Negative & Directed Numbers (Cambridge (CIE) IGCSE Maths) : Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

Negative & Directed Numbers

What are negative numbers?

  • Negative numbers are any number less than zero

    • They may also be referred to as directed numbers

  • 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

  • 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

    • 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 it as follows:

      • A room is -5°C, then -6°C of cold air is 'removed'

      • The room now warms to 1°C

  • Money and debt is another common context for negative numbers

    • A negative sign means you owe money

    • 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 only $100 in debt

      • This is equivalent to (-300) + 200 = -100

Examiner Tips and Tricks

  • 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

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 is negative

-18

(-9) × (-2)

9 × 2 = 18
both are negative

18

(-9) ÷ (-3)

9 ÷ 3 = 3
both are negative

3

(-10) ÷ 5

10 ÷ 5 = 2
one is negative

-2

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Jamie Wood

Author: Jamie Wood

Expertise: Maths Content Creator

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.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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