Negative number arithmetic

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Using the annihilation model

Subtraction is modelled as take away using the annihilation model.

We have already discussed some ideas relating to subtraction in the earlier sections. The difference between the negative sign in front of a number and the subtraction sign was discussed in Key Ideas.

Using the annihilation model in Addition of Negative Numbers we saw that addition of a negative number is the same as subtraction of the positive value of that number. So 4 - 7 is equivalent to 4 + (-7).

Example 1: (-4) - (-2)

We can represent (-4) - (-2) by starting with 4 negative chips and taking away 2 negative chips. This leaves 2 negative chips.

(-4) - (-2)

-2

Here we have taken away 2 negative chips. Using the model again we can show that we could have got the same result by adding 2 positive chips.

Example 2: Using the model to show (-4) -(-2) = (-4) + (+2)

-4

(+2)

-2

So we can see that -4 - (-2) is equivalent to -4 + (+2). Taking away a negative number is the same as adding the positive value of the number

Example 3: (+3) - (+5)

+3 - (+5)

I want to take away +5 from +3, but I haven't got enough positive chips to do this. One way I could approach the problem is to represent +3 in another way, as I have done below. +3 can be represented by many arrangements of positive and negative chips so long as when all positives and negatives annihilate one another there are 3 positive chips left.

To represent +3 I have added another 5 positive chips and 5 negative chips. I now have 8 positive chips and 5 negative chips (which still represents +3). I now have enough positive chips to take away 5.

3 - (+ 5)

= -2

Instead of taking away 5 positive chips we could have added 5 negative chips to get the same result. So we can see again that taking away +5 is the same as adding -5. We call this method of subtraction adding opposites.

Example 4: Using the model to show (+3) - (+5) is also equal to (-2)

-5

-2

Subtraction of Integers - Adding opposites

If a and b are any integers, then a - b = a + (-b)

These explanations may seem unnecessarily complicated for the fairly straightforward examples we have used here; however the point is to illustrate how and why the rules which we apply in the addition and subtraction of directed numbers work. In order to be able to teach these concepts we need to understand the principles underlying the general rules which are often stated with little justification in school textbooks.

Addition and Subtraction of directed numbers

Example 5: movie - using the annihilation model, 12 - (-4)

Click on the image to view the movie

Example 6: movie - using the number line model, 5 - 3 + 7

Click on the image to view the movie

Quick quiz

Using the annihilation model, or another model if you prefer, find answers to the following examples:

a) 6 - (+3)

b) -7 - 4

c)(-8) - (-5)

d) -8 + 5

Without using a model find answers to the following examples:

a) -4 - 16

b) 10 - 19

c) -22 - 13

d) -7 - (-14)

e) -9 - (-5) + 15 - 7 - (-4)

f) 12 - (-4) + (-3) - 9 + (-11) - 5

g) -6 + 8 - (-9) + 7 -(-10)

h) -5 - (-17) - 25 + (-13) + 26

Lance wants to check the final balance of all the transactions listed on his monthly savings account statement. What is his final balance?

- 1235.99

583.00

299.00

- 25.75

- 60.00

784.21

201.00

The local shopping centre has just been expanded and now has a ground level, 5 levels above this and a car park which is on levels B1, B2 and B3 below ground level.

a) From level B2, how many levels do you travel to get to level B1?

b) From level, B2, how many levels do you travel to get to ground level?

c) If I am shopping on level 5 of the shopping centre and I have parked on level B3, how many levels do I need to travel to get to my car?

d) If I park on level B2, how many levels do I need to travel to get to level 2?

Write each of the problems above as an addition or subtraction of negative numbers. The first is done for you.

a) (-2) - (-1) = ? or (-2) + ? = (-1)

(-2) - (-1) = 1 or (-2) + 1 = (-1)

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.

1. Find the answer to 9 - (- 4) - 8

This reads, positive nine take away negative four take away positive eight.

Using the annihilation method, we have 9 positive chips and we wish to take away 4 negative chips. However, we do not have any negative chips to take away, so we must add 4 negative chips and 4 positive chips to 9. This gives 13 positive chips and 4 negative chips. When we take four negative chips away from this we are left with 13 positive chips. 13 positive chips take away 8 positive chips leaves us with 5 positive chips.

So, 9 - (- 4) - 8 = 13 - 8 = 5