Algebra Know-It-ALL

Signs in multiplication and division

Here is a set of rules for multiplication and division by any integer except −1, 0, or 1. Remem-

ber that “more” always means more positive, or moving upward on the integer line, and “less”

always means more negative, or moving down. Study these rules carefully. If any of them

confuse you, “plug in” some actual numbers and test them.

• When you multiply a positive integer by 2 or more, the result stays positive and the
  absolute value increases (it gets farther from 0).


• When you divide a positive integer by 2 or more, the result stays positive and the abso-
  lute value decreases (it gets closer to 0).


• When you multiply a negative integer by 2 or more, the result stays negative and the
  absolute value increases (it gets farther from 0).


• When you divide a negative integer by 2 or more, the result stays negative and the
  absolute value decreases (it gets closer to 0).


• When you multiply a positive integer by −2 or less, the result becomes negative and
  the absolute value increases (it gets farther from 0).


• When you divide a positive integer by −2 or less, the result becomes negative and the
  absolute value decreases (it gets closer to 0).


• When you multiply a negative integer by −2 or less, the result becomes positive and
  the absolute value increases (it gets farther from 0).


• When you divide a negative integer by −2 or less, the result becomes positive and the
  absolute value decreases (it gets closer to 0).

“Homogenize” these!

You would do well to memorize (or, as my dad would always say, “homogenize”) two general

facts. For any two integers a and b

a× (−b)=−ab=−(ab)

and

a/(−b)=−a/b=−(a/b)

Are you confused?

Suppose you see an expression with addition, subtraction, multiplication, and division all mixed up, but

with no parentheses. Here’s an example.

2 + 48/4 × 6 − 2 × 5 + 12/2 × 2 − 5

You’ll get an answer that depends on which operations you do first. Do not approach a mixed-operation

problem like this by simply grinding out the arithmetic from left to right. You must use certain rules of

precedence. In this order:

• Group all the multiplications


• Do all the multiplications from left to right

72 Multiplication and Division