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

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Homework Help Overview

The discussion revolves around the proof of the statement that 0! equals 1, a concept often accepted without question in mathematics. Participants explore the definition of factorial and its implications in various contexts, including combinatorial interpretations and recursive relationships.

Discussion Character

  • Conceptual clarification, Assumption checking, Exploratory

Approaches and Questions Raised

  • Some participants discuss the recursive definition of factorial and its application to 0!. Others question the necessity of proving this statement, suggesting it is a matter of definition. Several participants explore the implications of defining 0! as 1 in mathematical formulas and functions.

Discussion Status

The discussion is active, with various interpretations and approaches being explored. Some participants have provided insights into the gamma function as a means to understand the factorial definition, while others emphasize the importance of definitions in mathematics. There is no explicit consensus, but multiple lines of reasoning are being examined.

Contextual Notes

Participants note that the definition of factorial is often extended beyond positive integers, raising questions about the foundational axioms and definitions used in mathematics. The discussion includes references to combinatorial reasoning and the need for consistency in mathematical equations.

  • #31


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?
 

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