Algebra Know-It-ALL

56 Addition and Subtraction

Grouping with parentheses

A couple of minutes ago, you saw some expressions containing parentheses around a negative

number or quantity. Here’s one of those expressions, with the right-hand side changed to a

variable:

− 3 − (−5)=a

These parentheses indicate that you should consider −5 as “negative 5,” not “the subtraction

of 5.” If you don’t write the parentheses, the above expression is

− 3 −− 5 =a

This is an example of improper mathematical grammar. It’s a problem because it’s ambigu-

ous. People can’t be sure what it means. Should you subtract 5 from −3? That would give you

−8, not 2. Should you subtract 5 from −3 twice? That would give you −13. Does the double

minus sign mean something different from plain old subtraction or addition?

Pay attention to parentheses when you see them. When you write an expression, be sure

to include parentheses when you need them. You can add extra parentheses to an expression

as long as you don’t change the meaning. Always be sure the number of opening parentheses

in an expression is the same as the number of closing parentheses.

Signs in addition and subtraction

Here’s a summary of how addition and subtraction work for negative integers as well as for

positive ones:

• When you add a positive, the result grows larger.


• When you subtract a positive, the result grows smaller.


• When you add a negative, the result grows smaller; it’s like subtracting a positive.


• When you subtract a negative, the result grows larger; it’s like adding a positive.

You might want to memorize two general facts. For any two integers a and b,

a+ (−b)=a−b

and

a− (−b)=a+b

Are you confused?

Here’s a real-life situation where the idea of subtracting a negative number makes pretty good sense.

Suppose the people in your town are trying to reduce their driving because of high gasoline prices.

They’ve started a support group. People go to meetings once a month to share information about how

much less they’ve driven in the past month compared with the month before that. One after another,

the members of the group tell their stories. William says, “I drove 45 fewer miles.” Anna says, “I cut

back by 65 miles.” Maria says, “I drove 200 miles less last month!” Then it comes to you. You have