2.3Multiplication and Division of Integers

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

Each step down adds

3

, so the line after

0

has to be

+3

$(-3) \times 3 = -9$

$(-3) \times 2 = -6$

$(-3) \times 1 = -3$

$(-3) \times 0 = 0$

$(-3) \times (-1) = \;?$

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

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 $\displaystyle \$1.50$ , recorded as $\displaystyle -1.50$ . What is the total change to her card balance?

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Four deductions of the same size:

4 \times (-1.50)

Positive

\times

negative

=

negative.

4 \times 1.50 = 6

, so

4 \times (-1.50) = -6

Her balance changed by

-\$6

, that is

\$6

less than before.

Sharing a loss equally

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

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Total loss

= -270

Each partner:

(-270) \div 3 = -90

Each partner loses

\$90

Finding the unknown number (reverse problem)

A number is multiplied by $\displaystyle -4$ and then $\displaystyle 7$ is added, giving $\displaystyle -13$ . Find the number.

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Work backwards. The last step added

7

, so undo it:

-13 - 7 = -20

The step before multiplied by

-4

, so undo that:

(-20) \div (-4) = 5

The number is

5

Check:

5 \times (-4) = -20

, and

-20 + 7 = -13

Now you try

Evaluate $8 \times (-10)$ .

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Positive

\times

negative

=

negative.

Ignore the signs and multiply:

8 \times 10 = 80

\therefore 8 \times (-10) = -80

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