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11
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Ch apter
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Number and Algebra
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Integers
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You have probably come across examples of negative numbers already. These
are the numbers that are less than zero. They are necessary in all sorts of
situations. For example, they are used in the measurement of temperature.
The temperature 0ºC is the temperature at which water freezes, known as the
freezing point. The temperature that is 5 degrees colder than freezing point is
written as −5ºC.
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In some Australian cities, the temperature drops below zero. Canberra’s lowest
recorded temperature is −10ºC. The lowest temperature ever recorded in
Australia is −23ºC, at Charlotte’s Pass in NSW. Here are the lowest recorded
temperatures at some other places:
Alice Springs
Paris
London
−7ºC
−24ºC
−21ºC
Negative numbers are also used to record heights below sea level.
For example, the surface of the Dead Sea in Israel is 424 metres below sea level.
This can be written as −424 metres. This is the lowest point on land anywhere
on Earth. The lowest point on land in Australia is at Lake Eyre, which is
15 metres below sea level. This is written as −15 metres.
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Negative integers
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11 A
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A debt subtracted from zero is a fortune.
A fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is zero.
The product of zero multiplied by zero is zero.
The product or quotient of two fortunes is a fortune.
The product or quotient of two debts is a fortune.
The product or quotient of a debt and a fortune is a debt.
The product or quotient of a fortune and a debt is a debt.
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Brahmagupta, an Indian mathematician, wrote important works on mathematics and
astronomy, including a work called Brahmasphutasiddhanta (The Opening of the Universe),
which he wrote in the year 628 CE. This book is believed to mark the first appearance of
negative numbers. Brahmagupta gives the following rules for positive and negative numbers
in terms of fortunes (positive numbers) and debts (negative numbers). By the end of this
chapter, you will be able to understand his words.
The whole numbers, together with the negative whole numbers, are called the integers. These are:
…, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …
The numbers 1, 2, 3, 4, 5, … are called the positive integers.
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The numbers …, −5, −4, −3, −2, −1 are called the negative integers.
The number 0 is neither positive nor negative.
The number line
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The integers can be represented by points on a number line. The line is infinite in both directions,
with the positive integers to the right of zero and the negative integers to the left of zero. The integers
are equally spaced.
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
An integer a is less than another integer b if a lies to the left of b on the number line. The symbol <
is used for less than. For example, −3 is to the left of −1, so −3 < −1.
An integer b is greater than another integer a if b lies to the right of a on the number line. The
symbol > is used for greater than. For example, 1 is to the right of −5, so 1 > −5.
a
a<b
b
and
b>a
A practical illustration of this is that a temperature of −8°C is colder than a temperature of −3°C,
and −8 < −3. Also, 0° is warmer than −5°C, and 0 > −5.
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Example 1
List all the integers less than 5 and greater than −3.
Solution
−2, −1, 0, 1, 2, 3, 4
a Arrange the following integers in increasing order.
−6, 6, 0, 100, −1000, −5, −100, 8
b Arrange the following integers in decreasing order.
−25, 1000, −500, −26, 53, 100, 56
Solution
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a −1000, −100, −6, −5, 0, 6, 8, 100
b 1000, 100, 56, 53, −25, −26, −500
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Example 2
Example 3
Draw a number line and mark on it with dots all the integers less than 6 and greater
than −5.
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Solution
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
Example 4
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a The sequence 10, 5, 0, −5, −10, … is ‘going down by fives’. Write down the next four
numbers, and mark them on the number line.
b The sequence −16, −14, −12, … is ‘going up by twos’. Write down the next four
numbers, and mark them on the number line.
Solution
a The next four numbers are −15, −20, −25, −30.
−45 −40 −35 −30
−25
−20
−15
−10
−5
0
5
10
0
2
4
6
b The next four numbers are −10, −8, −6, −4.
−16 −14 −12 −10
−8
−6
−4
−2
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The opposite of an integer
The number −2 is the same distance from 0 as 2, but lies on the opposite side of zero. We call −2 the
opposite of 2. Similarly, the opposite of −2 is 2.
The operation of forming opposites can be visualised by putting a pin in the number line at 0 and
rotating the number line by 180°.
The opposite of 2 is −2.
−5
−4
−3
−2
−1
0
1
The opposite of −2 is 2.
2
3
4
5
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−6
Notice that the opposite of the opposite is the number we started with. For example, −(−2) = 2.
Example 1
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Exercise 11A
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Note: The number 0 is the opposite of itself. That is, −0 = 0. No other number has this property.
1 a List the integers less than 3 and greater than −5.
b List the integers greater than −8 and less than −1.
c List the integers less than −4 and greater than −10.
Example 2
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d List the integers greater than −132 and less than −123.
2 a Arrange the following integers in increasing order.
−10, 10, 0, 100, −100, −6, −1000, 5
b Arrange the following integers in decreasing order.
−30, 45, −45, −550, −31, 26, −26, 55
Example 3
3 a Draw a number line and mark on it the numbers −2, −4, −6 and −8.
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b Draw a number line and mark on it the numbers −1, −3, −5 and −7.
c Draw a number line and mark on it the integers less than 0 and greater than −8.
d Draw a number line and mark on it the integers less than 3 and greater than −3.
Example 4
4 The sequence −15, −13, −11, … is ‘going up by twos’. Write down the next three terms.
(Draw a number line to help you.)
5 The sequence 3, 1, −1, … is ‘going down by twos’. Write down the next three terms. (Draw
a number line to help you.)
6 The sequence −50, −45, −40, … is ‘going up by fives’. Write down the next three terms.
(Draw a number line to help you.)
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7 Give the opposite of each integer.
a 5
b −4
c −10
d −12
e 7
f −8
g −4
h −3
8 Simplify:
a −(−2)
b −(−7)
c −(−20)
d −(−(−10))
e −(−(−30))
f −(−(−(−40)))
9 Insert the symbol > or < in each box to make a true statement.
a 3 □ 5
c −7 □ −4
b 3 □ −5
d 2 □ − (−3)
a
b
°c
c
°c
100
90
90
80
80
70
70
60
60
d
°c
100
°c
100
90
80
80
70
70
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0
0
0
-10
-10
-10
-10
-20
-20
-20
-20
-30
-30
-30
-30
-40
-40
-40
-40
50
40
30
20
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10
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90
G
100
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10 Write down the reading for each thermometer shown below.
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11 B
Addition and subtraction
of a positive integer
If a submarine is at a depth of −250 m and then rises by 20 m, its final position is −230 m. This can
be written −250 + 20 = −230.
Joseph has $3000 and he spends $5000. He now has a debt of $2000, so it is natural to interpret this
as 3000 − 5000 = −2000.
These are examples of adding and subtracting a positive integer.
The number line and addition
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The number line provides a useful picture for addition and subtraction of integers.
Addition of a positive integer
−6
−5
−4
−3
−2
−1
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When you add a positive integer, move to the right along the number line.
0
1
2
3
4
5
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For example, to calculate −3 + 4, start at −3 and move 4 steps to the right. We see that
−3 + 4 = 1.
We can interpret −3 + 4 in terms of money:
I start with a debt of $3 but I then earned $4. I now have $1.
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Subtraction of a positive integer
We will start by thinking of subtraction as taking away.
When you subtract a positive integer, move to the left along the number line.
For example, to calculate 2 − 5, start at 2 and move to the left 5 steps. We see that
2 − 5 = −3.
−5
−4
−3
−2
−1
0
1
2
3
4
5
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−6
We can interpret 2 − 5 in terms of money:
I had $2 and I spent $5. I now have a debt of $3.
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1 1 B A d d i t i o n an d s u bt r a c t i o n o f a p o s i t i v e i nte g e r
Example 5
Write the answers to these additions.
a −5 + 6
c −11 + 20
b −7 + 12
Solution
a
−5
−4
−3
−5 + 6 = 1
−2
−1
0
1
(Start at −5 on the number line and move 6 steps to the right.)
(Start at −7 on the number line and move 12 steps to the right.)
c −11 + 20 = 9
(Start at −11 on the number line and move 20 steps to the right.)
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b −7 + 12 = 5
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Example 6
Find the value of:
b 6 − 9
Solution
a
−6
−5
−4
−3
c −4 − 11
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a −2 − 3
−2
−1
0
1
2
3
d 3 − 12 − 8
4
5
Start at −2 and move 3 steps to the left. We see that −2 − 3 = −5.
(Start at 6 and move 9 steps to the left.)
c −4 − 11 = −15
(Start at −4 and move 11 steps to the left.)
d 3 − 12 − 8 = −17
(Start at 3 and move 12 steps to the left and then 8 steps to the left.)
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b 6 − 9 = −3
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Exercise 11B
You may wish to visualise these calculations on a number line.
Example 5
Example 6
1 Calculate these additions.
a −5 + 7
b −2 + 3
c −5 + 10
d −1 + 4
e −12 + 16
f −5 + 2
g −6 + 12
h −5 + 10
i −11 + 4
j −12 + 4
k −8 + 10
l −1 + 12
2 Calculate these subtractions.
a 5 − 6
b 6 − 12
c −5 − 10
d −11 − 10
e −7 − 16
f −5 − 2
g −6 − 2
h 5 − 10
i −11 − 4
j −12 − 5
k −10 − 9
l −5 − 8
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3 Work from left to right to calculate the answer.
Example 6d
a 15 − 6 − 8
b 6 − 12 − 5
c −8 − 10 − 11
d −11 + 10 − 20
e −7 − 16 − 20
f 5 − 2 − 10
g −6 − 2 − 20
h 5 − 10 + 20
i −11 − 4 − 30
j −12 − 5 + 20
k −20 − 30 − 10
l −5 + 6 − 7
4 Work from left to right to calculate the answer.
a 11 − 10 − 20 − 15
b −2 − 3 − 4 − 5
d −11 + 1 + 2 + 8 + 1
e −20 − 2 − 4 + 6
c 20 − 9 − 7 − 4
5 a Johanne has a total amount of $3400 and spends $5000. What is Johanne’s debt?
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rancis has a debt of $4670, but earns $3456 and pays off a portion of the debt. How
b F
much does Francis owe now?
c David has a debt of $3760, but earns $4000 and pays off the debt. How much does David
have now?
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6 a A submarine is at a depth of −320 m and then rises by 40 m. What is the new depth of the
submarine?
b T
he temperature in a freezer is −17°C. The freezer is turned off and in 10 minutes the
temperature has risen by 8°C. What is the temperature of the freezer now?
a 1 □ 2 □ 3 □ 4 = −2
b 3 □ 10 □ 9 □ 5 = 7
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7 Place either a plus (+) or a minus (−) sign in each box to make these statements true.
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c 1 □ 2 □ 3 □ 4 □ 5 = −1
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11 C
Addition and subtraction of a
negative integer
In the previous section, we considered addition and subtraction of a positive integer. In this section,
we will add and subtract negative integers.
Addition of a negative integer
Adding a negative integer to another integer means that you take a certain number of steps to the left
on a number line.
The result of the addition 4 + (−6) is the number you get by moving 6 steps to the left, starting at 4.
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
4 + (−6) = −2
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1 1 C A d d i t i o n an d s u bt r a c t i o n o f a ne g at i v e i nte g e r
Example 7
Find −2 + (−3).
Solution
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
−2 + (−3) is the number you get by moving 3 steps to the left, starting at −2. That is, −5.
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Notice that −2 − 3 is also equal to −5.
All additions of this form can be completed in a similar way. For example:
4 + (−7) = −3
and note that
−11 + (−3) = −14
and note that
4 − 7 = −3
−11 − 3 = −14
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This suggests the following rule.
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To add a negative integer, subtract its opposite
For example:
4 + (−10) = 4 − 10
= −6
−7 + (−12) = −7 − 12
= −19
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Subtracting a negative integer
We have already seen that adding −2 means taking 2 steps to the left. For example:
0
5
7
7 + (−2) = 5
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We want subtracting −2 to be the reverse of the process of adding −2. So to subtract −2, we take 2
steps to the right. For example:
0
7
9
7 − (−2) = 9
There is a very simple way to state this rule:
To subtract a negative number, add its opposite
For example:
7 − (−2) = 7 + 2
=9
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1 1 C A d d i t i o n an d s u bt r a c t i o n o f a ne g at i v e i nte g e r
Example 8
Evaluate:
a 12 + (−3)
d −12 − (−6)
b −3 + (−7)
e 4 − (−15)
c 6 − (−18)
f −25 − (−3)
Solution
Calculate:
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b −14 + (−7) − (−15)
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a 6 − (−3) + (−8)
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Example 9
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a 12 + (−3) = 12 − 3
=9
b −3 + (−7) = −3 − 7
= −10
c 6 − (−18) = 6 + 18
= 24
d −12 − (−6) = −12 + 6
= −6
e 4 − (−15) = 4 + 15
= 19
f −25 − (−3) = −25 + 3
= −22
Solution
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a 6 − (−3) + (−8) = 6 + 3 − 8
=9−8
=1
b −14 + (−7) − (−15) = −14 − 7 + 15
= −21 + 15
= −6
Example 10
The temperature on Saturday at 1 a.m. was −13°C and the temperature at 2 p.m. was −2°C.
Calculate the rise in temperature.
Solution
Rise in temperature = −2 − (−13)
= −2 + 13
= 11°C
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Exercise 11C
b −6 + 2
c −5 + 10
d −11 + (−4)
e −12 + 16
f −5 + (−2)
g −6 + (−2)
h −5 + (−10)
i 11 + (−4)
j −12 + 4
k −20 + (−30)
l −110 + 100
2 Find:
a 5 − (−6)
b 6 − (−12)
c −5 − (−10)
d 11 − (−4)
e −12 − (−16)
f −5 − (−2)
g −6 − (−2)
h 5 − (−10)
j −12 − (−4)
k −15 − (−20)
3 Evaluate:
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Example 9
a 5 + (−2)
i −11 − 4
l −30 − (−100)
a 15 − 26 + (−25)
b −10 − 12 + 8
c −39 + 54 − 1
d 31 − 41 − (−9)
e 6 + 12 − 16
f −28 − (−35) − (−2)
g −36 − 17 + 26
h 5 − (−21) + 45
i 16 + (−4) − (−4)
j −92 + 54 − (−82)
k −900 + 1000 − (−100)
l −500 + 2000 − (−50)
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Example
8c, d, e, f
1 Find:
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Example
7, 8a, b
4 Write the answers to these subtractions.
a 23 − (−20)
b −78 − (−56)
c −65 − (−78)
a 45 − 50
b 30 − (−5)
c 60 − (−5)
d 4 − 11 − 21 + 40
e 12 − 20 + 30
f 7 − 10 − 20
g 7 − (−15) + 20
h −11 − 10 − (−4)
i −30 + 50 − 45 − (−6)
j −34 + 60 − (−5) + 10
k 43 + 50 − (−23)
l −10 − 45 + 30
6 The temperature in Moscow on a winter’s day went from a minimum of −19°C to a
maximum of −2°C. By how much did the temperature rise?
7 The temperature in Ballarat on a very cold winter’s day went from −3°C to 7°C. What was
the rise in temperature?
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Example 10
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5 Evaluate:
8 The table below shows temperatures at 5 a.m. and 2 p.m. for a number of cities. Complete
the table.
Temperature at 5 a.m.
Temperature at 2 p.m.
−10
−15
−25
−20
−7
−11
−13
5
5
−3
−15
−5
2
5
Rise in temperature (°C)
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9 A meat pie in the microwave rises in temperature by about 9°C for each minute of heating.
If you take a frozen meat pie out of the freezer, where it has been stored at −14°C, how long
does it have to be in the microwave before it reaches 40°C?
10 The temperature in Canberra on a very cold day went from 11°C to −3°C. What was the
change in temperature?
11 The table below shows the temperatures inside and outside a building on different days.
Temperature inside (°C)
Temperature outside (°C)
M
20
25
T
13
18
W
24
20
T
10
−5
F
−5
S
3
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Day
−10
−6
G
For each day, calculate (Temperature inside − Temperature outside). What does it mean if
the result of this calculation is negative?
PA
12 Jane has just received her first credit card, and has already used it to buy some clothes. The
balance is −$140. She spends another $70 at the grocery store the next day. At the end of
the week, she will be paid $280. If she uses this to pay off her credit card, how much will
Jane have left?
13 Find 1 − 2 + 3 − 4 + 5 − 6 + ....... + 99 − 100.
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11 D
Multiplication involving
negative integers
Multiplication with negative integers
5 × (−3) means 5 lots of −3 added together. That is:
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5 × (−3) = (−3) + (−3) + (−3) + (−3) + (−3)
= −15
Just as 8 × 6 = 6 × 8, we will take −3 × 5 to be the same as 5 × (−3).
All products such as 5 × (−3) and −3 × 5 are treated in the same way.
For example:
−6 × 3 = 3 × (−6)
−15 × 4 = 4 × (−15)
= −18
= −60
The question remains as to what we might mean by multiplying two negative integers together.
We first investigate this by looking at a multiplication table.
In the left-hand column below, we are taking multiples of 5. The products go down by 5 each time.
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In the right-hand column, we are taking multiples of −5. The products go up by 5 each time.
2 × 5 = 10
2 × (−5) = −10
1×5=5
1 × (−5) = − 5
0×5=0
0 × (−5) = 0
−1 × 5 = −5
−1 × (−5) = ?
−2 × 5 = −10
−2 × (−5) = ?
The pattern suggests that it would be natural to take −1 × (−5) to equal 5 and −2 × (−5) to equal 10
so that the pattern continues in a natural way.
All products such as −5 × (−2) and −5 × (−1) are treated in the same way. For example:
−3 × (−8) = 24
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−6 × (−2) = 12
The sign of the product of two integers
G
We have the following rules.
• The product of two positive numbers is a positive number.
−4 × 7 = −(4 × 7)
= −28
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• The product of a negative number and a positive number is a negative number. For example:
• The product of two negative numbers is a positive number. For example:
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−4 × (−7) = 4 × 7
= 28
Example 11
Evaluate each of these products.
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a 3 × (−20)
d 15 × (−40)
b −6 × 10
e −12 × 8 × 2
c −25 × (−30)
f −4 × (−8) × (−3)
Solution
a 3 × (−20) = −60
b −6 × 10 = −60
c −25 × (−30) = 25 × 30
= 750
e −12 × 8 × 2 = (−96) × 2
= −192
d 15 × (−40) = −600
f −4 × (−8) × (−3) = 32 × (3)
= −96
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Exercise 11D
1 Calculate each multiplication.
a −1 × 5
b 6 × (−1)
c −6 × (−1)
d 16 × (−1)
e 0 × (−3)
f −3 × (−2)
g 6 × (−3)
h −3 × 8
i −7 × 8
j −9 × 0
k −2 × (−5)
l −4 × 6
2 Calculate each multiplication.
a 5 × (−2)
b 6 × (−2)
c 5 × (−1)
d 11 × (−4)
e 12 × (−16)
f −5 × 2
g −6 × 2
h −5 × 10
i −11 × 4
j −12 × 4
k −20 × (−6)
l 16 × (−3)
m −7 × (−18)
n −13 × (−13)
o −19 × 8
p 15 × (−4)
q −17 × (−9)
r −6 × (−17)
s −14 × 20
t −12 × (−15)
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3 Evaluate:
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Example 11
b −4 × (−7) × (−6)
c 60 × (−4) × (−10)
d −45 × (−7) × 20
e 45 × (−3) × (−20)
f −34 × (−3) × (−2)
g −3 × (−1) × (−4)
h 6 × (−3) × (−1)
i −1 × (−1) × (−6)
PA
a 3 × (−2) × (−6)
4 Copy and complete these multiplications.
b 5 × … = −65
c −7 × … = 42
d … × (−8) = 56
e 5 × … = −30
f … × (−6) = 30
g −7 × … = −84
h (−1) × … = −8
i … × (−9) = 0
j −8 × … = −96
k … × 3 = −81
l −5 × … = 80
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a 2 × … = −30
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11 E
Division involving negative
integers
Division with negative integers
Every multiplication statement, for non-zero numbers, has equivalent division statements. For
example, 7 × 3 = 21 is equivalent to 21 ÷ 3 = 7 or 21 ÷ 7 = 3. We will use this fact to establish the
rules for division involving integers.
Here are some more examples:
7 × 6 = 42 is equivalent to 42 ÷ 6 = 7
7 × (−6) = −42 is equivalent to −42 ÷ (−6) = 7
×6
× (−6)
7
42
÷6
312
−42
7
÷ (−6)
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1 1 E D i v i s i o n i n v o lv i n g ne g at i v e i nte g e r s
−7 × 6 = −42 is equivalent to −42 ÷ 6 = −7
−7 × (−6) = 42 is equivalent to 42 ÷ (−6) = −7
×6
× (−6)
−7
−42
−7
÷6
42
÷ (−6)
The sign of the quotient of two integers
• The quotient of a positive number and a negative number is a negative number. For example:
ES
28 ÷ (−7) = −4
• The quotient of a negative number and a positive number is a negative number. For example:
−28 ÷ 7 = −4
• The quotient of two negative numbers is a positive number. For example:
G
−28 ÷ (−7) = 4
Example 12
PA
Notice that the rules for the sign of a quotient are the same as the rules for the sign of a product.
Evaluate each of these divisions.
b −20 ÷ (−4)
N
AL
a −45 ÷ 9
c 63 ÷ (−9)
Solution
a −45 ÷ 9 = −5
b −20 ÷ (−4) = 5
c 63 ÷ (−9) = −7
FI
As before, we use another way of writing division. For example, −16 ÷ 2 can be written as
−16
.
2
Example 13
Evaluate:
−45
a
9
b
−36
−4
c
60
−12
b
−36
=9
−4
c
60
= −5
−12
Solution
a
−45
= −5
9
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1 1 E D i v i s i o n i n v o lv i n g ne g at i v e i nte g e r s
Exercise 11E
1 Calculate each division.
b −26 ÷ 2
c −35 ÷ 7
d −21 ÷ 3
e −120 ÷ 3
f 15 ÷ (−3)
g 36 ÷ (−2)
h 45 ÷ (−5)
i 21 ÷ (−7)
j 456 ÷ (−1)
k −51 ÷ (−3)
l −72 ÷ (−12)
n −121 ÷ 11
o −64 ÷ (−4)
p −144 ÷ (−6)
r −500 ÷ (−10)
s −162 ÷ 6
t −396 ÷ 11
m −100 ÷ (−50)
q −39 ÷ (−13)
2 Evaluate:
a
5
−1
b
−5
−1
c
6
−2
e
−1
−1
f
1
−1
g
−50
−1
i
−10
2
j
12
−3
G
Example 13
ES
a −15 ÷ 3
k
−9
−3
PA
Example 12
3 Calculate each division.
d
8
−4
h
−2
1
l
−6
6
−48
−12
b
−52
13
c
−60
12
d
−112
−8
e
132
−4
f
−600
5
g
−225
15
h
292
−4
i
−80
10
j
696
−24
k
−196
14
l
−1000
−100
−144
6
n
256
−4
o
−98
−7
p
−288
−16
m
N
AL
a
4 Copy and complete these divisions.
b −45 ÷ … = 9
c −312 ÷ … = −3
d …. ÷ (−13) = −13
e …. ÷ (−20) = 20
f …. ÷ (−25) = 0
g −60 ÷ …. = −5
h …. ÷ 15 = −30
i 121 ÷ … = −11
a 45 × (−10) ÷ 3
b −34 × (−3) × (−2)
c 6 × (−10) ÷ 5
d −10 × 20 ÷ (−5)
e −5 × 12 ÷ (−6)
f 16 ÷ (−8) × (−25)
FI
a 50 ÷ … = −10
5 Evaluate:
6 Put multiplication (×) or division (÷) signs in each box to make each statement true.
a 8 □ 4 □ (−2)
314
□ 3 = −12
b 3 □ 4 □ (−5) □ 6 = −10
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11 F
Indices and order of
operations
You need to be particularly careful with the order of operations when working with negative integers.
For example, −42 = −16 and (−4) 2 = 16. In the first case, 4 is first squared and then the opposite is
taken. In the second case, −4 is squared.
Notice how different the two answers are.
Remember that multiplication is done before addition unless there are brackets. For example:
Order of operations
• Evaluate expressions inside brackets first.
G
ES
(−12 + 2) × 5) = −10 × 5
−12 + 2 × 5 = −12 + 10
and
= −50
= −2
The same general conventions that we have previously stated for whole numbers also apply when
dealing with negative integers.
PA
• In the absence of brackets, carry out operations in the following order:
–– powers
–– multiplication and division from left to right
N
AL
–– addition and subtraction from left to right.
Example 14
Evaluate:
a (−6) 2
e −20 ÷ 2 × 10
b −62
f 2 × (−4) ÷ 8
c −6 − 5 + 4
g −6 + 3 × (−2)
d 6 × (−2) + 8
h 11 − (−3) × (−5)
FI
Solution
a (−6) 2 = −6 × (−6)
= 36
b −62 = −(6 × 6)
= −36
c −6 − 5 + 4 = −11 + 4
= −7
d 6 × (−2) + 8 = −12 + 8
= −4
e −20 ÷ 2 × 10 = −10 × 10
= −100
f 2 × (−4)
g −6 + 3 × (−2) = −6 + (−6)
= −12
h 11 − (−3) × (−5) = 11 − 15
= −4
÷
8 = −8 ÷ 8
= −1
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315
1 1 F In d i c es an d o r d e r o f o p e r at i o ns
Example 15
Evaluate:
a 3 × (−6 + 8)
c −3 + 6 × (7 − 12) 2
b 6 − (5 + 4)
d 3 × (−6) + 3 × 8
b 6 − (5 + 4) = 6 − 9
= −3
c −3 + 6 × (7 − 12) 2 = −3 + 6 × (−5) 2
= −3 + 6 × 25
= −3 + 150
= 147
d 3 × (−6) + 3 × 8 = −18 + 24
=6
Example 16
Evaluate:
Solution
b −7 + 36 ÷ (−2) 2 + 4
PA
a 4 × (−6) ÷ 2 + 3
ES
a 3 × (−6 + 8) = 3 × 2
=6
G
Solution
N
AL
a 4 × (−6) ÷ 2 + 3 = −24 ÷ 2 + 3
= −12 + 3
= −9
b −7 + 36 ÷ (−2) 2 + 4 = −7 + (36 ÷ 4) + 4
= −7 + 9 + 4
=6
Exercise 11F
1 Evaluate:
a −6 + 20 − 15
b −4 − (−10) + 20
c −6 + 12 − 15
d −4 + 11 − (−15)
e −15 + 7 − 8
f 65 − (−34) + 50
g −12 + 20 − 50
h −50 − 23 − 47
i −20 − (−25) + 60
FI
Example 14
2 Evaluate:
316
a −3 × (−16) + 8
b −4 + 6 × 11 − 14
c −6 − 18 × 4
d −6 × (−3) + 12
e −2 × (−6 + 16) − 25
f −15 + 5 × (−3) + 12
g −11 + 5 × 12 + (−15)
h −18 − 4 × 26 − (−12)
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Example 15
1 1 F In d i c es an d o r d e r o f o p e r at i o ns
3 Evaluate:
a −(3 − 17)
b −(27 − 54)
c 12 + (4 − 16)
d −43 + (6 − 11)
e 15 − 21 + 4 × (−3)
f −3 × (56 − 87)
g −14 × (2 − 11)
h 5 × (13 − 41)
i −7 × (11 − 18)
j (34 + 34) − (−5) × (−120) k (50 + 70) × (−3) − 5 × (−2)
4 Evaluate:
b −(11) 2
c 2 × (−4) 2
d −9 × (−3) 2
e (−10) 2 × − (3) 2
f (−12) 2
g (−5) 3
h (−2) 4
i (−2) 5
j (−2) 6
k (−1) 3
l (−1) 4
a 2 × (−2) 6
b 3 × (−2) 5
c 4 × (−4) 3
d 5 × (−2) 2
e 3 × (−4) 2
f 2 × (−1) 5
g 4 × (−3) 3
h 7 × (−1) 23
5 Evaluate:
G
6 Evaluate:
a −3 × (−16 + 8)
b −4 + 6 × (11 − 12)
c −6 − (15 + 4)
d −6 × (−2 + 12)
f −89 + 5 × (−32 + 12)
PA
e −2 × (−6 + 16) − 20
g −71 + 5 × (51 + (−35))
7 Evaluate:
a 40 ÷ (−5) ÷ 8
h −18 − 4 × (26 − (−12))
b 80 × (−3) ÷ 10
d 60 × (−5) ÷ 25
N
AL
c 50 ÷ 10 × 2
ES
a (−8) 2
8 Evaluate:
b (−10) 2 × (−10) 3
c 2 × (−10) 3 + 102
d −2 × (−10) 2 × (−10)
9 Evaluate:
a 3 × (−12) ÷ 4 + 1
b −5 + 49 ÷ (−7) 2 + 2
c −4 × 6 ÷ 8 − 5
d 3 − 50 ÷ (3 − 8) 2 − 2
e 14 − 3 × 6 ÷ (−2)
f 7 − 32 × (1 − 3) 2
g 5 × (−14) ÷ (−7) − 3
h 16 + 12 ÷ (−2) 2 − 4
FI
Example 16
a (−10) 2 + 2 × (−10)
10 A shop manager buys 200 shirts at $16 each and sells them for a total of $3000. Calculate
the total purchase price, and subtract this from the total amount gained from the sale. What
does this number represent?
11 A man puts $1000 into a bank account every month for 12 months. Initially, he had $3000
in the bank.
a How much does he have in the account at the end of 12 months, given that he has not
withdrawn any money?
b At the end of the 12 months, he writes a cheque for $20 000. How much does he now
have left?
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317
12 A pizza shop runs a delivery van at a cost of $200 a day to deliver pizzas from the shop to
its customers. Each pizza costs $3 to make and sells for $9.
a If the pizza shop delivers 90 pizzas in a day, how much money does it make?
b The price of a pizza increases to $10 and the cost of making a pizza is unchanged. How
much money does the pizza shop make if 90 pizzas are delivered?
c If the price of a pizza decreases to $8 and the cost of making it increases to $4, how much
does the pizza shop make or lose if it delivers 45 pizzas in a day?
13 The local charity is planning to run a fair, and is trying to decide how much to charge for
entry. The hall where it plans to hold it will cost $500 to rent for the day. It plans to charge
$5 per person for entry, and to give each person a show bag that costs $2 to produce.
ES
a If 120 people come to the fair, how much money will the charity make?
b I f the charity decides instead to charge $8 per person, and 120 people attend, how much
will it make or lose?
PA
G
c If it charges $5 per child and $8 per adult, and 60 children and 60 adults attend, how
much will it make or lose?
Review exercise
N
AL
1 Complete each addition.
a 25 + (−2)
b −36 + 22
c −35 + 50
d −51 + (−44)
e −32 + 16
f −45 + (−23)
g −160 + (−20)
h −50 + (−10)
i 110 + (−40)
j −120 + 40
k 35 + (−3)
l −72 + 22
n −91 + (−44)
o −65 + 59
p −60 + (−25)
q −165 + (−25)
r −55 + (−10)
s 115 + (−45)
t −125 + 43
u −332 + (−215)
FI
m −75 + 50
2 In an indoor cricket match, a team has made 25 runs and lost 7 wickets. What is the team’s
score? (A run adds 1 and a wicket subtracts 5.)
3 The temperature in June at a base in Antarctica varied from a minimum of −60°C to a
maximum of −35°C. What was the value of:
a maximum temperature − minimum temperature?
b minimum temperature − maximum temperature?
318
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Review exercise
4 The temperature in Canberra had gone down to −3°C. The temperature in a heated house
was a cosy 22°C. What was the value of:
a (inside temperature) − (outside temperature)?
b (outside temperature) − (inside temperature)?
5 Complete each multiplication.
b −36 × 11
c −35 × 50
d −51 × (−40)
e −3 × 16
f −50 × (−23)
g −160 × (−20)
h −50 × (−10)
i 11 × (−40)
j −120 × 20
k −20 × (−5)
l −25 ×(−4)
6 Complete each division.
ES
a 125 × (−2)
a 125 ÷ (−5)
b −36 ÷ 9
d −51 ÷ (−3)
e −16 ÷ (−4)
g −160 ÷ (−20)
h −1500 ÷ (−10)
j −120 ÷ 20
k −196 ÷ (−14)
l 625 ÷ (−25)
b 7 × (11 − 20)
c −3 × (5 + 15)
e −12 × (−6 + 20)
f −(−4) 2
h (10 − 3) × (−3 + 10)
i (−5 − 10) × (10 − 4)
d −6 × (−4 − 6)
g (3 − 7) × (11 − 15)
G
f −50 ÷ (−10)
PA
7 Evaluate each expression.
a −4 × (6 − 7)
c −35 ÷ 5
i 110 ÷ (−40)
N
AL
8 Start with the number −5, add 11 and then subtract 20. Multiply the result by 4. What is the
final result?
9 Start with −100, subtract 200 and then add −300. Divide the result by 100. What is the final
result?
10 Evaluate:
b −82
c −11 − 15 + 14
d 16 × (−2) + 10
e −3 × (−8) + 100 − 150
f −200 ÷ 2 × 10
g 2 × (−6) ÷ 8
h 4 × (−6) ÷ (−3)
FI
a (−8) 2
11 Evaluate:
a 5 × (−7 + 18)
b −13 + 16 × (7 − 12)
c 16 − (15 + 14)
d 16 × (−12 + 8)
e −3 × (−16 + 20) − 25
f 3 × (−8) + 3 × 18
g 5 ×
h 7 × (−3) 2 + 3 × (−4)
(−72)
+ 3 × 42
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319
Challenge exercise
1 What is the least product you could obtain by multiplying any two of the following
numbers:
−8, −6, −1, 1 and 4?
2 Evaluate:
b (−1) 1001
ES
a (−1) 1000
3 The integers on the edges of each triangle below are given by the sum of integers which
are to be placed in the circles. Find the numbers in the circles.
a
−6
−1
PA
−1
G
b
−9
−6
d
N
AL
c
−7
0
−18
−18
−4
−34
−10
FI
4 Put the three numbers 4, −2 and −7 into the boxes below
□+□−□=□
so that the answer is:
a −1
b 9
c −13
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c h a l l en g e e x e r c i se
5 Put the three numbers 5, −5 and −4 into the boxes below
□+□−□=□
so that the answer is:
a 6
b −14
c 4
6 Find the number that must be placed in the box to make the following statement true.
3−
□ + (−5) = 0
ES
7 Place brackets in each statement below to make the statement true.
a 5 + (−3) × 3 + 4 = 14
b 5 + (−3) × 3 + 4 × 2 = 4
G
c 5 − 5 × 6 + 7 × 6 − 5 = 37
2
−5
0
N
AL
−4
PA
8 This is a magic square. All rows, columns and diagonals have the same sum. Complete
the magic square.
9 a Find the value of 2 − 4 + 6 − 8 + 10 − 12 by:
i working from left to right
ii pairing the numbers ((2 − 4) + (6 − 8) + (10 − 12))
b Evaluate 2 − 4 + 6 − 8 + 10 − 12 + 14 − 16 + … + 98 − 100.
FI
10 Evaluate 100 + 99 − 98 − 97 + … + 4 + 3 − 2 − 1.
11 The average of five numbers was 2. If the smallest number is deleted, the average is 4.
What is the smallest number?
12 Find the value of:
a (1 − 3) + (5 − 7) + (9 − 11) + (13 − 15) + (17 − 19)
b 1 − 3 + 5 − 7 + 9 − 11 + … + 101 − 103
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