Denominator = 2 or 1/2

Date: 01/16/97 at 13:17:47
From: Ray
Subject: Fractions

Dear Dr. Math,

I think I understand fractions, and how denominators and numerators
work, but there is still one problem.  My teacher says that when 
the denominator is 2, that means I cut the pie (or whatever) in 2. 
Same with 3 or 4, or any number.  

My teacher also says that the denominator can be another fraction! 
She says that if the denominator is 1/2, then the pie is multiplied 
by 2.  I think she's from Mars!  Can you help me?

Date: 01/20/97 at 11:20:11
From: Doctor Toby
Subject: Re: fractions

What you have here are two different concepts (very closely related to 
each other) which are both being called `fractions'.

One concept is that of a *rational number*. This is something whose 
numerator and denominator are both integers (ordinary numbers: 
positive, negative, or - in the numerator - zero). This is what people 
usually mean when they say `fraction'. To get 3/5 of a pie, you cut it 
into 5 pieces and take 3 of the pieces.

The other concept is that of *division*. You probably already know 
about division; it's one of the four basic operations, along with 
addition, subtraction, and multiplication. People often write division 
like `25 @ 5';  except instead of `@' they use a symbol like this:
                              .
                            _____

                              .

But that is not the only way to write division! Another way people 
write 25 divided by 5 is as a fraction, with 25 in the numerator and 
5 in the denominator. So when your teacher is talking about 1/2 in the 
denominator, she's not talking about cutting a pie into pieces any 
more; she's talking about dividing something by 1/2.

Cutting a pie and dividing numbers are different concepts, but they 
use the same notation. 

You might wonder why this confusing state of affairs exists. But when 
you think about it, it actually makes some sense.

Suppose you see `3/5' written down somewhere. Does this mean the 
rational number 3/5 or does it mean 3 divided by 5? You might spend 
all day trying to decide, but it doesn't really matter; the answer to 
3 divided by 5 IS the rational number 3/5! These two kinds of 
fractions, rational numbers and division, are really the same thing 
being thought about in different ways. So it makes sense that they use 
the same notation.

Now you know that your teacher really meant dividing by 1/2 when she 
talked about 1/2 being in the denominator: `If the denominator is 1/2, 
then the pie is multiplied by 2' just means that dividing by 1/2 is 
the same as multiplying by 2.

If you don't understand this, think about what division means;
37 (for example) divided by 1/2 is whatever number you have to 
multiply by 1/2 in order to get 37. If you get 37 when you multiply 
this mystery number by 1/2, the number must have been twice as big as 
37 to begin with. That's why dividing by 1/2 is the same as 
multiplying by 2.

-Doctor Toby,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/