## Multiplication of Signed Numbers

The rules for multiplying signed numbers may be formulated from the fact that multiplication serves as a shorthand notation for addition. For example, 4 x (−3), which means "4 times negative −3" is the same as the following:

(−3) + (−3) + (−3) + (−3) = −12

As (4)(−3) = −12 and the order of factors in multiplication does not matter, it follows that (−3)(4) = −12. Next, we will examine the product of (−4)(−3). It has already been seen that (4)(3) = +12 and (4)(−3) = −12, which resulted in opposite answers. Consequently, since 4 times −3 equals −12, the product resulting from −4 times −3 should be the opposite of −12, which is +12. Therefore:

(−4)(−3) = +12

From the above example, it becomes apparent that:

- Mulitplication of two numbers of different signs results in a negative number
- Multiplication of two numbers of the same sign results in a positive number

For multiplication of several signed numbers, it may be shown that:

- Product will be positive if there is an even number of negative numbers or zero negative numbers
- Product will be negative if there is an odd number of negative numbers

Example #1:

Solution #1:

Example #2:

Solution #2:

Evaluate (2)(−3)(−1)(2)(−2)

Solution #1:

Since
there are three negative numbers, the product
will be negative. Therefore, performing the multiplication and
inserting a negative sign in front of the answer results in:

(2)(−3)(−1)(2)(−2) = −24

(2)(−3)(−1)(2)(−2) = −24

Example #2:

Is the following product positive or negative?

(122)(−33)(21)(−162)(322)(−31)(−112)(443)

(122)(−33)(21)(−162)(322)(−31)(−112)(443)

Solution #2:

Since
there are four negative numbers, the
product will be positive. We do not need to multiply the numbers
in order to determine the product will be positive.