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Integers, Rational Numbers and Real Numbers

Chapter 2 Study Notes

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Section 4 of 8
Contents▾
Why Negative Numbers Matter2.1Negative Numbers and Ordering2.2Addition and Subtraction of Integers2.3Multiplication and Division of Integers2.4Combined Operations (BODMAS)2.5Operations on Fractions and Decimals with Negatives2.6Rational and Irrational NumbersChapter Summary

2.3Multiplication and Division of Integers

Why does a negative times a negative give a positive? Look at this pattern:

(−3)×3=−9(-3) \times 3 = -9

(−3)×2=−6(-3) \times 2 = -6

(−3)×1=−3(-3) \times 1 = -3

(−3)×0=0(-3) \times 0 = 0

(−3)×(−1)=  ?(-3) \times (-1) = \;?

Each step down in the second number adds 33 to the result, so (−3)×(−1)(-3) \times (-1) must be +3+3 and (−3)×(−2)=+6(-3) \times (-2) = +6.

Sign rules for multiplication and division

  • Positive ×\times positive == positive
  • Negative ×\times negative == positive
  • Positive ×\times negative == negative
  • Negative ×\times positive == negative

Card deducted four times

Siti taps her stored-value card on 4 bus rides. Each ride deducts $1.50\$1.50, recorded as −1.50-1.50. What is the total change to her card balance?

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Four deductions of the same size: 4×(−1.50)4 \times (-1.50).
Positive ×\times negative == negative.
4×1.50=64 \times 1.50 = 6, so 4×(−1.50)=−64 \times (-1.50) = -6.
Her balance changed by −$6-\$6, that is $6\$6 less than before.

Sharing a loss equally

Three business partners share a loss of $270\$270 equally. How much does each partner lose?

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Total loss =−270= -270.
Each partner: (−270)÷3=−90(-270) \div 3 = -90.
Each partner loses $90\$90.

Finding the unknown number (reverse problem)

A number is multiplied by −4-4 and then 77 is added, giving −13-13. Find the number.

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Work backwards. The last step added 77, so undo it: −13−7=−20-13 - 7 = -20.
The step before multiplied by −4-4, so undo that: (−20)÷(−4)=5(-20) \div (-4) = 5.
The number is 55.
Check: 5×(−4)=−205 \times (-4) = -20, and −20+7=−13-20 + 7 = -13.

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