Why do two negatives make a positive?

Let us first consider two examples.

How did you get to your answer? How would you describe your method to another person?

Let’s take a look:

Solution 1

As there is a positive sign and a negative sign next to each other, this makes a negative. So, we can replace the two signs (+ -) with a single sign (-).

Solution 2

A negative and a negative gives a positive. Hence, the answer is a positive number.

Was your thought process similar to mine?

Why did we arrive to these conclusions and why are these assumptions true?

In school, your teacher may have told you these three mathematical statements:

A positive and a positive gives a positive.

A negative and a negative gives a positive.

A negative and a positive gives a negative.

Well, I found myself teaching this word for word and as the poor darlings ate my rant up I found myself pausing and asking myself why this was true.

Let’s take a look at our trusty number line.

number line

As we approach the right, the numbers are getting larger and we consider this the positive direction. Similarly, the left arrow is considered to be the negative direction. Let’s consider a person on a number line. Let’s name him Peter.

Let’s take a look at Example 1: -1 + (-2) = -3

eg1

Peter will start from the position -1. Here comes the tricky bit- the positive and the negative sign next to each other. Peter is facing the positive direction on the number line (This is given by the first sign). Peter will move in the opposite direction to which he is facing (This is given by the second sign). This means Peter will move in the negative direction two positions The digit two tells us how many positions Peter moves. Another way to think about it is that Peter takes two steps backwards. Hence, Peter finishes at -3.

In the diagram, Peter finishes facing the negative direction, however, this is not important. Really, there is no direction at the end. Peter simply ends up at position -3.

In summary, when two signs of addition or subtraction are next to each other, the first sign indicates which direction an object is facing and the second sign indicates the movement relative to the direction in which the object is facing.

Let’s take a look at Example 2: (-2) * (-4) = 8

eg2

In multiplication and division, the rules are different. Peter starts at 0 this time. He is facing the negative direction. Peter then moves two positions in the opposite direction. In other words, he takes two steps backwards. This means Peter will move in the positive direction He repeats this four times. Hence, Peter finishes at 8.

Behind these flawless mathematical truths are the explanations that are just as elegant. Mathematics is merely another language to describe phenomena in the natural world. It may be worthwhile teaching students mathematics using language and analogy as the primary focus, similar to what we have just done.

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