Multiplying Integers
Objective Learn how to multiply both positive
and negative integers.
We will be studying multiplication in this lesson.
Multiplication may be the most confusing and nonintuitive of the
operations when applied to negative integers. We will try to demonstrate
how the idea behind multiplication is applied to examples.
Multiplying with Negative Integers
Take a look at the following rules for multiplying negative
integers. Have the definition of opposites in mind.
Key Idea
1. When we multiply a positive integer and a negative integer
together, the result is the opposite of the number we would get
if both integers had been positive.
For example, 2 Ã— ( 3) = (2 Ã— 3) = 6 and  5 Ã— 4 = (5 Ã—
4) = 20.
2. When we multiply two negative integers, the result is the
positive integer we would get if both integers had been positive.
For example, 3 Ã— (4) = 3 Ã— 4 = 12.
In summary, (1) multiplying a positive integer times a
negative integer gives a negative integer, and (2) multiplying
two negative integers gives a positive integer.
Try to answer the following questions in order to make sure
you understood the previous properties.
• Will 3 Ã— (7) be positive or negative?
positive
• Will 457 Ã— ( 325) be positive or negative?
negative
The rules for multiplying two integers can also be remembered
using the phrase same signs, positive product; different signs,
negative product.
Why Are These Rules True?
Multiplying any integer (positive or negative) by a positive
integer is the same as adding “copies” of that integer,
where the number of copies is equal to the positive integer. Multiplication
by a positive integer can be viewed as repeated addition. That
is, 5 Ã— 2 is the same as 2 + 2 + 2 + 2 + 2. Then, 5 Ã— (2) is
then equal to (2) + (2) + (2) + (2) + (2). The following
example uses this thinking.
Example
What is 5 Ã— (3)?
Solution
The product 5 Ã— (3) can be found by adding 5 copies of the
integer 3. So,
5 Ã— (3) = (3) + (3) + (3) + (3) + (3) = 15.
The same answer can be obtained by applying the first rule in
the Key Idea shown earlier.
5 Ã— (3) = (5 Ã— 3) = 15
The idea that the product of two negative integers is positive
is many times harder to understand.
Why is 5 Ã— (3) = 15?
You do not need a formal proof of this fact. Simply notice
that the Example just showed that 5 Ã— (3) = 15. It is
reasonable for the products 5 Ã— (3) and 5 Ã— (3) to be
opposite of each other since they cannot be the same. So 5 Ã— (3)
should be the opposite of 15 or 15.
Try to give reasons for why the following fact is true.
Given any number x , 1 Â· x is the opposite of x. By the
rules for multiplying negative integers, if x is positive, then 1
Â· x is negative and its value is the opposite of the value of 1
Â· x = x . Similarly, if x is negative, then 1 Â· x is positive
and its value is the opposite of the value of 1 Â· x .
The only way you will fully understand the concept of
multiplication of negative integers is if you become skilled at
the calculations.
