative number and a positive number together. Since there are more negative ones than positive ones, the answer is negative.

∴ (−5) + 2 = −3

In fact, we can write (−5) + 2 = −5 + 2 = −3.
We can also write (−5) + 2 = 2 + (−5) = 2 − 5 = −3.

② Evaluate each of the following.

(a) 7 + (−3) = (b) (−4) + 6 = (c) 3 + (−5) = (d) (−5) + 2 =

③ Evaluate the following. What do you observe?

(a) 2 + 3 = and 3 + 2 =

(b) (−5) + 4 = and 4 + (−5) =

观察一下： a + b 和 b + a 结果有什么关系？

自评：

From Activity 2, ③(a) and (b), we observe the following.
Addition of integers is commutative, e.g., 3 + 2 = 2 + 3, (−5) + 4 = 4 + (−5).

📖 WORKED EXAMPLE 4

For each of the following, predict whether the answer is positive or negative and then evaluate it without using a calculator or algebra discs.

(a) (−8) + (−7) (b) 25 + (−11) (c) (−16) + 9 (d) 0 + (−4)

Solution

(a) Since both numbers are negative, the answer is negative.
(−8) + (−7) = −8 − 7 = −15

(b) Since there are more positive ones than negative ones, the answer is positive.
25 + (−11) = 25 − 11 = 14

(c) Since there are more negative ones than positive ones, the answer is negative.
(−16) + 9 = −16 + 9 = −7 or (−16) + 9 = 9 + (−16) = 9 − 16 = −7

(d) Since there are only negative ones, the answer is negative.
0 + (−4) = 0 − 4 = −4

✏️ TRY IT YOURSELF 4

For each of the following, predict whether the answer is positive or negative and then evaluate it without using a calculator or algebra discs.

(a) (−16) + (−14) = (b) 17 + (−9) = (c) −13 + 21 =

(d) (−23) + 12 = (e) 0 + (−25) = (f) 14 + (−26) =

📖 WORKED EXAMPLE 5

If you deposit $10 into a savings account and then transfer $12 to another account, how will the balance in the first savings account change?

Solution

Let the amount deposited be represented by a positive integer and the amount transferred by a negative integer.

Amount deposited = $10
Amount transferred = −$12
Change in balance = $10 + (−$12)
= −$2

The balance in the first savings account will change by −$2 or decrease by $2.

✏️ TRY IT YOURSELF 5

(a) The temperature of a substance rises by 6 °C in the first hour of recording and then drops by 7 °C in the next hour. How much has the temperature changed after the first two hours?

Change = °C

(b) Jane owes Peter $18. She returns him $13. How much money does Jane owe Peter now?

Jane owes Peter $

BSubtraction

The negative of a number is obtained by changing its sign. For example, the negative of 3 is −3. The negative of −3 is −(−3) = 3. We can use algebra discs to represent the negative of a number by flipping the discs.

111

→ flip →

−1−1−1

−(3) = −3

−1−1−1

→ flip →

111

−(−3) = 3

Let us see how we can make sense of subtraction of integers using algebra discs.

Activity 3

Objective: To perform subtraction of integers using algebra discs.

Subtracting a positive integer

Example (i): Evaluate 7 − 3.

1111 111 Take away 3 1 discs to represent subtraction of 3.

∴ 7 − 3 = 4

In fact, we can write 7 − 3 = 7 + (−3) = 4.

Example (ii): Evaluate 3 − 5.

111

Observe that we do not have 5 1 discs to take away from 3 1 discs. So, we can add two zero pairs to obtain the following.

111

1 −1

1 −1 2 zero pairs

Take away 5 1 discs to show subtraction of 5.

Remaining: −1−1 = −2

∴ 3 − 5 = −2

We can also write 3 − 5 = 3 + (−5) = −2.

Example (iii): Evaluate (−2) − 4.

−1−1

Observe that we do not have 4 1 discs to take away from 2 −1 discs. So, we add four zero pairs to obtain the following.

−1−1

1−1

1−14 zero pairs

Take away 4 1 discs to show subtraction of 4.

Remaining: −1−1−1−1−1−1 = −6

∴ (−2) − 4 = −6

We can also write (−2) − 4 = −2 − 4 = −6 or (−2) − 4 = (−2) + (−4) = −6.

① Evaluate each of the following.

(a) 5 − 2 = (b) 3 − 4 = (c) (−3) − 1 = (d) (−6) − 3 =

Subtracting a negative integer

Example (i): Evaluate (−4) − (−1).

−1−1−1 −1 Take away 1 −1 disc to represent subtraction of −1.

∴ (−4) − (−1) = −3

We can also write (−4) − (−1) = −4 + 1 = −3.

Example (ii): Evaluate 4 − (−3).

1111

Observe that we do not have 3 −1 discs to take away from 4 1 discs. So, we add three zero pairs to obtain the following.

1111

1−1

1−13 zero pairs

Take away 3 −1 discs to show subtraction of −3.

Remaining: 1111111 = 7

∴ 4 − (−3) = 7

We can also write 4 − (−3) = 4 + 3 = 7.

② Evaluate each of the following.

(a) (−9) − (−7) = (b) (−3) − (−5) = (c) 6 − (−3) = (d) 2 − (−4) =

From Activity 3, we observe the following.

When we subtract a number, we can see it as adding the negative of the number. For example,
3 − 5 = 3 + (−5)
(3 subtract 5) (3 add negative 5)
The negative of a negative number is a positive number. For example,
−(−5) = 5
(negative of negative 5) (positive 5)

📖 WORKED EXAMPLE 6

Evaluate each of the following without using a calculator or algebra discs.

(a) 0 − (−17) (b) 5 − 8 (c) −16 − (−9)

Solution

(a) 0 − (−17) = 0 + 17 = 17
Since the sum is zero and positive ones, the answer is positive.

(b) 5 − 8 = 5 + (−8) = −3
Since there are more negative ones than positive ones, the answer is negative.

✏️ TRY IT YOURSELF 6

Evaluate each of the following without using a calculator or algebra discs.

(a) 13 − (−21) = (b) −19 − (−4) = (c) 3 − (−26) =

(d) 13 − 21 = (e) −14 − 24 = (f) −11 − (−25) =

📖 WORKED EXAMPLE 7

Singapore is in the GMT+8 time zone and New York is in the GMT−4 time zone in August.

(a) How many hours is the local time in Singapore ahead of the local time in New York in August?
(b) When it is 3 a.m. on 9 August in Singapore, what is the date and the local time in New York?

Solution

(a) The number of hours that Singapore is ahead of New York is the difference between their time zones.
8 − (−4) = 8 + 4 = 12 hours
The local time in Singapore is 12 hours ahead of the local time in New York in August.

(b) We can use a diagram to represent time as shown. Singapore is 12 hours ahead of New York in August.

We can see from the diagram that the date and local time in New York is 3 p.m. on 8 August.

✏️ TRY IT YOURSELF 7

Sydney is in GMT+10 time zone and Vancouver is in GMT−7 time zone in July.

(a) How many hours is the local time in Vancouver behind the local time in Sydney in July?

(b) When it is 4 p.m. on 3 July in Vancouver, what is the date and local time in Sydney?

(c) Jane lives in Vancouver and she is planning to visit Sydney. If the plane departs at 4 p.m. on 3 July, and the flight duration is 15 hours 30 minutes, what is the date and local time in Sydney when she lands?

📝 PRACTICE EXERCISE 2.2

Do not use a calculator for this exercise.

BASIC MASTERY

Evaluate the following.

(a) −13 + 5 = (b) (−27) + (−3) = (c) (−4) + 18 = (d) 18 + (−7) =

(e) 0 + (−12) = (f) −37 + 37 = (g) −16 + (−25) = (h) 24 + (−30) =
Evaluate the following.

(a) 6 − 13 = (b) (−17) − 8 = (c) 29 − (−7) = (d) −12 − (−23) =

(e) (−5) − (−5) = (f) −6 − 24 = (g) (−19) − (−8) = (h) 11 − (−24) =

INTERMEDIATE

Find the values of the following.

(a) 3 − (−8) + (−4) = (b) (−2) − (−5) + (−3) = (c) (−9) + (−2) − (−7) = (d) 8 − (−2) + (−8) =

(e) 7 + (−12) + (−6) = (f) −19 − (−15) + 10 = (g) 15 − (−7) + (−7) = (h) −12 + (−9) + (−3) =
Find the missing numbers.

(a) 7 + = 3 (b) 11 + = −5 (c) −9 − = −13

(d) −8 − = 10 (e) −8 + = 6 (f) 4 − = 7
The freezing point of water is 0 °C. After adding salt, the freezing point of water decreases by 6 °C. Find the freezing point of water after adding salt.

Freezing point = °C

Note: Road salting is one of the ways to decrease freezing point to allow vehicles to move safely without slipping during winter.
A helicopter flying 120 m above sea level detects a submarine. If the submarine is 39 m below sea level, what is the vertical distance between the helicopter and the submarine?

Vertical distance = m
James had $300 in his bank account yesterday. His father deposited $600 into his account today. James will withdraw $200 from it tomorrow. If the amount withdrawn is represented as −$200, find the balance in the account

(a) today = $

(b) tomorrow = $

ADVANCED

A car travels 16 km due South, 33 km due North and then 12 km due South. What is its final position from the starting point?

N ↑

positive

S ↓

negative

Distance from start = km

Direction =
[OPEN] Determine whether each statement below is TRUE or FALSE. Give a counterexample if the statement is false.

(a) Positive number + Positive number = Negative number

Counterexample (if FALSE):

ENFALSE. Counterexample: 2 + 3 = 5 (positive). Adding two positive numbers always gives a positive number.

⏱ 20 秒后查看中文版

中文错误。反例：2 + 3 = 5（正数）。两个正数相加永远是正数。

自评：

(b) Negative number + Positive number = Positive number

Counterexample (if FALSE):

ENFALSE. Counterexample: (−5) + 2 = −3 (negative). When the negative has a larger size, the answer is negative.

⏱ 20 秒后查看中文版

中文错误。反例：(−5) + 2 = −3（负数）。当负数的"绝对值"更大时，结果是负数。

自评：

(c) Negative number − Positive number = Negative number

（这条是 TRUE — 不需要反例。如果你想验证，先自己想一个例子，再展开下面的说明对照。）

💡 思考后再展开

负数 − 正数 = 负数 + (−正数) = 两个负数相加 → 永远是负数。例如：(−3) − 5 = −8。

(d) Negative number − Negative number = Positive number

Counterexample (if FALSE):

ENFALSE. Counterexample: (−5) − (−3) = −5 + 3 = −2 (negative). The sign of the answer depends on which number has the larger size.

⏱ 20 秒后查看中文版

中文错误。反例：(−5) − (−3) = −5 + 3 = −2（负数）。结果的正负取决于两个数哪个绝对值更大。

自评：

Note: A counterexample is a special case that can be used to show that a statement is false.
Ali has been running a small business for 4 years. The following table shows his profits (positive) and losses (negative).

Year Profit/Loss

1 −$23 000

2 −$6 000

3 $9 000

4 $17 000

(a) Find his total profit or loss in these 4 years. $

(b) If his targeted total profit is $18 000 in the first 5 years, what should his profit be in the 5th year? $
Louis' flight to New York from London departed at London time 8.30 a.m. on 8 November. The plane arrived in New York at New York time 11.10 a.m. on 8 November. Given that London is in the GMT+0 time zone and New York is in the GMT−5 time zone, find the flight time from London to New York.

Flight time =
Find the sum of the following:

1 − 2 + 3 − 4 + 5 − 6 + … + 99 − 100 =

Year	Profit/Loss
1	−$23 000
2	−$6 000
3	$9 000
4	$17 000

章末概念检查 · Concept Checkpoints

5 道封闭题，自动判分。如果做错，回到上面对应章节复习再来。