Algebra Know-It-ALL

It works when you multiply

The fact that you can multiply two integers in either order and get the same result is called the

commutative law for multiplication. For any two integers a and b

ab=ba

If both factors have the same sign (positive or negative), the product is positive. If the factors

have opposite signs, the product is negative. The commutative law also works if there are three

or more factors. You can rearrange them in any order you want. If there are an even number

of negative factors, the product is positive. If there are an odd number of negative factors, the

product is negative.

It fails when you divide

When you divide an integer by another integer, the order is important. If you divide 20 by 4,

you get 5. But if you divide 4 by 20, you don’t get 5. You don’t even get an integer. A more

dramatic example is the division of 0 by any other integer. If you divide 0 by −3, you get 0.

But if you divide −3 by 0, you get an undefined quantity!

Cross-multiplication

Here’s a fact that you will find useful in algebra. It’s called the rule of cross-multiplication. Sup-

pose you have two ratios of integers, a/b and c/d, and you’re told that they’re equal. You’re also

assured that neither b nor d is equal to 0. You write

a/b=c/d

You can multiply the dividend on the left-hand side of this equation (here, that’s a) by

the divisor on the right-hand side (d), and get the same result as when you multiply the

divisor on the left-hand side (b) by the dividend on the right-hand side (c). In formal

terms,

If a/b=c/d, then ad=bc

This rule works in the reverse sense, too. For any four integers a, b, c, and d, where b is not

equal to 0 (written b≠ 0) and d≠ 0,

If ad=bc, then a/b=c/d

When a rule works in both logical directions, mathematicians use the expression “if and only

if ” and abbreviate it as “iff.” Now we know that for any four integers a, b, c, and d,where

b≠ 0 and d≠ 0,

ad=bc iff a/b=c/d

74 Multiplication and Division