# 1.02.01 Negative numbers

First of all it is important to make sense of what negative numbers mean. In some situations negative numbers don’t make sense. If you have £100 pounds in your wallet and you keep removing money to pay for things you will have less and less money until it is empty (ie you will eventually have £0). You obviously cannot remove any more money once your wallet is empty and it makes no sense to say something like you have -£20 in your wallet. If however you spend all of the money in your bank account you may still be able to take out more, in this case the bank lends extra money to you. A bank balance of -£20 does make sense it means you owe the bank £20!

**Other examples of uses of negative numbers**

- When travelling along a road in a vehicle, if you regard your velocity as positive then you will regard the velocity of cars travelling in the opposite direction as being negative
- A location on a map that has a positive altitude is above sea level, one with a negative altitude is below sea level
- A temperature of 10
^{o}C is 10^{o}C above the temperature of ice (at atmospheric pressure). A temperature of – 10^{o}C is 10^{o}C lower than the temperature of ice

# 1.02.02 Adding and subtracting positive and negative numbers

## Adding negative numbers

Imagine you have two bank accounts one with a balance of £100 in it and a second account with a balance of -£20. If you add the balance of the second account to the first you would have a total balance of £80. (This is the same balance you would have had if you had just withdrawn £20 from your first account.) This illustrates rule 1:

**Adding a negative number is the same as subtracting a positive number of the same magnitude.**

Examples

- 4 + (-3) is the same as 4 – 3 = 1
- 20 + (-15) is the same as 20 – 15 = 5
- 12 + (-5) is the same as 12 – 5 = 7

## Subtracting negative numbers

If your account has a balance of -£50 and you bank manager kindly agrees to “write off” your debt i.e. to remove the balance of -£50 this would leave you with a balance of £0. (If you had deposited £50 into your account this would have achieved the same result). This illustrates rule 2:

**Subtracting a negative number is the same as adding a positive number of the same magnitude**

- 6 – (-2) is the same as 6 + 2 = 8
- 14 – (-9) is the same as 14 + 9 = 23
- 20 – (-13) is the same as 20 + 13 = 33

In the above examples once you have applied either of the rules to rearrange the sum you are left with a simple familiar calculation. However this is not always the case and you may be faced with several possible unfamiliar procedures.

you may be required to:

- subtract a large positive number from a smaller positive number
- subtract a positive number from a negative number
- add a positive number to a negative number

The next page shows how to deal with these situations

# 1.02.03 Methods for dealing with calculations that involve or result in negative numbers

## 1. Subtracting large positive numbers from smaller positive numbers

If you subtract a larger positive number from a smaller positive number it will produce a negative number. The answer you will get will be the same as you would get by subtracting the small number from the larger one except that the answer will be negative

Example

- 3 – 8 = ?
- (8 – 3 = 5)
- Therefore 3 – 8 = -5

## 2. Subtracting a positive number from a negative number

If you subtract a positive number from one that is already negative your result will be a negative number of larger magnitude. (if you draw out more money from the bank when you are already overdrawn you will be further in debt!). The answer you will get will be the same as if you added two positive numbers of the same magnitude except that the answer will be negative.

Example

- (-3) – 6 = ?
- (3 + 6 = 9)
- Therefore (-3) – 6 = -9

## 3. Adding a smaller positive number to a larger negative number

If you add a smaller positive number to a larger negative number you will reduce the magnitude of the negative number. (if you pay off some of but not all of your overdraft you will reduce the amount you owe). Change the negative number to a positive one and subtract the positive number instead of adding it. Change the answer to a negative number to get the answer to the original sum.

Example

- (-11) + 6 = ?
- 11 – 6 = 5
- Therefore (-11) + 6 = -5

## 4. Adding a larger positive number to a smaller negative number

If you add a larger positive number to a smaller negative number you will produce a positive number. (if you your account is overdrawn and you pay in more than you owe you will have some money left in your account. The answer is found as described above but this time the answer is positive not negative.)

Example

- (-12) + 16 = ?
- 16 – 12 = 4
- Therefore (-12) + 16 = 4

# 1.02.04 Multiplying and dividing positive and negative numbers

These are a fairly simple calculations to deal with. To find the value of the answer you simply multiply or divide the numbers as normal ignoring their signs. To decide whether the answer is positive or negative:

- if there is an odd number of negative numbers in the calculation the answer will be negative
- if there is an even number of negative numbers in the calculation the answer will be negative

Examples

- 2 x (-3) x (-2) x 4 = ?
- 2 x 3 x 2 x 4 = 48
- two negative numbers therefore answer is positive
**2 x (-3) x (-2) x 4 = 48**

- (-2) x (-3) x (-2) x 4 = ?
- 2 x 3 x 2 x 4 = 48
- three negative numbers therefore answer is negative
**(-2) x (-3) x (-2) x 4 = -48**

- (-2) x (-3) x (-2) x (-4) = ?
- 2 x 3 x 2 x 4 = 48
- four negative numbers therefore answer is positive
**(-2) x (-3) x (-2) x (-4) = 48**

- (-8)/4 = ?
- 8/4 = 2
- one negative number therefore answer is negative
**(-8)/4 = -2**

- (-8)/(-4) = ?
- 8/4 = 2
- two negative numbers therefore answer is positive
**(-8)/(-4) = 2**

- (2 x (-3))/4 = ?
- (2 x 3)/4 = 6/4 =1.5
- one negative number therefore answer is negative
**(2 x (-3))/4 = -1.5**

- (2 x (-3))/(-4) = ?
- (2 x 3)/4 = 6/4 =1.5
- two negative numbers therefore answer is positive
**(2 x (-3))/(-4) = 1.5**