Algebra Know-It-ALL

and

a/1=a

The sign-changing element

When you multiply or divide any integer by −1, you reverse the sign but do not change the
absolute value. A positive integer becomes negative, and a negative integer becomes positive,
but the distance from 0 on the integer line stays the same. For any integer a

a(−1)=−a

and

a/(−1)=−a

Conversely,

−a(−1)=a

and

−a/(−1)=a

Note that a(−1) here means a times −1, not a minus 1. Those parentheses are important! The inte-
ger−1 can be called the multiplicative sign-changing element or the divisive sign-changing element.

Parentheses in simple products and quotients

Look at these expressions:

3 × (−5)=a

and

15/(−3)=b

If you don’t write the parentheses, the above expressions are

3 ×− 5 =a

and

15/− 3 =b

Expressions like these might be clear enough to you, but they would confuse some people.
Don’t be afraid to add parentheses to an expression if you think they will prevent ambiguity.
Just be sure that for every opening parenthesis you put in, you include a corresponding closing
parenthesis later in the expression.

Identity, Grouping, and Signs 71