## Why is the Product of Two Negative Numbers a Positive Number?

When I ask most adults or high school students what is -5 times -10, I usually get the correct response of positive 50 or just 50. When I ask them if they can give me simple explanation or an example to show why the product or two negative numbers is a positive number, they can’t.  Students seem to remember the rule “two negatives make a positive”, but some forget that this rule applies to the product or quotient of two real numbers, but not to the sum or difference of two real numbers. I believe it’s fair to say that many lower level math textbooks and math courses treat this rule as just a simple mathematical fact of life, and rarely if ever give an intuitive explanation of why this rule is true.

The purpose of this post is to give examples of intuitive explanations as to why the product of two negative real numbers is a positive real number. It’s important that mathematics makes sense to students, and math should not be just a bunch of somewhat arbitrary rules that can be used to get the right answer. I have used and still use many of these examples when I explain why the product of two negative numbers is positive number. Some of these examples are probably familiar to many readers of this post. The first three examples are my favorites, and I use them exclusively. In all of the examples, it is assumed that students understand multiplication is repeated addition, and students have a basic understanding of the addition rules for positive and negative numbers. Of course, this is a big assumption.

Before I show you my examples, I will give you a brief description of the state of math education on the 1960’s and 1970’s in the United States. The Russian launch of the satellite Sputnik on October 4, 1957 caused a national panic which led to in complete revision of math and science curriculums. The revised math curriculum was called “new math” which was based on the advice of university research mathematicians and other professional mathematicians; not on the advice of wise and experienced math educators. New math placed emphasis on set theory, the fundamental properties of real numbers, functions, relations and the symbols of modern abstract mathematics. The new math approached mathematics from a more rigorous and abstract point of view as opposed to an intuitive and practical point of view. Students and parents found the new math strange and mystifying. Like most new waves in education, the new math was eventually replaced by another new wave. I clearly remember one nationally renowned math educator at a NCTM national convention in the 1980’s state (almost word for word), “Our research shows that a more rigorous abstract approach of teaching math only works with very bright students.” Of course, every experienced math teacher in the audience already knew this. The new math failed because you can’t make an abstraction (identify common core properties of different objects/systems) if you don’t have a base of knowledge and experience from which to make an abstraction. Since elementary, middle school, high school students, and normal adults don’t have this crucial base of knowledge and experience, it should be no surprise that the new math failed. By the early 1980’s, math education started to move in a new direction.

My first example is taken from the book Why Johnny Can’t Add: the Failure of the New Math written by Morris Kline (1908-1992) who was a scientist and professor of mathematics at NYU. One of Professor Kline’s core beliefs is that math concepts should be explained by using concrete examples that students can relate to. The example below is similar to the example Kline used to explain the product rule for positive and negative numbers.

Suppose homeowner Bob hires neighbor boy Bill to do general yard work at \$10.00/hour. We have four situations to consider; two from Bob’s point of view and two from Bill’s point of view. Bill is gaining \$10 every hour he works and Bob is losing \$10 every hour every hour Bill works.

• 4 hours in the future, Bill will be \$40 richer. (+4 * +10 = +40)
• 4 hours ago, Bill was \$40 poorer. (-4 * (+10) = -40)
• 4 hours in the future, Bob will be \$40 poorer. (+4 * (-10) = -40)
• 4 hours ago, Bob was \$40 richer. (-4 * (-10) = +40)

The next example involves filming a person walking forward at the rate of 4 ft/sec for 10 seconds, and then filming the same person walking backwards at the rate of 4 ft/sec for 10 seconds. By running the two films forwards and backwards in a film projector, we have four cases to consider. Let film A show the person walking forward, and film B show the person walking backwards. (It’s fun to pace back and forth across the room to illustrate this example.)

• If film A is run forward in the projector for 10 seconds, we will see the person walk forward 40 ft. (+4 * +10 = +40)
• If film A is run backwards in the projector for 10 seconds, we will see the person walk backwards 40 ft. (4 * (-10)) = -40)
• If film B is run forward in the projector for 10 seconds, we will see the person walk backwards 40 ft. (-4 * 10 = -40)
• If film B is run backwards in the projector for 10 seconds, we will see the person walk forward 40 ft. (-4 * (-10)) = 40)

The next example is probably familiar to many readers. From my personal experience, a few students find the two previous examples somewhat confusing, but the pattern approach illustrated in the text box below seems to make the most sense to students. The first 4 rows of the table follow from the fact that the product of a positive number and a negative number is a negative number which makes perfect sense to most students. As the value of numbers in column A decrease by 1, the product of A and B gets bigger by 5. When A decreases from 0 to -1, I tell by students they can’t change horses in midstream; so the pattern must be maintained by increasing the product by 5 when A is decreased by 1.

The next example hinges on the idea that multiplication is repeated addition under the following rules: (Rules are easier to understand if m and n are integers.)

• If m is positive, then m * n equals n added to itself m times.
• If m is negative, then m * n equals the opposite of n added to itself |m| times.

The text box below illustrates how these rules work then n = ±4 and m = ±6.

My last example uses the properties of real numbers and mathematical reasoning to demonstrate  (-3)(-5) equals (3)(5) = 15. The demonstration hinges on the following properties of real numbers:

• The distribute property
• A negative number times a positive number is negative. (Previously established)
• m + n = 0 if and only if m and n are opposites of each other.

Because this demonstration requires a higher level of mathematical maturity, I advise against showing this demonstration to younger learners.

I will close this post with a discussion of the concept of positive and negative numbers by looking at two different number systems. You may be surprised to learn that in some number systems, the concept of positive and negative numbers does not exist. My post, A Simple Way to Introduce Complex Numbers, discusses the basics of complex numbers.

The set of real numbers:

• For every real number x: x = 0, x < 0, or x > 0.
• Every real number x not equal to zero has a unique opposite which is denoted by the symbol –x.
• The opposite of a real number is the same as the additive inverse of a real number.
• Real numbers x and y are opposites of each other if and only if x + y = 0.
• If x > 0, then x is a positive number and –x is a negative number.
• If x < 0, then x is a negative number and –x is a positive number.
• The angular direction of all positive numbers is to the right or 00.
• The angular direction of all negative numbers is to the left or 1800.
• For all real numbers x and y, x*y = (-x)(-y). Note: This is true for any pair of real numbers. The expression (-x)*(-y) does not indicate we are multiplying two negative real numbers.
• The symbol –x means the opposite of x; not negative x.

The set of complex numbers:

• All complex number z can be expressed in the form z = a + bi where a and b are real numbers and i is the unit imaginary number such that i2 = -1.
• Every complex number z not equal to zero has a unique opposite which is denoted by –z.
• If z = a + bi, then –z = -a – bi.
• The opposite of a complex number is the same as the additive inverse of a complex number.
• Complex numbers w and z are opposites of each other if and only if w + z = 0.
• The angular direction of complex number z can range from 00 to 3600.
• In general, complex numbers are neither positive or negative because the angular direction of a complex number can range from 00 to 3600; not just 00 or 1800.
• For all complex numbers w and z, w*z = (-w)(-z). Note: This is true for any pair of complex numbers. The expression (-w)*(-z) does not indicate we are multiplying two negative complex numbers because complex numbers in general don’t have a positive or negative property.
• The symbol –z means the opposite of z; not negative z.