Proving 0!=1: An Analysis of the Fundamental Mathematical Concept

Thread starter rock.freak667
Start date Nov 4, 2007
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SUMMARY

The discussion centers on the mathematical proof that 0! = 1, a fundamental concept in combinatorics and calculus. Participants highlight that 0! is defined as 1 to maintain consistency in mathematical equations, particularly in permutations and combinations. The recursive definition of factorial, n! = n * (n-1)!, leads to the conclusion that 0! must equal 1 when n = 1. Additionally, the gamma function, Γ(z), provides a formal basis for this definition, as Γ(1) = 0! = 1.

PREREQUISITES

Understanding of factorial notation and definitions
Familiarity with recursive functions in mathematics
Basic knowledge of the gamma function, Γ(z)
Combinatorial principles, specifically permutations and combinations

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Study the properties and applications of the gamma function, Γ(z)
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Students of mathematics, educators teaching combinatorics, and anyone interested in the foundations of mathematical definitions and proofs.

Jul 13, 2011

Dick

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stefounet said:

If I understand correctly, you define n!=Γ(n+1) for all integer and I define n!=n(n-1)...1. In the scope of my definition, 0! is NOT defined.

Depending on the OP's definition of n!, either the Γ function argument is (obviously) a satisfactory answer to his question, or there is no absolute answer because it is a matter of convention. In this later case, agreed, the Γ function argument would be one among many other justifications of the convenient convention 0!=1.

Considering "The attempt at a solution" of the OP, it seemed obvious to me that he was working in the set of integers, and that he was looking for an (impossible) arithmetic proof. I agree that your analytic definition is on top of the pure arithmeric one, but it does rely on a much heavier set of axioms.

Apologies if I'm starting to sound like a troll :)
Cheers

Everybody is right here. So this is beating an old dead horse. Defining n!=n(n-1)...1 clearly makes no sense if n=0. Making 0! an arbitrary definition. But then the notation n!=n(n-1)...1 is also vague. Best to define it inductively by (n+1)!=n!*(n+1). But where to start the induction? If you start it by defining 1!=1 that works. But you could also start it by defining 0!=1 and get the same function. Clearly there's a problem with defining (-1)! equal to some number and trying to proceed from there. Starting by defining 0!=1 and going inductively from there agrees with the gamma function definition and gives you the usual definition of n!. So why not 0!=1?