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, ...

Negative numbers are useful in all kinds of scenarios. A common example is money – if you owe money to a person or a bank, that debt is often considered to be negative money, i.e. if Percy owes Ruby £20, then Percy has £-20.

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.

Example: Using a number line, find a) 5+3, b) 8-4.

a) Firstly, locate 3 on a number line. Then, since we are adding 5, hop over 5 spaces to the right.

We can see that we end up at 8, therefore we know that 5+3=8.

b) Locate 8 on a number line. Then, since we’re subtracting 4, hop over 4 spaces to the left.

We can see that we end up at 4, so we know that 8-4=4.

Sometimes a subtraction will result in counting below zero, but the process is exactly the same. The difference is that the outcome will be a negative number.

Example: Using a number line, find the result of 5-7.

As before, we’re going to find 5 on a number line and move 7 spaces to the left. This looks like:

We end up at -2, so our answer is negative. This happens when we subtract a bigger number from a smaller one. So, we can conclude that the answer is 5-7=-2.

We’ve seen how we can get negatives as an answer, but we need some rules to work with them.

Calculating with Negative Numbers

The rules we need to know for calculations involving positives and negatives are as follows.

• Subtracting a positive number is subtraction. In practice, this means that

7-(+3)=7-3=4• Adding a negative number is subtraction. In practice, this means that

14+(-5)=14-5=9• Subtracting a negative number is addition. In practice, this means that

20-(-10)=20+10=30The rules can be summarised as follows:

• Two different signs, +- or -+, results in a subtraction.

• Two of the same signs, ++ or --, results in an addition.

Example: Using a number line, find a) 13 + (-5), b) 1-(-4).

a) These two signs are different, so 13+(-5)=13-5. So, starting and 13 on the number line and going 5 spaces left, we get

So, the result is 13+(-5)=8.

b) These two signs are the same, so 1-(-4)=1+4. Starting at 1 on the number line and going 4 spaces right, we get

So, the result is 1-(-4)=5.

Multiplying & Dividing

There are similar rules for multiplying and dividing with negative numbers. They are

• If you multiply/divide two numbers with the same sign, + and + or - and -, the result is positive.

• If you multiply/divide two numbers with different signs, + and - or - and +, the result is negative.

Example: Calculate: a) -3\times (-12), b) 28 \div (-7)

a) Both values have the same sign, so the result should be positive. So, given that we know

3 \times 12 = 36,

The answer must just be 36.

b) Both values have different signs, so the result should be negative. So, given that we know

28\div 7=4The answer must be -4.

### Example Questions

1) Evaluate

a) 3-(+7)

b) 7-(-2)

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

Note: you are encouraged to use number lines to help, but if you feel comfortable without them then give it a go.

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.

2) Evaluate

a) -5\times 7

b) -18 \div 3

c) -4 \times (-10)

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\times 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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