Algebra Know-It-ALL

136 Irrational and Real Numbers

Product of signs

When two real numbers with plus signs (meaning positive, or larger than 0) or minus signs

(meaning negative, or less than 0) are multiplied by each other, the following rules apply:

(+)(+)= (+)

(+)(−)= (−)

(−)(+)= (−)

(−)(−)= (+)

Quotient of signs

When two real numbers with plus or minus signs are divided by each other, the following

rules apply:

(+) / (+)= (+)

(+) / (−)= (−)

(−) / (+)= (−)

(−) / (−)= (+)

Power of signs

When a real number with a plus sign or a minus sign is raised to a positive integer power, n,

the following rules apply:

(+)n= (+)

(−)n= (−) if n is odd

(−)n= (+) if n is even

Reciprocal of reciprocal

For all nonzero real numbers, x, the reciprocal of the reciprocal is equal to the original number:

1/(1/x)=x whenx≠ 0

More Rules for Real Variables

Here are some more sophisticated rules for expression morphing. As with the facts in the

previous section, you don’t have to try to memorize these. It can help if you work out a few

examples using numbers in place of the variables. Some of the restrictions here, in which

variables are not allowed to equal 0, are a little stronger than necessary to keep things straight-

forward and to be sure we stay safe!