Rules of Signs
36. Multiplication and Division of Signed numbers
Rule of signs. In algebra, it is understood that placing the variables next to each other, as ab, or numbers next to a variable, as 3a, or two expressions enclosed in a pair of parentheses, as (+5) (-3) or as a(-6 + (-2) (8 - 4), it means to multiply. Thus, ab means a × b, 3a means 3 × a, and a(-6 + (-2) (8 - 4) means a × (-6 + (-2) × (8 - 4).
The properties of multiplication and division for whole numbers may be extended to the signed numbers. this means that multiplication and division of signed numbers is carried out by thesame process as multiplication and division in arthmetic. However, the product or quotient of two numbers with like signs is positive and the product or quotient oftwo numbers with unlike signs is negative. Using multiplication as repeated addition, we can show on the number line that 3(+6) means 6 + 6 + 6 = 18 and 3(-6) means -6 +(-6) + (-6) = -18.
Thus, we may state that 3(+6) means three positive sixes or 18 and 3(-6) means three negative sixes or -18. Since multiplication and division are inverse operations that undo one another, then the following examples of multiplication of signed numbers demonstrates the validity of the premise that "the product or quotient of two numbers with like signs is positive and with unlike signs negative."
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