What is the rationale for defining 0 factorial to be 1?

Answer 1

What is the rationale for defining 0 factorial to be 1?
Answer
The defining 0 factorial to be 1 is not a rationale.
"Why is zero factorial equal to one?" is a problem that one has to prove.
When 0 factorial to be 1 to be proved,
the defining 0 factorial to be 1 is unvaluable.
One has only one general primitive definition of a factorial number:
n! = n x (n-1) x (n-2) x (n-3) x ... x 2 x 1.
After that zero factorial denoted 0! is a problem that one has to accept
by convention 0!=1 as a part of definition.
One has to prove zero factorial to be one.
Only from the definition of a factorial number and by dividing both sides
by n one has: n!/n (n-1)! or (n-1)! = n!/n
when n=2 one has (2-1)! = 2!/2 or 1! = 2x1/2 or 1! = 1
when n=1 one has (1-1)! = 1!/1 or 0! = 1/1 or 0! = 1. =
This is a proof that zero factorial is equal to one to be known.
But a new proof is:
A Schema Proof Without Words
That Zero Factorial Is Equal To One.

... ... ...








Now the expression 0! = 1 is already a proof, not need a definition
nor a convention. So the defining 0 factorial to be 1 is unvaluable.
The proof "without words" above
that zero factorial is equal to one is a New that:
*One has not to accept by convention 0!=1 anymore.
*Zero factorial is not an empty product.
*This Schema leads to a Law of Factorial.
Note that the above schema is true but should not be used in a formal proof for 0!=1.
The problem arises when you simplify the pattern formed by this schema into a MacLauren Series, which is the mathematical basis for it in the first place. Upon doing so you arrive with,

. This representation illustrates that upon solving it you use 0!.
In proofs you cannot define something by using that which you are defining in the definition. (ie) 0! can't be used when solving a problem within a proof of 0!.
For clarification, the above series will represent the drawn out solution for the factorial of a number, i. (ie) 1×76 -6×66 +15×56 -20×46 +15×36 -6×26 +1×16 , where i=6.