- 208
- 0
Why is zero factorial equal to 1?
___________________________"Man has pondered
Since time immemorial
Why 1 is the value
Of zero-factorial."
This is the* definition when x is a positive integer. With the usual convention that an empty product is 1, it is consistant with our usual definition for 0! when x=0. However, x=0 is usually left as a special case and explicitly defined as 0!=1.Joffe said:Is this the correct definition of factorial, or is it inconsistant with 0! ?
[tex]x! = \prod_{n=1}^{x} n[/tex]
It would just change how we express the power series, it would not have any effect whatsoever on the mathematics. This is the same situation as adopting the convention x^0=1 for x=0, without this your power series as expressed doesn't make sense either. This is not a mathematically compelling reason for either convention/definition, rather a notational one.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.
Step back, take a breath and try to think what the person who first came up with such Taylor/McLaurin series might have done if 0! weren't 1, because surely he or she wouldn't have used something that was undefined or unsuitable?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.
And I'm sure the gamma function is fascinated by you.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.