Why do we get a positive number when we multiply two negative numbers?

Other than the question about how to use PEMDAS, the next most asked question that students have is the question “Why do we get a positive number when we multiply two negative numbers?” Why do they ask this question? Because they are usually only taught the operations in the following image without an explanation.

*Multiplication of positive and negative numbers*

What is missing from their instruction is why the multiplication of two negative numbers results in a positive number. “There are different possible answers to this question, depending on the standard of proof one needs and the background knowledge one brings to the question.” [1]

The following are some explanations and hopefully one will spark an Aha! moment [2] with you.

Khan Academy

When we multiply or divide two negative numbers, the result is a positive number. This might seem strange at first, but it’s important to remember that a negative sign in math is really just an instruction to change the direction of a number on a number line. So when we multiply or divide two negative numbers, we’re reversing the direction twice, which brings us back to a positive number. [3]

The Reflective Educator

Imagine we represent multiplication as jumps on a number line.

For 3 × 3, we draw 3 groups of 3 moving to the right. Both the number of groups and the direction of each group are to the right.

But what about 3 × -3? Now we have 3 groups of the number still, but the number is negative.

If we find -3 × 3, the size and direction of the number we multiply are the same, but now we are finding -3 groups of that number. One way to think of this is to think of taking 3 groups of the number away. Another is to think of -3 times a number as being a reflection of 3 times the same number.

So -3 × -3 is, therefore, a reflection of 3 × -3 across the number line.

In one sense though, this visual argument is just mathematical consistency represented using a number line. If multiplication by a negative is a reflection across 0 on the number line, and we think of negative numbers as being reflections across 0 of the number line, then multiplication of a negative number times a negative number is a double-reflection. [1]

Simplified Explanation

The following is a simplified explanation of the more rigorous proofs of the multiplication of two negative numbers that can be found in the Additional Reading and Videos below.

We know that

3 * 6 = 6 + 6 + 6 = 18,

and

3 * -6 = -6 + -6 + -6 = -18,

but what does -3 * -6 equal?

Let’s use the fact that the negative of a number is a reflection on the number line, i.e., “To locate the opposite (or negative) of any whole number, first locate the whole number on the number line. The opposite is the reflection of the whole number through the origin (zero)”. [4]

For example, to find its opposite, reflect the number 5 through the origin. This will be the location of the opposite (negative) of the whole number 5, which we indicate by the symbol −5.

Let’s rewrite our expression -3 * -6 as follows.

(-1 * 3) * -6 = (-1)(3 * -6) = (-1)(-18)

The above final expression can be stated as find the reflection (i.e., -1) of the number (-18) through the origin. Therefore, the reflection of -18 is 18 just as we demonstrated above using the number 5.

References

[1] ⭐ “Why Is a Negative Times a Negative Positive?” 2023. The Reflective Educator. Accessed December 23. https://davidwees.com/content/why-is-a-negative-times-a-negative-positive/.

Given that the goal of an argument that something is true is to leave the other person convinced of the truth of the argument, whenever anyone uses any justification, representation, or proof, it behooves one to check that one’s audience is left convinced.

[2] The eureka effect (also known as the Aha! moment or eureka moment) refers to the common human experience of suddenly understanding a previously incomprehensible problem or concept. Some research describes the Aha! effect (also known as insight or epiphany) as a memory advantage.

[3] ⭐ “Negative Numbers: Multiplication and Division FAQ (Article).” 2023. Khan Academy. Khan Academy. Accessed December 23. https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-negative-numbers-multiply-and-divide/x6b17ba59:multiply-with-negatives/a/negative-numbers-multiplication-and-division-faq.

[4] “1.5: Positive and Negative Numbers.” 2023. Mathematics LibreTexts. Libretexts. September 1. https://math.libretexts.org/Courses/Western_Technical_College/PrePALS_Math_with_Business_Apps/01%3A_Whole_Numbers/1.05%3A_Positive_and_Negative_Numbers.

Additional Reading

Hemanth. 2022. “Why Is Negative Times Negative Really Positive?” Medium. Street Science. December 22. https://medium.com/street-science/why-is-negative-times-negative-really-positive-89f773e0b2a.

Müller, Kasper. 2024. “The Best Explanation of Why Negative Times Negative Is Positive.” Medium. Cantor’s Paradise. March 8. https://www.cantorsparadise.com/the-best-explanation-of-why-negative-times-negative-is-positive-dc42c73ad352.

In this article, I will give you two intuitive reasons behind this that don’t require any mathematical training whatsoever: a real-world example and a geometrically flavored explanation. At the end of the article, we will also see a more numeric/arithmetic approach that is still easy to understand.

“Multiplying Positive and Negative Numbers: 3 Simple Rules.” 2023. K5 Learning. Accessed December 23. https://www.k5learning.com/blog/multiplying-positive-and-negative-numbers-3-simple-rules.

If you look at it on the number line, walking backwards while facing in the negative direction, you move in the positive direction.

Penny. 2023. “Negative Times Negative Is Positive.” Math Central. Accessed December 23. https://mathcentral.uregina.ca/QQ/database/QQ.09.01/mary1.html.

Peterson, Dave. 2022. “Dave Peterson.” The Math Doctors. February 10. https://www.themathdoctors.org/negative-x-negative-positive-abstract-proofs/.

“Why Is a Negative Times a Negative a Positive?” 2023. Negative Times a Negative. Accessed December 23. http://academic.sun.ac.za/mathed/174/MinMin.htm.

“Why Is a Negative Times a Negative Always Positive?” 2023. Illustrative Mathematics. Accessed December 23. https://tasks.illustrativemathematics.org/content-standards/tasks/1667.