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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/
```
Associated Topics:
Elementary Fractions
Middle School Fractions

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