An Informal Argument for Why 0! = 1

Date: 01/13/2008 at 07:33:30
From: Miggy
Subject: why is 0 factorial 1

I don't understand why 0 factorial is 1.

Date: 01/13/2008 at 10:36:01
From: Doctor Tom
Subject: Re: why is 0 factorial 1

Hi Miggy,

Thanks for writing to Doctor Math!

There are some good technical why 0! = 1, but you may not find any of
them convincing.  You can read about some of them on our Frequently
Asked Questions page:

  Why Does 0! = 1?
    http://mathforum.org/dr.math/faq/faq.0factorial.html 

Here's a more informal argument that might make sense.  People often
have trouble with 0! being 1 because they think that they are
multiplying no numbers, or only 0, so it must be 0.  How can an "empty
product" be worth 1?

Let me go at this by analogy, starting with an "empty sum".  You know
how to add 4 things or 3 things or 2 things, right?  What if you just
add 1 thing?  Shouldn't that just be the thing?  How about zero 
things?  Shouldn't that just be zero?  For example, suppose you have 5
numbers and you divide them between two friends.  What is the sum of
the numbers?  It should be the sum of the numbers A has plus the sum
of the numbers B has, right?  What if you give them all to A? 
Shouldn't the same method work?  Well, that means that you need to say
that the sum of B's zero numbers is 0 so that the total sum of A and B
is still correct.  So we agree that an "empty sum" has a value of 0.

Similarly, suppose you again divide 5 numbers between two friends. 
What is the product of the numbers?  It should be the product of the
numbers A has times the product of the numbers B has.  But if you give
them all to A, you'd better call the product of the zero numbers that
B has 1.  If you call it 0, then multiplying A's product by B's
product will be 0, not the correct total product.  So it makes sense
to agree that an "empty product" must have a value of 1.

So 4! = 4x3x2x1: you're multiplying 4 numbers.  1! = 1: you're
multiplying 1 number.  What should 0! be?  You're multiplying zero
numbers, and as we just saw, that had better be 1, right?

I hope this helps you understand why 0! must be 1.  Please write back
if you have questions about what I've said.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/