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: zero plus open two minus close. If you add a number to zero, you end up with the same number. For example, zero plus six equals six. Similarly, equation left hand side zero plus open two minus close equals right hand side open two minus close.

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:

equation sequence 30 plus open two minus times zero close equals 30 minus 20 equals 10 and equation sequence five minus plus open negative 10 close equals five minus minus 10 equals one minus times five.

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 30 minus 20, 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, zero minus open two minus close 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, zero minus open two minus close equals two.

This is a number line from –3 to 0 with the scale marked on in units. An arrow starts at 0 on the scale and moves 2 units down to end at –2.

Subtracting a negative number is the same as adding the corresponding positive number. For example, equation sequence 30 minus open two minus times zero close equals 30 plus 20 equals 50 and equation sequence three minus times zero minus open two minus times zero close equals three minus times zero plus 20 equals one minus times zero.

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 symbol Activity: Adding and Subtracting Negative Numbers

Work out the following calculations in your math notebook.

(a) 20 plus open five minus close

Hint symbol

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.

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Answer

(a) equation sequence 20 plus open five minus close equals 20 minus five equals 15

(b) two minus times zero plus open five minus close

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Answer

(b) equation sequence two minus times zero plus open five minus close equals two minus times zero minus five equals two minus times five

(c) 20 minus open five minus close

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Answer

(c) equation sequence 20 minus open five minus close equals 20 plus five equals 25

(d) two minus times zero minus open five minus close

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Answer

(d) equation sequence two minus times zero minus open five minus close equals two minus times zero plus five equals one minus times five

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5.2.6 Working with Negative Numbers

5.2.8 Rules for Negatives