Negative Numbers

We often use positive and negative numbers when telling the temperature. In winter, for example, a temperature -2℃ is 2℃ below zero.

When doing arithmetic, it is very helpful to regard positive numbers as something we have in our possession and the negative numbers as something we owe (a debt) and need to pay back.

A negative number has a minus sign in front of it, as in -2℃. Positive numbers are written with or without the positive sign in front of the number.

In 10+3, we regard the 10 as something we possess and the +3 as something we possess. So in total, we possess 13. As 10+3=13.

In -10 + 3, we regard the -10 as a debt and the +3 as something we have. So, after selling the account, we shall still have 7 as a debt. Therefore, -10+3=-7.

In -10-3, we regard -10 as a debt and the -3 again as a debt. So in total, we have 13 as a debt. Therefore, we should have -10-3=-13.

Another way to calculate 10+3 is by using the number line. Starting from the first number, we move to the right if we are adding and to the left if we are subtracting.

For 10+3, start at 10 on the number line

Since we are adding, we move to the right from 10 by 3 steps. This gives 13.

For (-10)+3, start at -10 and move 3 steps to the right, since we are adding.

For (-10)-3, start at -10 and move 3 steps to the left since we are subtracting.

This take us to -13. Hence, (-10)-3=-13

For -10+11, we start at -10 and we move 11 steps to the right

We are at +1, so -10+11=1.

We should also know some rules:

+ + = +

+ - = -

- + = -

- - = +

With bigger numbers, using the number line becomes impractical. We should then visualise what is happening on the number line. For example, consider -57+123.

On the number line we will start at -57, because we are adding 123. We should move 123 steps to the right of -57. With some efforts we see that this will take us to 123-57=66. Therefore -57+123=66.

To calculate (-57)-123, we start at -57 on the number line and because we are subtracting 123, we should move to the left of -57 by 123 steps. Again, with some effort, we see that this will take us to  -(57+123) on the number line. Now 57+123=180. Therefore (-57)-123=-180.

With practice, we shall become more fluent at these and will not even need to visualise the number line.

When we are multiplying numbers (positive, negative), we should remember these rules:

A positive\timesa positive=a positive

A positive\timesa negative=a positive

A negative\timesa positive=a negative

A negative\timesa negative=a positive










The same rules apply for division









-15\div-3 = 5