Negative & Directed Numbers (Cambridge International IGCSE Mathematics)

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

    • (−12)÷(−4)=3
    • (−6)×(−4)=24

  • Two numbers with different signs make a negative

    • (−12)÷4=−3
    • 6×(−4)=−24

• 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

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

Exam Hint

• 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)×(−3)=9 On many calculators, $-3^{2}=-9$ but (−3)²=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

Answer:

Calculation	Working	Answer
3 + (-4)	3 - 4	Final answer: -1
(-5) + (-8)	(-5) - 8	Final answer: -13
7 - (-10)	7 + 10	Final answer: 17
(-8) - (-6)	(-8) + 6	Final answer: -2
(-3) × 6	3 × 6 = 18 one is negative	Final answer: -18
(-9) × (-2)	9 × 2 = 18 both are negative	Final answer: 18
(-9) ÷ (-3)	9 ÷ 3 = 3 both are negative	Final answer: 3
(-10) ÷ 5	10 ÷ 5 = 2 one is negative	Final answer: -2