Algebra Know-It-ALL

60 Addition and Subtraction

This works whether the numbers are positive, negative, or 0. It also works if there are four or

more numbers in a sum.

It fails when you subtract

In subtraction, the grouping does matter. It’s easy to see why you can’t apply the associative law

to subtraction and get away with it. Consider this:

(3 − 5) − 7 =− 2 − 7 =− 9

but

3 − (5 − 7) = 3 − (−2)

= 3 + 2 = 5

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

(a−b)−c=a− (b−c)

The associative law hardly ever works with subtraction.

Mixing the signs

How about mixed addition and subtraction? Let’s try it with some integers.

(3 + 5) − 7 = 8 − 7 = 1

and

3 + (5 − 7) = 3 + (−2)= 1

It works in this case. What happens if we switch the positions of the signs?

(3 − 5) + 7 =− 2 + 7 = 5

but

3 − (5 + 7) = 3 − 12 =− 9

This time, it fails! Now we know that with mixed signs, the associative law sometimes works,

but not always. If something is to be called a law in mathematics, “sometimes” does not suf-

fice. “Usually” won’t do the job either. Even “almost always” is not good enough. In order to

be a law, something has to work all the time.

It’s easy to prove that a law does not hold in every possible case. You only have to find one

case where it fails, called a counterexample, to show that something can’t be called a law. Prov-

ing that a law always works is more difficult. You can’t do it using specific integers, because

you’d have to try an infinite number of cases one at a time. You have to use airtight logic.

That’s what proofs are all about.