Download Add, subtract, multiply, divide negative numbers

Add, subtract, multiply, divide
negative numbers
Introduction
3
What is a directed number?
4
The idea of opposites
4
What is an integer?
4
Adding and subtracting directed numbers
Double negative
7
8
Multiplying and dividing directed numbers
11
Order of operations with directed numbers
13
Using a calculator
Suggested responses to activities
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
15
18
1
2
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
Introduction
In Topic 1 you learnt that the number system starts with counting numbers
{1, 2, 3, 4, …….} then extends to include zero and negative numbers.
In this topic you will learn how to add, subtract, multiply and divide
negative numbers. You will also apply the rule of ‘order of operations’ to
negative numbers.
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
3
What is a directed number?
A directed number has both size and direction. For example, the number
+ 3 means ‘3 units to the right of zero’, while the number – 4 means ‘4 units
to the left of zero’.
All numbers to the right of zero are called positive numbers, and all
numbers to the left of zero are called negative numbers.
We usually leave the + sign off positive numbers. When you see the number
10, it means + 10.
The idea of opposites
Just as our English language contains pairs of opposite words, (hot-cold, updown, in-out), our Mathematical language uses + and – signs to signal
opposites. So:
+ 23 is the opposite of – 23
– 5.9 is the opposite of + 5.9
Opposite numbers are an equal distance from 0.
What is an integer?
Integers are simply positive or negative whole numbers. The set of integers
is the set {….– 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5, …….}.
A number line can be used to give a picture of the set of integers as shown
in Figure 1.
Figure 1: Integers on a number line
Numbers to the right of zero are greater than numbers to the left of zero. For
example 3 is greater than – 3. This is the same if you think of a temperature
of 3C.
4
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
In general, when comparing 2 integers, the number on the right is bigger
than the number on the left. For example,
– 2 is bigger than – 3.
Write ‘is greater than’ or ‘is less than’ in the space provided to make these
statements true.
1
4______________________2
2
–1______________________+2
3
0______________________–2
4
–1______________________–3
1 4 is greater than 2
2 –1 is less than +2
3 0 is greater than –2
4 –1 is greater than –3
Activity 2A
Write the answers to these problems in your workbook.
1
What is the opposite of?
(a) 5
(b) 7.2
(c) –1
(d)
2
Write down all the integers between:
(a) – 1 and + 3
(b) – 11 and – 5
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
5
You may use this number line to help answer the next two questions.
3
Write ‘is greater than’ or ‘is less than’ to make these statements true.
(a) + 1_________________________– 2
(b) – 5_________________________– 2
(c) – 14________________________+ 12
(d) – 1_________________________– 6
(e)
4
0__________________________8
What number is?
(a) 2 units to the right of + 3?
(b) 5 units to the left of + 1?
(c) 3 units to the left of –2?
(d) 12 units to the right of –3?
(e) 20 units to the left of +2?
(f) 6 units to the left of +5?
(g) 4 units to the right of –8?
(h) 9 units to the right of –9?
Check your answers with the suggested responses at the end of the topic.
6
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
Adding and subtracting directed
numbers
Consider the following examples.
Example 1
If the temperature is 2C and falls by 5C, what is the temperature then?
It is easy to solve this problem by using a number line.
Using a number sentence, we can write:
So the temperature is –3C.
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
7
Example 2
Start 2 floors below ground level and then go up one floor. What floor are
you now on? Once again, a number line helps you to see the solution.
Using a number sentence, we can write
Example 3
Your bank account is overdrawn by $10 and you withdraw another $5. What
is the balance of your account?
Once again, a number line helps you to see the solution.
Using a number sentence, we can write
Directed numbers can be added or subtracted using a number line.

to add, hop to the right

to subtract, hop to the left
Double negative
has two minus, or negative, signs directly next to each other (ie there
is no number separating the two minus signs). When this is the case, the
‘
‘ becomes a ‘+’. That is:
8
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
Example 4
Consider the expression
. Once again this question involves two
minus signs next to each other. The ‘
‘ becomes a ‘+’ so:
Don’t forget you can still use a number line.
(Start from –5, take 3 hops to the right).
This rule might seem a little strange at first, however we also make use of
the double negative in our everyday English language.
For example, the prefix ‘un’ is used to make a ‘positive word’ into a
‘negative word’.
Happy is a positive word. Unhappy is a negative word.
‘Not’ is a negative word used to change a positive expression into a negative
one. What happens when we put two negative words together?
‘Not unhappy’ changes the meaning back to being positive. This is the
double negative.
To subtract a negative number, change the
two minus signs into a plus sign.
Activity 2B
Don’t forget to write your answers in your workbook.
1
Use a number line to help answer these questions:
(a) 15 – 20
(b) – 10 + 25
(c) – 1 – 4
(d) – 7 + 3
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
9
(e) 6 + 20
(f) 0 – 9
(g) – 12 + 12
(h) – 4 – 4
2
Rewrite each of these number sentences as additions, then use a number line to help
answer the questions. The first one has been done for you.
(a)
(b) 4 – – 6
(c) 0 – – 2
(d) – 2 – – 5
(e) – 10 – – 6
3
Calculate the following.
(a) 100 – 200
(b) 16 – – 7
(c) – 4 – 0
(d) – 7 – – 7
(e) 0.1 – 0.4
(f) – 5 – 6
Check your answers with the suggested responses at the end of the topic.
10
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
Multiplying and dividing directed
numbers
There are just a couple of simple rules to remember for multiplication and
division questions:
Multiplying and dividing two numbers with the same sign gives a positive
result.
Multiplying and dividing two numbers with different signs gives a negative
result.
Example 1
Solve: 4 × – 3
Step1: 4 × 3 = 12
Step2: signs are different (4 is positive, 3 is negative) so the answer will
be negative. Therefore the answer is:
4 × – 3 = – 12
Example 2
Solve: – 5 × – 6
Step1: 5 × 6 = 30
Step2: signs are the same (– × – → +). Therefore the answer is:
– 5 × – 6 = 30
Example 3
Solve: – 32 ÷ 8
Step1: 32 ÷ 8 = 4
Step2: signs are different (– ÷ + → –). Therefore the answer is:
– 32 ÷ 8 = – 4
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
11
Example 4
Solve: – 6 ÷ – 4
Step1: 6 ÷ 4= 1.5
Step2: signs are the same (– ÷ – → +). Therefore the answer is:
– 6 ÷ – 4 = 1.5
Activity 2C
Solve the following problems. Don’t forget to write your answers in your workbook.
1
4 × __ 5
2
36 ÷ __ 3
3
– 7 × __ 6
4
–4×3
5
– 50 ÷ 10
6
(– 8)2
7
8
– 1.4 × 100
9
100 ÷ __ 5
10 45 ÷ 3
Check your answers with the suggested responses at the end of the topic.
12
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
Order of operations with directed
numbers
Do you remember learning about ‘order of operations’ in Recognising
Everyday Mathematics? The principles used in order of operations also
apply to directed numbers.
Review: flowchart showing the order of operations in a number sentence.
Example 1
Example 2
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
13
Example 3
Example 4
You do the top line first, because the fraction bar acts in the same way as
brackets.
Note: If you were to do this question on a calculator you would have to put
brackets around the top line.
Activity 2D
Solve the following problems without using a calculator. Don’t forget to write your
answers in your workbook.
1
2
– 10 – 4 – 8
3
(4 – 7) × – 8
4
16 ÷ – 4 × – 5
5
(– 5)3
6
(– 12 × 4) ÷ (– 4 + 10)
7
14
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
8
9
10
Check your answers with the suggested responses at the end of the topic.
Using a calculator
Depending on the model, your scientific calculator has a
key or a
key. This is called the sign change key.
To enter – 6, put in the number 6 then press
If your calculator has
.
as the sign change key, then you have to enter it
first followed by the number 6. Let’s try some calculations now.
Example 1
Find:
Both methods will give the correct answer of – 32.
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
15
Example 2
Find: – 99 – 11.2
Your answer is – 110.2
Example 3
Find:
Use a calculator to find the answer to the following problems. Try to
do them without looking at the answers.
1
– 1.5 + 2.5 – 10
_______________
2
3
16
_______________
(– 12 × 4) ÷ 20
_______________
4
_______________
5
_______________
6
_______________
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
1 –9
2 53
3 – 2.4
4
5 5
6 2
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
17
Suggested responses to activities
Activity 2A
1
(a) – 5
(b) – 7.2
(c) 1
(d)
2
3
4
18
(a) 0, 1, 2
(b)
(a)
(b)
(c)
(d)
– 10, – 9, – 8, – 7, – 6
+2 is greater than – 4
– 5 is less than – 2
– 14 is less than +12
– 1 is greater than – 6
(e)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
0 is less than 8
+5
–4
–5
+9
– 18
–1
–4
0
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
Activity 2B
1
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
–5
+15 (or 15)
–5
–4
26
–9
0
–8
2
(a)
(b)
(c)
(d)
(e)
(a)
(b)
(c)
(d)
(e)
9 + 2 = 11
4 + 6 = 10
0+2=2
– 2 + 5 = + 3 (or 3)
– 10 + 6 = – 4
– 100
16 + 7 = 23
–4
–7+7=0
– 0.3
3
(f) – 11
Activity 2C
1
– 20
2
– 12
3
42
4
– 12
5
–5
6
– 8 x – 8 = 64
7
5
8
– 140
9
– 20
10 15
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
19
Activity 2D
1
1
2
– 22
3
24
4
20
5
– 125
6
–8
7
1
8
–2
9
–3
10 4
20
4930CF: 2 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996