Negative Numbers (Edexcel GCSE Maths: Foundation)

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Jamie W

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Jamie W

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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
  • 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
      • open parentheses negative 12 close parentheses divided by open parentheses negative 4 close parentheses equals 3
      • open parentheses negative 6 close parentheses cross times open parentheses negative 4 close parentheses equals 24
    • Two numbers with different signs make a negative
      • open parentheses negative 12 close parentheses divided by 4 equals negative 3
      • 6 cross times open parentheses negative 4 close parentheses equals negative 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 divided by open parentheses negative 4 close parentheses, calculate 12 divided by 4 equals 3, then, as the signs are different, make the answer negative, so 12 divided by open parentheses negative 4 close parentheses equals negative 3
  • When adding and subtracting with negative numbers
    • Subtracting a negative number is the same as adding the positive
      • e.g.  5 minus open parentheses negative 3 close parentheses equals 5 plus 3 equals 8
    • Adding a negative number is the same as subtracting the positive
      • e.g.  7 plus open parentheses negative 3 close parentheses equals 7 minus 3 equals 4

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
  • 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 open parentheses negative 3 close parentheses cross times open parentheses negative 3 close parentheses equals 9
        On many calculators, negative 3 squared equals negative 9 but open parentheses negative 3 close parentheses squared equals 9
        The second one is the required calculation

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

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Jamie W

Author: Jamie W

Expertise: Maths

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.