MULTIPLICATION
To explain the rules for multiplication of signed
numbers, we recall that multiplication of
whole numbers may be thought of as shortened addition. Two types of
multiplication problems must be examined; the
first type involves number8 with unlike signs, and the second involves numbers
with like signs.
Unlike Signs
Consider the example 3(4), in which the multiplicand
is negative. This means we are to add 4
three times; that is, 3(4) is equal to (4)
+ (4) + (4), which is equal to 12. For example,
if we have three 4dollar debts, we owe 12
dollars in all.
When the multiplier is negative, as in 3(7), we
are to take away 7 three times. Thus, 3(7)
is
equal to (7)  (7)  (7) which is equal to 21. For
example, if 7 shells were expended in one firing,
7 the next, and 7 the next, there would be a
loss of 21 shells in all. Thus, the rule is as
follows: The product of two numbers with unlike
signs is negative.
The law of signs for unlike signs is sometimes stated as follows: Minus times
plus is minus; plus times minus is minus.
Thus a problem such as 3(4) can be reduced
to the following two steps:
1. Multiply the signs and write down the sign
of’ the answer before working with the numbers
themselves.
2. Multiply the numbers as if they were unsigned numbers.
Using the suggested procedure, the sign of the
answer for 3(4) is found to be minus. The product
of 3 and 4 is 12, and the final answer is
12. When there are more than two numbers to
be multiplied, the signs are taken in pairs until
the final sign is determined.
Like Signs
When both factors are positive, as in 4(5), the
sign of the product is positive. We are to add
+5 four times, as follows:
4(5) = 5 + 5 + 5 + 5 = 20
When both factors are negative, as in 4(5), the
sign of the product is positive. We are to take
away 5 four times.
Remember that taking away a negative 5 is the same
as adding a positive 5. For example, suppose
someone owes a man 20 dollars and pays him
back (or diminishes the debt) 5 dollars at a
time. He takes away a debt of 20 dollars by
giving him four positive 5dollar bills, or a total
of 20 positive dollars in all.
The rule developed by the foregoing example is
as follows: The product of two numbers with like
signs is positive.
Knowing that the product of two positive numbers
or two negative numbers is positive, we can
conclude that the product of any even number of negative numbers is positive.
Similarly, the product of any odd number of
negative numbers is negative.
The laws of signs may be combined as follows: Minus times plus is minus; plus
times minus is minus; minus times minus is
plus; plus times plus is plus. Use of this
combined rule may be illustrated as follows:
4(2)  (5)  (6)  (3) = 720
Taking the signs in pairs, the understood plus on
the 4 times the minus on the 2 produces a minus.
This minus times the minus on the 5 produces
a plus. This plus times the understood plus on the 6 produces a plus. This plus
times the minus on the 3 produces a minus, so
we know that the final answer is negative. The
product of the numbers, disregarding their signs,
is 720; therefore, the final answer is 720.
Practice problems. Multiply as indicated:
1. 5(8) = ?
2. 7(3) (2) = ?
3. 6(1)(4) = ?
4. 2(3)(4)(5)(6) = ?
Answers:
1. 40
2. 42
3. 24
4. 720
DIVISION
Because division is the inverse of multiplication, we can quickly develop the
rules for division of signed numbers by
comparison with the corresponding
multiplication rules, as in the following
examples:
1. Division involving two numbers with unlike signs is related to
multiplication with unlike signs, as follows:
Therefore,
Thus, the rule for division with unlike signs is: The
quotient of two numbers with unlike signs is
negative.
2. Division involving two numbers with like signs
is related to multiplication with like signs, as
follows:
3(4) =
12
Therefore,
Thus the rule for division with like signs is: The
quotient of two numbers with like signs is
positive.
The following examples show the application of
the rules for dividing signed numbers:
Practice problems. Multiply and divide as indicated:
Answers:
1. 3
2. 1
3. 2
4. 9
