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© Copyright 2003, Jim Loy
A couple of readers told me they didn't see why division by 1/2 is the same as multiplying by 2:
a/(1/2) = 2a
Certainly common speech, "division in half," "division by half," and things like that, encourage confusion. In mathematics, it's pretty simple, however. If we are confused, we go back to the definition. One definition of division is this:
a/b means, "How many times does b go into a?"
If we have 3/(1/2), then what is the answer? How many times does 1/2 go into 3? Obviously, the answer is 6. The process is easy. It involves multiplication, like this 1/2 x ??? = 3. This works for any fraction, not just 1/2, and also including improper fractions like 7/4. It also works when the answer does not come out even.
So, based upon the definition of division, we verify the rule:
x/(a/b) = xb/a
or:
When dividing by fractions, we invert and multiply.
By the way, we can state the above definition of division as this: a/b=c means a=bc. Here we state division as the inverse operation of multiplication. That is what we did above, but in a wordier form.
Addendum:
Almost identical reasoning helps us deal with subtraction of negative numbers. We know that a-(-b) = a+b. We cancel the negative signs, and the reason is the definition of subtraction. a-(-b) asks what do we add to -b, to get a. The answer is obviously a+b.
One reader of my pages wanted to change the rules of arithmetic so that a-(-b) = a-b. Well, then subtraction would no longer be the inverse operation of addition. And the definition of subtraction would actually become more complicated. And all kinds of formulas (in physics and elsewhere) would no longer work.