 Why is a negative number times a negative number a positive number?

By: Diana Brown

This is a common question by many students taking mathematics courses in middle school and even high school.   It is not surprising to most that this is difficult concept for students to understand. Negative numbers are not easily understood by most people.  The difficulty with understanding a negative times a negative is that this is not something we do in our everyday lives. Below are several methods that may be used to help students better understand the meaning of a negative times a negative.

Method One:  Money representations

We can start of explaining negative multiplication by helping the student understand the easier concepts such as a positive times a positive or a positive times a negative.  For example in the case of money, we can represent a positive times a negative by saying \$700 is being deducted each month to pay ones mortgage payment. After six months, how much money has been taken out of the pay for the mortgage? We can figure out the answer by doing multiplication.

6 * -\$700 = -\$4,200.  This is an illustration of a positive times a negative resulting in a negative.

We can use this same method to represent a negative times a negative:

Now suppose that, as a bonus, the employer decides to pay the mortgage for one year. The employer removes the mortgage deduction from the monthly paychecks. How much money is gained by the employee in our example? We can represent "removes" by a negative number and figure out the answer by multiplying.

-12 * -\$700 = \$8,400

This is an illustration of a negative times a negative resulting in a positive. If one thinks of multiplication as grouping, then we have made a positive group by taking away a negative number twelve times.

Method Two:  A mathematical illustration

Most students are quick to agree that a negative number can be represented as a number times -1.  The illustration of a negative times a positive is easier to understand.  So if we take a number and multiply the number times -1 it is represented below:

(-1)x = -x.  Two negative numbers being multiplied together can be represented as followed:

(-x) (-y) = (-1) (x) (-1) (y) = (-1) (-1) (x) (y), So what is (-1)(-1)?

First we will start with things that we know.  For example, we know that -1(0) = 0.

We can rewrite (-1)(0) = (-1)(-1 + 1), then using the distributive property on the right side of the equation we get:

(-1) (-1) + (-1) (1)

Now we know that (-1)(1) = -1, but we aren’t sure what (-1)(-1) is, but we do know that whatever it is must be the equation is equal to zero, so since it can’t be -1 for that would make the equation equal to -2, then it must be +1. See the math below:

0 = (-1) (0) = (-1)(-1 + 1) = (-1) (-1) + (-1) (1) = ? + (-1) therefore

0 = ? + (-1), from our statement above (-1)(-1) must be +1 to complete the statement: 0 = ? + (-1).  Which may help to conclude that a negative times a negative equals a positive.

Method Three:  A proof

Let a and b be any two real numbers. Consider the number x defined by

x = ab + (-a)(b) + (-a)(-b).

We can write

`x = ab + (-a)[ (b) + (-b) ]       (factor out -a)`
`  = ab + (-a)(0)`
`  = ab + 0`
`  = ab.`

Also,

`x = [ a + (-a) ]b + (-a)(-b)      (factor out b)`
`  = 0 * b + (-a)(-b)`
`  = 0 + (-a)(-b)`
`  = (-a)(-b).`

So we have

x = ab
and
x = (-a)(-b)

Hence, by the transitivity of equality, we have

ab = (-a)(-b).

Method Four:  Using words to represent negative numbers

Some people think of the word “NOT” as a negative meaning.  One might say I am NOT going to my friend’s house.  This seems like a negative version of saying I AM going to my friend’s house.  So what if I said this with two NOTS.  I am not going to not go to my friend’s house.  It seems the two NOTS cancel each other out and I am going to my friends house is derived.  This seems that a double negative statement really derives a positive statement.  More examples:

 Negative * Negative Positive He cannot just do nothing He must do something a not infrequent visitor a frequent visitor I don't never go I go

Method Five:  Pattern recognition

Lets look at the sequence below.  Notice what happens to the right side of the equation as the first number goes down by one:

4 x 5 = 20

3 x 5 = 15

2 x 5 = 10

1 x 5 = 5

0 x 5 = 0                        We notice that the numbers are going down by 5.  Lets keep going:

-1 x 5 = -5

-2 x 5 = -10

-3 x 5 = -15

-4 x 5 = -20.                  Now lets do this same kind of sequence replacing 5 with -5.

4 x -5 = -20

3 x -5 = -15

2 x -5 = -10

1 x -5 = -5

0 x -5 = 0                      We notice that the numbers are going up by 5.  If we use the same logic as before then we will see that:

-1 x -5 = 5

-2 x -5 = 10

-3 x -5 = 15

-4 x -5 = 20.                  Therefore it seems that a negative times a negative is again a positive.

Method Six: Using technology

We can use the coordinate system in Geometer’s Sketchpad to create a line. Remember that the slope of a line is rise over run.  If we plot a point in the coordinate plane and use a slope to plot a second point we will look at the slope to determine if the line has a positive slope of a negative slope.

In the following diagram we plotted a point (2, 2): Let’s use a slope of 1/2 to find the next three points. Now if we construct a line through the points we will see that the result is a line with a positive slope. Let’s try this same thing starting at the point (2, 2) and using the slope -1/-2.  See the diagram below for results. Notice that the line is the same as before, a positive slope.  Therefore this could be a graphical approach to showing the relationship of two negative numbers.

Websites used to help the discovery of different methods:

www.mathforum.org

plato.stanford.edu