Algebra Know-It-ALL

It works when you multiply

Here’s how a mathematician would formally state the associative law in its most basic form.

For any three integers a, b, and c

(ab)c=a(bc)

For instance:

(3× 5) × 2 = 15 × 2 = 30

and

3 × (5 × 2) = 3 × 10 = 30

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

than three numbers in a product, as long as there aren’t infinitely many.

It fails when you divide

In division, as in subtraction, the way in which you group the integers or variables is impor-

tant. Consider this:

(16/4)/2= 4/2 = 2

but

16/(4/2)= 16/2 = 8

For any three integers a, b, andc, it is not necessarily true that

(a/b)/c=a/(b/c)

The associative law hardly ever works with division.

Are you confused?

Look at another expression where we have to divide more than once, and see what happens when we insert

parentheses in different places. Let’s try this:

4,000/40/10/5

Going straightaway from left to right, we get 4,000/40 = 100, then 100/10 = 10, and finally 10/5 = 2.

Now let’s think of it this way:

(4,000/40) / (10/5)

76 Multiplication and Division