- #1
quasi426
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Why is zero factorial equal to 1?
___________________________"Man has pondered
Since time immemorial
Why 1 is the value
Of zero-factorial."
Joffe said:Is this the correct definition of factorial, or is it inconsistant with 0! ?
[tex]x! = \prod_{n=1}^{x} n[/tex]
Quadratic said:For instance, if 0! was anything other than 1, the cosine function wouldn't make any sense. Consider: f(x) = Cos(x) = x^0/0! + x^2/2! -x^4/4!... where x is a radian measure. So, if 0! was not equal to 1, then the first term in the series would not equal 1, and the Taylor series that derived it would be wrong, which would turn everything that we know about math upside down.
Quadratic said:For instance, if 0! was anything other than 1, the cosine function wouldn't make any sense. Consider: f(x) = Cos(x) = x^0/0! + x^2/2! -x^4/4!... where x is a radian measure.
benorin said:We should pin this thread, or one of the numerous others like unto it, to the top of the forum. Then again, why? After all: I dig the gamma function.
The definition of factorial is based on the concept of multiplying a number by all of the positive integers that are smaller than it. In the case of zero, there are no positive integers that are smaller, so by definition, the result is 1.
The value of zero factorial is not derived, but rather defined as 1 based on the concept of factorial and the mathematical rules for multiplying by zero.
The value of zero factorial being equal to 1 is significant in various mathematical calculations, such as in the binomial theorem, combinatorics, and probability. It also helps simplify equations and makes them more elegant.
While it may seem counterintuitive at first, the concept of zero factorial can be understood by thinking of it as the number of ways to arrange zero objects. Since there is only one way to arrange zero objects (i.e. not arranging them at all), the result is 1.
Yes, there are various mathematical proofs that explain why zero factorial is defined as 1, including using the definition of factorial, mathematical induction, and the gamma function. These proofs show that any other value for zero factorial would cause inconsistencies in mathematical equations and therefore, 1 is the most logical and consistent choice.