Why is a Negative Times a Negative Positive?

If a student is using a repeated addition model of multiplication then it certainly makes sense that a positive number times a positive number is positive. 3 × 5 means three copies of 5, and 5 + 5 + 5 = 15.

Using the repeated addition model, we can even argue that 3 × (-5) means sum three copies of -5, and therefore 3(-5) = (-5) + (-5) + (-5) = -15. (Note that this uses signed addition from 7.NS.1)

-3 × 5 is a bit trickier--what does it mean to have -3 groups of 5? But, here we can argue that, since multiplication is commutative, -3 × 5 = 5 × -3, which would mean 5 groups of -3 each, or (-3) + (-3) + (-3) + (-3) + (-3) = -15.

Finally, how do we deal with -3 × -5? In this case we can’t use the commutative law to justify an answer because that will yield -5 × -3 and we will have the same problem again. There are two possible choices:

  • Reason from patterns
  • Reason from mathematical properties

Reasoning from patterns

Three Pattern-based Arguments

Reasoning from patterns might look like the following.

1. Problem: -3 × -5

Starting from what we know, we can build this table:

Looking at this table, if we define -3 × -5 = 15, then the table above is symmetrical--there are two positive products and two negative products. Multiplication has other symmetrical properties (e.g., the commutative property), so maybe this should also be symmetrical.

 Admittedly this is a fairly weak pattern--three observations is a small data set from which to generalize.

2. Another popular way to explain that a negative times a negative ought to be positive is reasoning from a model showing change over time.

Watch this video showing a jug full of water draining.

Now, what would you see if the video of the water draining was played backward? Take a look:

This model doesn’t prove that a negative times a negative is a positive, but it does suggest how the combination of two negative quantities--draining water and a film playing in reverse--can combine to make an image of something moving in the positive direction (i.e., the bucket filling).

3. Another pattern-based argument can use the constant differences in multiplication tables. Look at the following multiplication tables:

Since each row is three less than the previous, this first chart helps show (again) why 3 × -5 = -15. Now let’s shift the order using the commutative property again and look at the -3 times tables:

Here, to move from one row to the next, we add three. And, if we continue the pattern past -3 × 0, the results become positive.

Taken together, these two charts provide a pattern-based explanation for why -3 × -5 = 15. On an extended multiplication chart, filling in these two tables would be the same as reading down the 3’s column and then left on the -3’s row.

Reasoning from Mathematical Properties

A Mathematical Argument

The explanations of why minus times a minus is plus may convince you that a negative number times a negative number yields a positive number. However, these are not complete proofs: observing a pattern is not the same as proving a mathematical statement. Let's show it's true using basic properties of our logical system.

We'll begin by showing that (-1) x (-1) = 1. We'll rely on the distributive property of multiplication over addition. It says that

a(b + c) = ab + ac for any real numbers a, b, and c.

We also know that any number plus its opposite is zero. So let's begin there:

1 + (-1) = 0

Now we multiply both sides by (-1). On the right-hand side, zero times (-1) is zero.

(-1) [ 1 + (-1) ] = 0

Now the key step: use the distributive property.

(-1)(1) + (-1)(-1) = 0

But (-1)(1) = (-1) because of identity (any number times 1 is itself), so

(-1) + (-1)(-1) = 0

Now we add one to both sides and simplify

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

(-1)(-1) = 1

Ta-daaa!

Now let's be more general and look at numbers other than -1. We'll look at arbitrary numbers a and b. Let's figure out what (-a)(-b) is. We know that a negative number is just (-1) times that number. So

(-a)(-b) = (-1)(a)(-1)(b)

Now we use the commutative property, rearranging the factors, to say that

(-a)(-b) = (-1)(-1)(a)(b) = (-1)(-1)(ab)

But we just showed that (-1)(-1) = 1, so

(-a)(-b) = ab

Which is what we were trying to show. The product of any two numbers is the product of their opposites.

 

meet the team!

Strategic Education Research Partnership
1100 Connecticut Ave NW #1310  •  Washington, DC  20036
serpinstitute.org  •  (202) 223-8555  •  info@serpinstitute.org

Project funding provided by The William and Flora Hewlett Foundation and S.D. Bechtel Jr. Foundation

Creative Commons License
Poster Problems by SERP is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

If a student is using a repeated addition model of multiplication then it certainly makes sense that a positive number times a positive number is positive. 3 × 5 means three copies of 5, and 5 + 5 + 5 = 15.

Using the repeated addition model, we can even argue that 3 × (-5) means sum three copies of -5, and therefore 3(-5) = (-5) + (-5) + (-5) = -15. (Note that this uses signed addition from 7.NS.1)

-3 × 5 is a bit trickier--what does it mean to have -3 groups of 5? But, here we can argue that, since multiplication is commutative, -3 × 5 = 5 × -3, which would mean 5 groups of -3 each, or (-3) + (-3) + (-3) + (-3) + (-3) = -15.

Finally, how do we deal with -3 × -5? In this case we can’t use the commutative law to justify an answer because that will yield -5 × -3 and we will have the same problem again. There are two possible choices:

  • Reason from patterns
  • Reason from mathematical properties

Reasoning from patterns

Three Pattern-based Arguments

Reasoning from patterns might look like the following.

1. Problem: -3 × -5

Starting from what we know, we can build this table:

Looking at this table, if we define -3 × -5 = 15, then the table above is symmetrical--there are two positive products and two negative products. Multiplication has other symmetrical properties (e.g., the commutative property), so maybe this should also be symmetrical.

 Admittedly this is a fairly weak pattern--three observations is a small data set from which to generalize.

2. Another popular way to explain that a negative times a negative ought to be positive is reasoning from a model showing change over time.

Watch this video showing a jug full of water draining.

Now, what would you see if the video of the water draining was played backward? Take a look:

This model doesn’t prove that a negative times a negative is a positive, but it does suggest how the combination of two negative quantities--draining water and a film playing in reverse--can combine to make an image of something moving in the positive direction (i.e., the bucket filling).

3. Another pattern-based argument can use the constant differences in multiplication tables. Look at the following multiplication tables:

Since each row is three less than the previous, this first chart helps show (again) why 3 × -5 = -15. Now let’s shift the order using the commutative property again and look at the -3 times tables:

Here, to move from one row to the next, we add three. And, if we continue the pattern past -3 × 0, the results become positive.

Taken together, these two charts provide a pattern-based explanation for why -3 × -5 = 15. On an extended multiplication chart, filling in these two tables would be the same as reading down the 3’s column and then left on the -3’s row.

Reasoning from Mathematical Properties

A Mathematical Argument

The explanations of why minus times a minus is plus may convince you that a negative number times a negative number yields a positive number. However, these are not complete proofs: observing a pattern is not the same as proving a mathematical statement. Let's show it's true using basic properties of our logical system.

We'll begin by showing that (-1) x (-1) = 1. We'll rely on the distributive property of multiplication over addition. It says that

a(b + c) = ab + ac for any real numbers a, b, and c.

We also know that any number plus its opposite is zero. So let's begin there:

1 + (-1) = 0

Now we multiply both sides by (-1). On the right-hand side, zero times (-1) is zero.

(-1) [ 1 + (-1) ] = 0

Now the key step: use the distributive property.

(-1)(1) + (-1)(-1) = 0

But (-1)(1) = (-1) because of identity (any number times 1 is itself), so

(-1) + (-1)(-1) = 0

Now we add one to both sides and simplify

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

(-1)(-1) = 1

Ta-daaa!

Now let's be more general and look at numbers other than -1. We'll look at arbitrary numbers a and b. Let's figure out what (-a)(-b) is. We know that a negative number is just (-1) times that number. So

(-a)(-b) = (-1)(a)(-1)(b)

Now we use the commutative property, rearranging the factors, to say that

(-a)(-b) = (-1)(-1)(a)(b) = (-1)(-1)(ab)

But we just showed that (-1)(-1) = 1, so

(-a)(-b) = ab

Which is what we were trying to show. The product of any two numbers is the product of their opposites.