| Using the annihilation model | Multiplying a negative number by a negative number |
| Mixed operations of directed numbers | Quick quiz |

Using the annihilation model

Minus times minus results in a plus,
The reason for this, we needn't discuss. - Ogden Nash

Ogden Nash may not have understood negative number arithmetic but by the end of this topic - you will!

In order to use the annihilation model for multiplication we will think of multiplication as repeated addition. For example, 3 x 2 can be thought of as 3 groups of 2.

Example 1: 3 x 2 = 2 + 2 + 2 = 6

3 groups of 2 positive chips gives 6 positive chips

2 x 3 = 2 + 2 + 2
= 6

Example 2: 4 x (-3)

4 groups of 3 negative chips gives 12 negative chips

4 x (-3)
= -12

Example 3: (-5) x 2

It does not make sense to say '-5 groups of 2 positive chips'. To use the model to solve this problem we apply the commutative law which tells us that (-5) x 2 = 2 x (-5). The commutative law can be explained for positive whole number multiplication by arrays. See Whole Numbers, Key Ideas, Associative and Commutative Laws. We generalise this property to negative numbers.

The problem is now in a form that we can easily represent using the model.

5 groups of 2 negative chips gives 10 negative chips.

5 x (-2)
= -10

 From the two examples above we can see that a negative number multiplied by a positive number, or a positive number multiplied by a negative number give a negative result. Let a and b be positive numbers, then (-a) x b = a x (-b) = -(a x b)

Multiplying a negative number by a negative number

Example 4: (-2) x (-3)

How can we model -2 x -3? We can't say -2 groups of 3 negative chips nor -3 groups of 2 negative chips. The annihilation model fails to provide a meaning and we need to look elsewhere. The best way to show the result of a negative number multiplied by a negative number, is to look for patterns.

The multiplication products on the pink shaded section are already known. We can extend the pattern into the green section.

Patterns of multiplication of directed numbers

 2 x 5 = 10 2 x 4 = 8 2 x 3 = 6 2 x 2 = 4 2 x 1= 2 2 x 0 = 0 2 x -1 = -2 2 x -2 = -4 2 x -3 = -6 2 x -4 = -8 2 x -5 = -10
a pattern develops of down by 2
 -2 x 5 = -10 -2 x 4 = -8 -2 x 3 = -6 -2 x 2 = -4 -2 x 1 = -2 -2 x 0 = 0 -2 x -1 = 2 -2 x -2 = 4 -2 x -3 = 6 -2 x -4 = 8 -2 x -5 = 10
a pattern develops of up by 2
the pattern continues below zero of down by 2 the pattern continues below zero of up by 2

By listing the 2 times table (we could have chosen any times table) and continuing below zero we can see that the pattern of down by 2 continues. If we also list the -2 times table we can see that a pattern develops of up by 2 and we can make this pattern continue if we agree that a negative number by a negative number is a positive.

Although the fact that this pattern develops is no proof, it is a strong indication that a negative number x a negative number = a positive number. Mathematicians prove this result by logical deduction from basic properties of numbers.

Basic multiplication facts

A table of basic multiplication facts also shows the patterns which evolve when we multiply negative and positive integers.

 x -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +5 -25 -20 -15 -10 -5 0 +5 +10 +15 +20 +25 +4 -20 -16 -12 -8 -4 0 +4 +8 +12 +16 +20 +3 -15 -12 -9 -6 -3 0 +3 +6 +9 +12 +15 +2 -10 -8 -6 -4 -2 0 +2 +4 +6 +8 +10 +1 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 0 0 0 0 0 0 0 0 0 0 0 0 -1 +5 +4 +3 +2 +1 0 -1 -2 -3 -4 -5 -2 +10 +8 +6 +4 +2 0 -2 -4 -6 -8 -10 -3 +15 +12 +9 +6 +3 0 -3 -6 -9 -12 -15 -4 +20 +16 +12 +8 +4 0 -4 -8 -12 -16 -20 -5 +25 +20 +15 +10 +5 0 -5 -10 -15 -20 -25

We can also see here that a negative number multiplied by a negative number gives a positive number result.

The distributive property

We could also use the distributive property of multiplication to see if it fits with the results we have seen so far. The distributive property says,

a x (b - c) = a x b - a x c

We can substitute some numbers for a, b and c.

Let a = -4, b = 6 and c = 8.

We will solve the LHS of the equation using the distributive law and solve the RHS using the results we have obtained so far using the annihilation model.

a x (b - c)

- 4 x (6 - 8) can be calculated using the distributive law

a x b - a x c

= (-4) x (6) - (-4) x (8)
= -24 - (-32)
= 8

Solving the RHS of the equation, we can avoid multiplying a negative number by a negative number and use the fact that subtracting a negative is the same as adding the positive value of the number (which we have seen demonstrated using the annihilation model).
Alternatively
- 4 x (6 - 8)

= (-4) x (-2)

Evaluating 6 - 8

Equating the two, we can see that

 -4 x (6 - 8) = 8 and therefore (-4) x (-2) = 8

Using the distributive property we have confirmed that

 a negative number x a negative number = a positive number (-a) x (-b) = a x b

Mixed operations of directed numbers

Example 5: movie, 2 x (1 - 3) + 1

Example 6: movie, 7 - 6 x (-4 - (-3))

Quick quiz

 1. Using the annihilation model, and the commutative law if necessary, solve the following problems: a) -3 x 2 b) 4 x -5 c) 7 x -1 d) -3 x 6 2. Without using a model, solve the following problems: a) 2(4 x 5) -1 b) -17 x 4 x -3 c) -6(-5) + 15 + (-6) d) 3 - 4 + 5 x -8 3. Using any method you like, show why -4 x -5 = 20. 4. Find the answers to: (a) 14 x (3-7) + (8 x -3) (b) ((-4) - (-9)) x 2 + (-6)

'Talking through' questions

The 'talking through' questions and answers below have been provided to enable you to see how an 'expert' might tackle these questions. The annihilation model has been used in the explanations where appropriate.

1. Solve 3 x (- 7)

This is 3 lots of 7 negative chips, which equals 21 negative chips

So 3 x (- 7) = - 21

2. Solve - 3((- 2) + (- 7)) + (- 2) - 5(- 4)

Here we must do the bracket first, so (- 2) + (- 7) means 2 negative chips plus another 7 negative chips = 9 negative chips. Hence (- 2) + (- 7) = - 9

The problem now becomes - 3 x - 9 + (- 2)- 5 x - 4

We must do the multiplications next.

We cannot use the annihilation model to work out -3 x -9 because it does not make sense to talk about 'negative 3 lots' of negative 9 chips.

- 3 x - 9 can be found by observing the pattern in a table below. In the multiplication table below, you know the answers to the turquoise multiplications. The pattern in the columns is given in pink, counting down in twos in the first column and down in ones in the second column, etc. The pattern in the rows is given in yellow. In the first row we are decreasing by threes. In the second row we are decreasing by twos, and in the third row by ones. In the fifth, sixth and seventh rows etc. we are increasing by ones, twos and threes, etc. respectively.

From the table, it can be seen that (- 3) x (- 9) = 27. We know that 5 x - 4 means 5 lots of four negative chips which equals 20 negative chips = - 20

The problem now becomes 27 + (- 2) - (- 20)

We have 27 positive chips and 2 negative chips. 2 of the negative chips will annihilate 2 of the positive chips, leaving 25 positive chips.

The problem now becomes 25 - (- 20)

This is 25 positive chips take away 20 negative chips. We have no negative chips to take away, so add 20 negative chips and 20 positive chips to the 25 positive chips. This gives 45 positive chips and 20 negative chips. When we take 20 negative chips away from this we are left with 45 positive chips.

So - 3((- 2) + (- 7)) + (- 2) - 5(- 4) = 45

 2 1 0 - 1 - 2 - 3 - 4 3 6 3 0 - 3 - 6 - 9 - 12 2 4 2 0 - 2 - 4 - 6 - 8 1 2 1 0 - 1 - 2 - 3 - 4 0 0 0 0 0 0 0 0 - 1 - 2 - 1 0 1 2 3 4 - 2 - 4 - 2 0 2 4 6 8 - 3 - 6 - 3 0 3 6 9 12 - 4 - 8 - 4 0 4 8 12 16 - 5 - 10 - 5 0 5 10 15 20 - 6 - 12 - 6 0 6 12 18 24 - 7 - 14 - 7 0 7 14 21 28 - 8 - 16 - 8 0 8 16 24 32 - 9 - 18 - 9 0 9 18 27 36