# Negative Numbers

## 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.

**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:

**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}

**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}

**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

**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

**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:

Answer: 5+3=8

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

Answer: 8-4=4

**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

**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:

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:

Answer: -105 + 28 = -77

**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

**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

**Question 1:** Evaluate the following:

a) 3-(+7)

b) 7-(-2)

c) -1-(-6) (tricky!)

**[3 marks]**

**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

**Question 2:** Evaluate the following:

a) -5\times 7

b) -18 \div 3

c) -4 \times (-10)

**[3 marks]**

**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

**Question 3: **Put the following numbers in increasing order of size:

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

**[2 marks]**

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