Algebra Know-It-ALL

Here’s another challenge!

Based on the commutative law for multiplication of two integers, and on the associative law for multiplica-

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

abc=cba

Solution

This works out just like the “challenge” at the end of Chap. 4. Simply change all the instances of addition

to multiplication. Table 5-2 is an S/R proof.

The Distributive Laws

When you come across a sum or difference that is multiplied by a single number or variable,

you will sometimes want to expand it into a sum or difference of products. To do this, you

can use the distributive laws.

Multiplication over addition

Suppose you have three integers a, b, and c arranged so that you must multiply a by the sum of

b and c. The left-hand distributive law of multiplication over addition tells you that this is equal

to the product ab plus the product bc. This comes out simpler if you write it down:

a(b+c)=ab+ac

Now imagine multiplying the sum of a and b by c. The right-hand distributive law of

multiplication over addition says that

(a+b)c=ac+bc

Table 5-2. Here is a proof that shows how you can reverse the order

in which three integers a,b, and c are multiplied, and get the same

product. As you read down the left-hand column, each statement is

equal to all the statements above it.

Statements Reasons

abc Begin here

a(bc) Group the second two integers

a(cb) Commutative law for the product of b and c

(cb)a Commutative law for the product of a and (cb)

cba Ungroup the first two integers

Q.E.D. Mission accomplished

78 Multiplication and Division