Algebra Know-It-ALL

58 Addition and Subtraction

but

5 − 3 = 2

In formal terms we would say that for any two integers a and b, it is not always true that

a−b=b−a

In fact, it is almost never true. It only works if a and b happen to be the same.

Turning it inside-out

We can use a trick that will make the commutative law “sort of work” with subtraction. This

trick is often used by accountants who must work with long columns of credits (money added)

mixed with debits (money taken away). This trick involves taking every subtraction and turn-

ing it into the addition of a negative number. Remember that adding a negative is the same as

subtracting a positive. That is, for any two integers a and b,

a−b=a+ (−b)

Because the right-hand side of this equation is an addition problem, we can apply the com-

mutative law and get

a+ (−b)=−b+a

Now we can combine the above two equations into a three-way equation:

a−b=a+ (−b)

=−b+a

Then we can get rid of the middle term and write

a−b=−b+a

Let’s call this the “inside-out commutative law for subtraction.” That’s not a formal name, but

you might find it useful as a memory aid.

Are you confused?

Imagine that you have a checking account and your balance on January 1 was exactly $700. Consider

that as a “starting deposit.” By the end of June, you’ve made 15 deposits and written 20 checks. You want

to figure out your balance as of June 30. You convert all the checks to “negative deposits.” For example,

a check for $25 becomes a “negative deposit” of −$25. Now you can add all the “positive deposits” and

“negative deposits” in any order, and you’ll always end up with the same final balance—if you don’t make

any calculation errors!