Why Is Negative Times Negative Really Positive?

Negative times negative is positive. Every school-going child learns this in grade school. It is one of the most fundamental pillars of mathematics on top of which more advanced concepts are built. One does not have to be a math expert to know it. But as soon as you ask the magic question, the ground starts shaking and insecurity starts leaking in.

What is this magic question, you ask? Well, it is as follows:

Why? Why is ‘negative times negative’ really positive?

In this essay, we will be answering this very question by covering certain fundamental mathematical elements in the form of incremental puzzles. In doing so, this essay will also help you strengthen and clarify (if not done already) some of the most fundamental mathematical operations. Let us begin.

How is the Additive Identity Related to Addition and Subtraction?

Addition is arguably the most fundamental mathematical operation. It is also the most intuitive. A child that has 2 candies in one box and 3 candies in another box starts counting them with her fingers as she shuffles them together into a third box. This counting involves adding increments of 1 each time to the previous ‘sum’.

The same child starts counting in reverse as she tries to split the candies between herself and her friend. This reversed addition is what she would later learn as subtraction. As children start learning more about addition and subtraction, they learn to do these operations using abstractions called numbers.

When dealing with addition and subtraction using numbers, the following question arises:

What number, when added to or subtracted from a target number, results in the same unchanged target number?

The answer turns out to be the number zero (0) — the additive identity. Now that we have covered the notion of the additive identity, let us move onto a more nuanced concept.

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The Difference Between Subtraction and the Negative

Eventually, children learn that addition and subtraction can also be performed on negative numbers. Here is an example:

3 + (-2) = 3–2 = 1

Here, you can see that a positive number (3) added to a negative number (-2) leads to the subtraction of the absolute value of the negative number (2) from the positive number (3).

Although this example appears trivial initially, there is a subtly tricky point hiding inside. It is the fact that the concept of subtraction is intuitive but the notion of negative numbers is not. If you think about it, “3–2” can be understood as follows:

How many candies does the child have left if she originally had 3 and gives 2 of them away?

But how do you make sense -2? “The child has -2 candies” does not make intuitive sense. Sure, we could say that the child owes 2 candies, but that would be an “assigned meaning”.

Essentially, negative numbers are abstract mathematical concepts that we need to assign meanings to in the context of reality. We cannot (normally) derive the meaning of negative numbers from reality as we can do for positive numbers. With this thought in mind, the key takeaway from this section is as follows:

When a positive number is added to a negative number, this leads to a subtraction operation where the absolute value of the negative number is subtracted from the positive number.

We now have a model for adding positive numbers and negative numbers. Let us go one step further and extend this to subtraction as well.

Subtracting Negative Numbers

When a negative number is subtracted from a positive number, it leads to an addition operation where the absolute value of the negative number is added to the positive number. Here is an example of this in action:

3 — (-2) = 3 + 2 = 5

Similarly, when a negative number is subtracted from another negative number, the subtraction is first resolved into addition, and then, we go back to a situation where a positive number is added with a negative number. Here is an example of this:

-2 — (-2) = -2 + 2 = 0

This example also holds a subtle takeaway:

A negative number (-x) is one that when added to its positive counterpart (x) results in the additive identity (zero).

With this in mind, let us now advance to the land of multiplication.

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Multiplication and the Multiplicative Identity

Multiplication is a more advanced mathematical operation, but it still lies in the domain of intuitive understanding. Multiplication could be understood as repeated addition.

For example, consider: 2 * 3 = 3* 2 = 6. The left-hand part of this equation answers the following question:

What happens when we add two, three times? (2 + 2 + 2)

The middle part of the equation answers the following question:

What happens when we add three, two times? (3 + 3)

As you can see, both questions deliver the same result. This property is known as commutativity. It is a fancy way of saying that the order in which the numbers are processed does not matter in the multiplicative operation. Now, here is an interesting question:

What number, when multiplied with a target number, results in the same unchanged target number?

The answer turns out to be one (1) — the multiplicative identity. Any number multiplied by 1 results in the same number. Now that we have covered the concept of the multiplicative identity, let us move on to the notion of multiplying negative numbers.

Multiplication involving negative numbers

It is quite clear that two positive numbers multiplied with each other result in a positive number as well (for example, 2 * 8 = 16). However, what happens when a positive number is multiplied by a negative number? Consider the following:

2 * (-8) = ?

The left-hand side of this equation could be understood as the following question:

What happens when we add two, (-8) times?

This is another way of asking what happens when -8 is added to itself 2 times (using the commutative property of multiplication). It would resolve as follows:

2 * (-8) = (-8) + (-8) = -16

So, it has now become clear that a negative number multiplied by a positive number results in a negative number:

(+) * (-) = (-)

Keeping this result in mind, we are now ready to tackle the core problem we started with.

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Why is Negative Times Negative Really Positive?

We already know that ‘1’ is the multiplicative identity. Any number multiplied with ‘1’ results in the original number. So, what is ‘-1’? It is the sign-wise opposite of the multiplicative identity, in the sense that it reverses the sign of any non-zero number that it is multiplied by. For example

1. (-1) * (1) = -1

2. (-1) * (25) = -25

3. (-1) * (a) = -a

Using the argument that ‘-1’ is the sign-wise opposite of the multiplicative identity, we could argue that ‘(-1) * (-a) = +a’. But that is not really convincing, is it? So, we are going to take a different approach.

Assume that we live in an era where nobody knows what ‘(-1) * (-1)’ could be. We are mathematicians trying to figure it out. Let us start by expressing the property that a number added to its negative counterpart results in the additive identity (zero) in the following form:

1 + (-1) = 0

Next, let us multiply both sides by ‘-1’ and apply the distributive law of multiplication (a * (b + c) = (a * b) + (a * c)):

Now, the first part on the left-hand side of the expression is something we have already seen. Any number multiplied by ‘-1’ reverses its sign. Therefore, the result is ‘-1’. The second part on the left-hand side is what we are after! The exclamation mark above the ‘equal to’ sign refers to our assertion that we require the expression to equate to zero.

Since we live in an era where we do not know what the value of ‘(-1) * (-1)’ could be, we treat it as an unknown (x). Consequently, we end up with the following expression:

Therefore:

(-1) * (-1) = +1

When we generalize this result, we get:

(-) * (-) = (+)

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Final Remarks

To conclude, here is a summary of how we illustrated that ‘negative times negative’ is positive:

1. We first covered addition, subtraction, and how the additive identity (0) is related to both operations.

2. Then, we established how the negative differs from the subtraction operation.

3. We then covered addition and subtraction using negative numbers. We saw that a negative number is one that when added to its positive counterpart leads to the additive identity (0).

4. Following this, we covered multiplication and the multiplicative identity (1).

5. Then, we established that (+) * (-) = (-).

6. We then started with the equation (1) + (-1) = 0, multiplied both sides by (-1), applied the distributive law, and treated ‘(-1) * (-1)’ as an unknown to arrive at the solution of ‘+1’.

7. We finally generalized the result as (-) * (-) = (+).

The reason why negative times negative is really positive is that our mathematical system requires it to be that way (logical consistency requirement).

When we build upon our knowledge of the fundamental mathematical operations incrementally, this fact reveals itself as it did in our little adventure in this essay.

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Further reading that might interest you: Why Exactly Is Zero Factorial Equal To One? and Why Is A Number Raised To The Power Zero = One?

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