Why is Zero Factorial Equal to One?

To understand why 0! = 1, we can begin with the fundamental definition of factorials. Through logical reasoning and mathematical proofs, we arrive at a clear conclusion.

Response 1: The factorial function is defined as n! = n(n-1)!. If we apply this to n = 0, we have 0! = 0(-1)!, which is undefined since we can't determine (-1)!. However, looking at the case for 1, we find:

1! = 1(1-1)! = 1(0!) = 1.

Thus, it follows that 0! must indeed equal 1.

Response 2: Some argue that while you mention the gamma function, which extends the concept of factorials, we find that ?(0) is not defined. We've previously established (in the article) that ?(1) = 1 through integration by parts and shown that ?(n+1) = n!. Therefore, by substituting n = 0, we get ?(1) = 0!, confirming that 0! = 1, even though ?(0) remains undefined since the gamma function converges only for Re(z) > 0.

Response 3: There’s a notion that mechanical iterations of the factorial formula lead to contradictions, suggesting factorials only exist for positive integers. However, the correct relation for transitioning from n! to (n-1)! is through n!/n, as opposed to n!/(n-1), which clarifies that our reasoning remains intact.

Response 4: A claim has been made that zero is not a number, and therefore 0! cannot exist. This topic warrants its own article for further exploration.

Response 5: Some remain unconvinced. The definition of factorial is that it represents the product of all integers less than or equal to the original number. Since there are no integers less than zero, yet zero is still a number, the only arrangement possible is that of zero itself, which leads to the conclusion that 0! must equal 1.

An example from combinatorics illustrates this further with the formula nCr = n!/(n-r)!r!. If we apply this to 5C5, we arrive at:

5C5 = 5!/(5-5)!5! = 5!/(0!)5!. If 0! were undefined, we would contradict Pascal’s Triangle because 5C5 = 1. Therefore, we must conclude that 0! = 1. This consistency across mathematical disciplines reinforces our understanding of zero factorial.

I hope this clarifies the concept! Thank you for your engagement. If you have any further insights or suggestions, feel free to share; I appreciate all feedback.

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