Except that no one has yet determined a method to actually perform this function, other than raw guess-and-check. Factorization is actually the basis of many modern cryptography systems for this very reason--it's one of the great unsolved (unsolvable?) problems of mathematics.

Interesting.
However, for any integer, eg 9, the factorial is a legitimate operation. Inverse factorial would only be possible for a very small percentage of numbers, ie those that factorise completely to give consecutive factors right down to 1. Apologies for butchering mathematical terms... I'm only an engineer after all.

people live with this daily, they even invented a number to represent the answers you "can't" get, is there any reason we can't define an answer to 5? or whatever.

In trig, the inverse of a function is denoted by a minus one exponent. But that would be ugly with the factorial.

One problem with ? is that it would be undefined for most numbers. So, an inverse gamma function would be better (and using the negative one exponent with gamma would look fine).

Sammy, what the hell are you talking about? Since factorial is already defined, and AntiQuark just defined the inverse, you can't define an answer to 5? or whatever.

5th Earth, what you say is true for
factorization, but factorials (whose
factors follow a much more regular
pattern than prime factors) surely do
not pose a problem.

Where a function is not reversible, mathmaticians need to invent a new class of numbers to cope. For example square-roots of negative numbers are 'imaginary' numbers. Numbers too small to make a difference are 'infinitesimals', while those too big to count are infinities.

Maybe there should be new classes of numbers for <non-integer>! and <non-factorial>?
I propose 'contrary' and 'pretend' respectively.

You can write Inverse factorial function as n? if you want to. I would use gamma^-1 (n), but that is just ascetics. I do however agree with Idischler using the gamma function to define non integer values would be good (do note that the gamma function is not defined for many non-positive values). Also inverse gamma is not a function (unless you limit the range), however this does not in any way stop one from using it (most clearly if you only define inverse factorial for positive values of x and y). Go to the link I posted above.

//I do however agree with Idischler using the gamma function to define non integer values would be good // And not just non integer, as n? would give you an error message for most integers.

The Gamma function is the obvious choice for an extension of the factorial function. However, as it isn't single valued, its inverse isn't a function at all.

That said, over the real domain [1,infinity], the Gamma function is single valued, and its inverse is therefore also a function.

[ldischler] I didn't know Google did that (cosh i pi). At least it gets it correct! 8~)

[madness] Gamma(i) is defined and its value can be calculated (to any desired accuracy), but InverseGamma(i) may well not exist, or possibly have more than one value - and in any case, it's likely to be extremely difficult to find other than by serendipity. There's no known algorithm to do it, as far as I know.

//There's no known algorithm to do it, as far as I know.// Look at forsooth's link above. He once had a calculator there, but now it's gone. Or at least, I don't see it.

[ldischler] There's no problem with finding an algorithm for InverseGamma(x) where x is real and greater than a minimum (approximately 0.885603194410888), and we're only looking for positive values of InverseGamma. (There are two positive values for InverseGamma(x) for all values of x greater than this minimum, one less than 1.461632.... and one greater than that. There are an infinite number of negative values of InverseGamma(x) for all non-zero real values of x.)

But InverseGamma(x) where x is not real (such as i) is much harder - it's not a function at all (it has zero, one or many values, depending on x) and I don't think there's any general way of finding its values.

Use a (small) look-up tsble - 70! is beyond what most people's calculators can manage - Including Complex or Simplex numbers, it's still not going to be a big LUT

<somewhat irrelevant aside> ! is only
calculable for integers. But presumably
there is some function that gives !
(something like x!=exp(x)/2+x^2 or
something) no? So if you plug a non-
integer x into the equation, would you
not get a quasi-meaningful answer to
"x!"? In effect you are interpolating
between the integers to get a value of x!
for non-integers.

In this case, shirley, x? would always
have a value for positive real x -
wouldn't it?

[MaxwellBuchanan] The gamma function we've been discussing is the function that gives x! for integer x, and a nice continuous function for any positive real x. It gives values for all positive, negative and indeed complex x as well, but it's got poles (infinities) at negative integers. See Wikipedia "Gamma function".

Would we then have to assume a lone ?
at the end of an exam question is an
an actual question, but the presence of
a second ?
would be asking for the inverse
factorial?

//The gamma function we've been
discussing is the function// Thanks for
the pointer. Sometimes I could weep that
there such pretty things out there to which
I will be, forever, mentally blind.