Negative Numbers

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Negative Numbers Revision

Negative Numbers

Negative numbers (or minus numbers), denoted by a minus sign (-), are what we use to count below zero. For example, counting backwards from 3 looks like

3,\,\,2,\,\,1,\,\,0,\,\,-1,\,\,-2,\,\,-3,\,\,-4, ...

Skill 1: Using a Number Line

Before we head further into understanding negative numbers, let’s recall the following:

• To add values on a number line, we have to move to the right.

• To subtract values on a number line, we have to move to the left.

 

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Skill 2: Ordering Negative Numbers

Just like positive numbers, negative numbers can be ordered by their size. The larger the number after the minus sign, the smaller the number.

This can be quite confusing, so it helps to use a number line:

Negative numbers number line
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Skill 3: Adding Negative Numbers

When we add a negative number, it is the same as taking away a positive number. For example:

\begin{aligned}5+(-2) &= 5-2 \\&=3\end{aligned}

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Skill 4: Subtracting Negative Numbers

When we subtract a negative number, it is the same as adding a positive number. For example:

\begin{aligned}100-(-25) &= 100+25 \\&=125\end{aligned}

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Skill 5: Multiplying Negative Numbers

When multiplying negative numbers, treat the calculation as if the numbers are positive. Then, there are two rules to remember:

  • When we multiply a positive number by a negative number (and vice-versa), the answer is always negative. For example:

2\times-3 = -6

 

  • When we multiply a negative number by a negative number, the answer is always positive. For example:

-5\times-7=35

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Skill 6: Dividing Negative Numbers

When dividing negative numbers, treat the calculation as if the numbers are positive. Then, there are two rules to remember:

  • When we divide a positive number by a negative number (and vice-versa), the answer is always negative. For example:

12 \div -4 = -3

 

  • When we divide a negative number by a negative number, the answer is always positive. For example:

-48 \div -6=8

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Example 1: Using a Number Line

Use a number line to work out the following:

a) 5+3

b) 8-4

[2 marks]

 

a) Locate the number 5 on a number line, then move 3 values to the right:

Addition number line

Answer: 5+3=8

 

b) Locate the number 8 on a number line, then move 4 values to the left:

Subtraction number line

Answer: 8-4=4

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Example 2: Ordering Negative Numbers

Put the following numbers in decreasing order of size:

-35, -8, -28, -40, -2, -17

[2 marks]

First, draw a suitable number line. We need our number line to include each of the numbers above:

 

Next, we have to add the numbers into their correct positions:

 

We have to put the numbers in decreasing order of size, which means we need to list them from right to left.

Answer:

-2, -8, -17, -28, -35, -40

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Example 3: Adding and Subtracting Negative Numbers

Calculate the following:

a) 55+(-33)

b) -105 - (-28)

[3 marks]

 

a) We can use a number line to make this calculation easier – remember that adding a negative number is the same as subtracting a positive number.  Our calculation can be rewritten as:

55-33

On the number line, starting from 55 we will move 33 values to the left:

 

Number line subtraction example

Answer: 55+(-33) = 55-33 = 22

 

b) Remember that subtracting a negative number is the same as adding a positive number. Our calculation can be rewritten as:

-105 + 28

On the number line, starting from -105 we will move 28 values to the right:

 

number line addition example

Answer: -105 + 28 = -77

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Example 4: Multiplying and Dividing Negative Numbers

Calculate the following:

a) 24 \times - 4

b) 90 \div -10

[3 marks]

 

a) Perform the calculation as if the numbers were positive: 24\times4 = 96

Then remember that multiplying a positive number by a negative number gives a negative, so the answer is:

24\times-4=-96

 

b) Perform the calculation as if the numbers were positive: 90\div10 = 9

Then remember that dividing a positive number by a negative number gives a negative, so the answer is:

90 \div -10 = -9

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Notes:

You are encouraged to use a number line to perform calculations with negative numbers, but you do not have to use them if you are comfortable without.

Negative Numbers Example Questions

a) These are two different signs so it must be a subtraction:

 

3-(+7)=3-7

 

Finding 3 on a number line and moving 7 to the left, we get

 

 

So, the answer is 3-(+7)=-4

 

 

b) These are two of the same signs so it must be an addition:

 

7-(-2)=7+2

 

Finding 7 on a number line and moving 2 to the right, we get

 

 

So, the answer is 7-(-2)=9

 

 

c) In this calculation we have two of the same signs so it must be an addition.

Note: the minus sign in front of the 1 does not change. It has no bearing on whether the calculation becomes an addition or a subtraction. So, the calculation is

 

-1-(-6)=-1+6

 

Finding -1 on a number line and moving 6 spaces to the right, we get

 

 

So, the answer is -1-(-6)=5

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a) We are multiplying one negative and one positive, so the answer must be negative. Given that

 

5\times 7=35

 

We get that

 

-5\times 7=-35

 

 

b) We are dividing one negative and one positive, so the answer must be negative. Given that

 

18\div 3=6

 

We get that

 

-18\div 3=-6

 

 

c) We are multiplying two negatives, so the answer must be positive. Given that

 

4\times 10=40

 

We get that

 

-4\times (-10)=40

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Draw a suitable number line:

 

 

Next, mark the numbers on the number line:

 

 

Finally, put the numbers in increasing order of size by listing them from left to right.

Answer:

-20, -13, -2, 0, 8, 15

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