Negative Numbers
What are negative numbers?
- Negative numbers are any number less than zero
- They may also be referred to as directed numbers
- You may come across them in real-life problems such as temperature or debt
- Negative numbers are indicated by a minus sign (-)
- To avoid confusion between subtraction and negative numbers, sometimes the following is used:
- negative numbers are written in brackets
- a longer dash is used for subtraction (—)
- the minus for a negative number is raised (superscript), e.g. -4
- To avoid confusion between subtraction and negative numbers, sometimes the following is used:
- Negative numbers are read by using the word 'negative' or 'minus' before the value
- e.g. -8 would be read/said as "negative eight" or "minus eight"
What are the rules for working with negative numbers?
- When multiplying and dividing with negative numbers
- Two numbers with the same sign make a positive
- Two numbers with different signs make a negative
- It can be easier to calculate ignoring signs, then make a decision about whether the answer should be positive or negative
- e.g. For , calculate , then, as the signs are different, make the answer negative, so
- Two numbers with the same sign make a positive
- When adding and subtracting with negative numbers
- Subtracting a negative number is the same as adding the positive
- e.g.
- Adding a negative number is the same as subtracting the positive
- e.g.
- Subtracting a negative number is the same as adding the positive
Where are negative numbers used in real-life?
- Temperature is a common context for negative numbers, and one that we are used to using
- If the temperature is 3°C, and it cools by 5°C, the new temperature will be -2°C
- This is equivalent to 3 - 5 = - 2
- If the temperature is -4°C, and it warms up by 6°C, the new temperature will be 2°C
- This is equivalent to (-4) + 6 = 2
- To explain why (-5) - (-6) = 1, you could think of this as a temperature of -5°C, and then -6°C of cold air is removed, which makes it warmer overall
- If the temperature is 3°C, and it cools by 5°C, the new temperature will be -2°C
- Money and debt is another context where negative numbers can be used
- A negative sign represents money that is owed
- If someone has a debt of $200, and they borrow another $400, their total debt is now $600
- This is equivalent to (-200) + (-400) = -600
- If someone is in debt by $300, but then pays off $200 of their debt, they are now $100 in debt
- This is equivalent to (-300) + 200 = -100
Examiner Tip
- It can help to think of negative numbers as temperature or hot and cold air
- Be careful to remember the rules when adding, subtracting, multiplying and dividing with negatives
- Your calculator isn't always as clever as you may think!
- Using brackets around negative numbers will always make sure the calculator is doing what you want
- e.g. The square of negative three is
On many calculators, but
The second one is the required calculation
- e.g. The square of negative three is
- Using brackets around negative numbers will always make sure the calculator is doing what you want
Worked example
Complete the following table.
Calculation | Working | Answer |
3 + (-4) | ||
(-5) + (-8) | ||
7 - (-10) | ||
(-8) - (-6) | ||
(-3) × 6 | ||
(-9) × (-2) | ||
(-9) ÷ (-3) | ||
(-10) ÷ 5 |
Calculation | Working | Answer |
3 + (-4) | 3 - 4 | -1 |
(-5) + (-8) | (-5) - 8 | -13 |
7 - (-10) | 7 + 10 | 17 |
(-8) - (-6) | (-8) + 6 | -2 |
(-3) × 6 | 3 × 6 = 18 one negative |
-18 |
(-9) × (-2) | 9 × 2 = 18 both negative |
18 |
(-9) ÷ (-3) | 9 ÷ 3 = 3 both negative |
3 |
(-10) ÷ 5 | 10 ÷ 5 = 2 one negative |
-2 |