# 5.2.7 Adding and Subtracting Negative Numbers

We use a similar process when adding and subtracting negative numbers. Let’s consider a simple case first: . If you add a number to zero, you end up with the same number. For example, . Similarly, .

On the number line, we start at 0 and move 2 units to the left, which is the same operation as subtracting 2.

Adding a negative number is the same as subtracting the corresponding positive number. So for example:

and .

This is an important rule to remember, so make a note of it in your math notebook now.

Now, what happens when you subtract a negative number? This isn’t easy to visualize, so try thinking of subtraction in a different way. For example, for the calculation , instead of saying “30 take away 20 is …” you can say, “What do I have to add on to 20 to get 30?”

In the same way, can be interpreted as “What do I have to add on to (2) to get 0?” From the number line, you have to move 2 units to the right to arrive at zero, which is the same as adding on 2. Thus, .

Subtracting a negative number is the same as adding the corresponding positive number. For example, and .

Another important rule for your math notebook.

It’s not easy to see why subtracting a negative number is equivalent to adding a positive number in real life, so think of driving a car along a road, going in a forward direction which we’ll call “positive”. If you wanted to go back in the opposite direction, which we will call backward, or “negatively”, you have two choices. You could either reverse, or turn round. Either is kind of equivalent to a negative number. But, if the car did both at the same time (that is, turned themselves around, and then reversed), then they would continue in my original direction — that is, positively. So, subtraction of a negative number is addition of the positive number.

## Activity: Adding and Subtracting Negative Numbers

Work out the following calculations in your math notebook.

(a)

### Discussion

You could use a number line to help visualize each problem. Also, remember that subtracting a negative number is the same as adding the corresponding positive number, and adding a negative number is the same as subtracting the corresponding positive number.

(a)

(b)

(b)

(c)

(c)

(d)