Algebra Know-It-ALL

62 Addition and Subtraction

We can use the associative law for addition to get

a+ [(−b)+ (−c)]

We can simplify this to

a+ (−b−c)

Now we know that when both operations are subtraction,

(a−b)−c=a+ (−b−c)

Are you confused?

We haven’t started to dissect the anatomy of multiplication yet. That will come in the next chapter. But

you’ve had basic multiplication in your arithmetic classes, so let’s “cheat” for a moment and take advantage

of that. What are you actually doing when you change c to −c? Here’s an alternative to the “number reflec-

tor” idea. When you want to find the negative (also called its additive inverse) of any integer, multiply by

−1. It works like this:

c × (−1)=−c

and

−c × (−1)=c

Here’s a challenge!

Based on the commutative law for the sum of two integers and the associative law for the sum of three

integers, show that for any three integers a, b, and c

a+b+c=c+b+a

Solution

If you can manipulate the left-hand side of this equation to get the expression on the right-hand side, that’s

good enough. Because these statements are not very complicated and the proof is not too hard, you can

write it as a table with statements on the left and reasons on the right. Table 4-1 shows how it’s done. This

is a simple statements/reasons (S/R) proof.