“Minus minus makes a plus.” You have all heard a version of this at some point in your life. Your teacher was trying to explain integers to you and this phrase was supposed to help you understand. The phrase stems from this problem:

4 – (-3) =

This question is often difficult for students. If the question was simply 4 – 3, then the answer would be 1. However, the extra negative sign confounds many students. A quick teacher response is to rattle off the phrase minus minus makes a plus. Then the student can transform the problem and solve it with ease:

4 – (-3)

4 + 3 = 7

Yesterday, I was trying to teach this concept to my wife and she kept asking why. She understood how to do the questions but she wanted a deeper explanation as to why the two minuses made a plus. Unfortunately, even though I have thought long and hard about integers, I have been unable to explain why the trick works. That is, until yesterday.

After going back and forth with her for a while, I suddenly had an idea! I will illustrate it in picture form. 4 can be represented as 4 individual +1s.

How can we subtract -3 from the above picture? There are no -1s to take away. Here was my epiphany. We need to create an environment in which this subtraction can take place. Consider the new picture below:

The picture still has the original 4 in the middle. However, there are now a bunch of +1 -1 pairs all around the 4. Since +1 – 1 = 0, these pairs do not change the answer and the picture still represents 4. Now the beauty of this approach.

We need to calculate 4 – (-3). That means we need to subtract -3. Hence, we need to subtract 3 -1s. Like so:

We have subtracted (or removed) 3 -1s. Count the total. The answer is clearly +7. This technique and be used with positive or negative numbers using addition or subtraction.

Here is another example:

-5 – (-7)

First the initial setup:

Then the subtraction:

The result is +2. I hope the above explanation gives you a deeper understanding of why subtracting a negative is equivalent to adding a positive. I know it did for me.

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Interesting. I was wrestling with the why of this problem recently, and I came to a different solution by thinking of – as a direction and then using vectors and a number line. When you add 3 you go right 3 units on the line. When you subtract 3 you flip the arrow around, causing it to go left. When you subtract (-3) you flip the arrow around again and go to the right, and thus achieve the same result as when you added. Every – you add to the equation flips the arrow.

Since writing this post, I have come across a similar analogy. The sign indicated which direction you are facing. So -(-3) means face backwards, and take 3 steps backwards, which is +3. The direction analogy seems particularly useful 🙂