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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:
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

Example #2:
Is the following product positive or negative?

(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.


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