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
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"

When multiplying and dividing with negative numbers
- Two numbers with the same sign make a positive
  - $(- 12) \div (- 4) = 3$
  - $(- 6) \times (- 4) = 24$
- Two numbers with different signs make a negative
  - $(- 12) \div 4 = - 3$
  - $6 \times (- 4) = - 24$
- It can be easier to calculate ignoring signs, then make a decision about whether the answer should be positive or negative
  - e.g. For $12 \div (- 4)$ , calculate $12 \div 4 = 3$ , then, as the signs are different, make the answer negative, so $12 \div (- 4) = - 3$
When adding and subtracting with negative numbers
- Subtracting a negative number is the same as adding the positive
  - e.g. $5 - (- 3) = 5 + 3 = 8$
- Adding a negative number is the same as subtracting the positive
  - e.g. $7 + (- 3) = 7 - 3 = 4$

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
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

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 $(- 3) \times (- 3) = 9$
    On many calculators, $- 3^{2} = - 9$ but ${(- 3)}^{2} = 9$
    The second one is the required calculation

Complete the following table.

Negative Numbers (Edexcel GCSE Maths: Foundation)