Are you confused?
Suppose you see an expression where you have to divide repeatedly, with no parentheses telling you which
division to do first. Here is an example:
200/2/5/4
In a case like this, proceed from left to right. Here, that means you should take 200 and divide it by 2,
getting 100. Then divide 100 by 5, getting 20. Finally, divide 20 by 4, getting 5.
Here’s a challenge!
Under what circumstances can we divide an integer a by an integer b, and get the same quotient (or ratio)
as when we divide b by a? Assume that a≠ 0 and b≠ 0.
Solution
We have two integers a and b, and we are told that
a/b=b/a
This equation is always true when a and b are the same. In that case, we can substitute a for b and get
a/a=a/a
which simplifies to1 = 1. That’s trivial!
Now suppose that a and b are additive inverses. Therefore, −a=b. Here, we can substitute −a for b in
the original equation and get
a/(−a)= (−a)/a
which simplifies to − 1 =−1. We know this because it is an application of one of those two facts we’re sup-
posed to have “homogenized” earlier in this chapter.
What’s the verdict? If we have two nonzero integers a and b whose absolute values are the same, meaning
that they’re either identical or are additive inverses of each other, then a/b=b/a. In symbols, we can write
If |a|= |b|, then a/b=b/a
The Associative Law for Multiplication
Another important rule that applies to multiplication involves how the factors are grouped
when you have three or more of them. You can lump the factors together any way you want,
and you’ll always end up with the same product. The simplest case, called the associative law
for multiplication, involves a product of three factors.
The Associative Law for Multiplication 75
