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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
Example
2: 4 x (-3)
4
groups of 3 negative chips gives 12 negative chips
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.
|
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)
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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
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| 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
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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: |
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a)
-3 x 2 |
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b)
4 x -5 |
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c)
7 x -1 |
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d)
-3 x 6 |
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| 2. |
Without
using a model, solve the following problems: |
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a)
2(4 x 5) -1 |
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b)
-17 x 4 x -3 |
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c)
-6(-5) + 15 + (-6) |
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d)
3 - 4 + 5 x -8 |
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| 3. |
Using
any method you like, show why -4 x -5 = 20.
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| 4. |
Find
the answers to: |
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(a)
14 x (3-7) + (8 x -3) |
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(b)
((-4) - (-9)) x 2 + (-6) |
To
view the quiz answers, click here.
'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.
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1.
Solve 3 x (- 7)
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This
is 3 lots of 7 negative chips, which equals 21 negative chips
So
3 x (- 7) = - 21
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| 2.
Solve - 3((- 2) + (- 7)) + (- 2) - 5(-
4) |
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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
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2
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1
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0
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-
1
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-
2
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-
3
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-
4
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3
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6
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3
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0
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-
3
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-
6
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-
9
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-
12
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2
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4
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2
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0
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-
2
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4
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6
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8
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1
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2
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1
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0
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-
1
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-
2
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-
3
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-
4
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0
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0
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0
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0
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0
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0
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0
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0
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-
1
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-
2
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-
1
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0
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1
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2
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3
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4
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-
2
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-
4
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-
2
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0
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2
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4
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6
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8
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-
3
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-
6
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-
3
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0
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3
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6
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9
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12
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-
4
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-
8
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-
4
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0
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4
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8
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12
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16
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-
5
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-
10
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-
5
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0
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5
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10
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15
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20
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-
6
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-
12
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-
6
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0
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6
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12
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18
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24
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-
7
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-
14
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-
7
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0
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7
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14
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21
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28
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-
8
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-
16
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-
8
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0
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8
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16
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24
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32
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-
9
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-
18
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-
9
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0
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9
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18
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27
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36
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