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Why Does 0! = 1 ?


Date: 12/8/95 at 8:10:32
From: Anonymous
Subject: factorial

Why does   0! = 1

A proof suitable for high school would be preferable.

Thanks.


Date: 12/15/95 at 0:47:16
From: Doctor Ken
Subject: Re: factorial

Hello!

Well, you're right that you can't just write down 0! from the 
"definition" of factorial.  In essence, you have to define a value for 
0!, and then stick to it.  Well, it just so happens that 1 is the most 
convenient number for 0! to be.  Here are a couple of reasons why.

Have you seen Pascal's Triangle?  It's a pretty important thing in math, 
and we have some pages about it if you want to look through our 
archives.  Anyway, the Nth number in the Mth row is given by the 
"choose" formula, also known as the binomial coefficient:  

              M!
           --------
           N!(M-N)!


And the first and last (when N = 0 and N = M) numbers are 0.  So to 
make everything work out right (it's actually very important to math 
that everything works out right here, since this is such an important 
formula), we choose 0! = 1.

There's another reason, too.  It turns out that there's a very 
complicated function called the Gamma function, and it's the same as 
the factorials except that you can plug non-integers into the Gamma 
function.  So you can do things like 3.5! by finding Gamma(3.5).  
Well, it turns out that if you plug in Gamma(0), you get 1.  Pretty neat, 
huh?

- Doctor Ken,  The Geometry Forum

    
Associated Topics:
High School Number Theory

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