It seems logical that the inverse of the factorial function, '!', should be '?'.
Then you would have meaningful identities like:
9! = 362880
362880? = 9
9?! = 9
362880?!?!?? = 9
3! = 6 = 720?
1! = 1? = 1?!??!?!!?
It would really clear up mathematics, especially for the novice.-- AntiQuark, Oct 28 2004 The Gamma function and the factorial http://en.wikipedia...wiki/Gamma_function [ldischler, Oct 28 2004, last modified Jun 06 2007] Minkowski's Question Mark Function http://mathworld.wo...onMarkFunction.html [ldischler, Oct 31 2004] Inverse Factorial http://forsooth42.g...%A7inversefactorial [forsooth, Jun 05 2007] e^(pi*i) http://xkcd.com/c179.html [bleh, Jun 06 2007] MathOverflow Inverse Gamma Function http://mathoverflow...12eb79cabeb39f955e3Scroll down to see a Python solution. [kwhitefoot, Jan 08 2014] 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.-- 5th Earth, Oct 28 2004 Just nod and smile [htj], just nod and smile.-- harderthanjesus, Oct 28 2004 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.-- david_scothern, Oct 28 2004 I really like this!? = I really like this
Croissant! for you.-- wagster, Oct 28 2004 "It would really clear up mathematics, especially for the novice." Sure about that, now?
3!=6 2!=2 1!=1 0!=1 (by definition)
1?=...
Careful, now. Not all functions have easy and exact inverses. Now that's an important lesson to learn early on in maths.-- st3f, Oct 28 2004 2^2 = 4 1^2 = 1 0^2 = 0 -1^2 = 1 2^2 = 4
sqrt(1) = ?
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.-- SammyTheSnake, Oct 28 2004 Would this allow me to divide by zero without fear of the ground opening up and swallowing me whole?-- vigilante, Oct 28 2004 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).-- ldischler, Oct 28 2004 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.-- yabba do yabba dabba, Oct 28 2004 Mathematician: "The answer is 9." Student [incredulously]: "Did you say 9???!!!!?!?!?!?" Mathematician: "That's right"-- phundug, Oct 28 2004 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.-- jutta, Oct 28 2004 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.-- Loris, Oct 28 2004 [Steve] The gamma function (it's the Greek gamma, but I'll use G) is the generalized factorial. You can input any number, integer or not.x! = G(x+1) See link.-- ldischler, Oct 28 2004 Cool link, Steve. It seems to work, except it goes to zero for anything less than .8957. (Probably where the function becomes two valued.)-- ldischler, Oct 31 2004 Jutta, you're right. I was misremembering how the ! function works.-- 5th Earth, Nov 01 2004 God but you people make my head hurt sometimes.Keep up the good work.-- 2 fries shy of a happy meal, Nov 01 2004 9?! = 9 is not valid. In order to perform the ? function, the operand must be a valid factorial.
All full factorials above 1 are even numbers. 9 is not even, therefore it is not a valid full factorial.
9!? = 9, however, would be valid.-- Freefall, Nov 01 2004 [Freefall], wouldn't a!? = a? !just be defined as an identity?-- RobertKidney, Nov 03 2004 a!? = a?! only where a? is defined.-- Freefall, Nov 03 2004 What about ? of odd numbers or negative numbers (or ! of negative numbers which my calculator claims to be ERROR 2)?-- chud, Nov 04 2004 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.-- forsooth, Jun 05 2007 42-- normzone, Jun 05 2007 sp. aesthetics, forsooth, [forsooth].-- pertinax, Jun 06 2007 I heard that the exclamation point and the question mark had a romantic fling and a few weeks later, the exclamation point missed her period.-- janbest, Jun 06 2007 [phundug] //Mathematician: "The answer is 9."
Student [incredulously]: "Did you say 9???!!!!?!?!?!?"
Mathematician: "That's right"//
Does this help:
? = Both eyebrows up
! = Furrowed brow
(Explains why Spock often had one brow-up and the other down)-- Dub, Jun 06 2007 //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.-- ldischler, Jun 06 2007 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.-- Cosh i Pi, Jun 06 2007 // a!? = a?! only where a? is defined [freefall]
By defined you mean reducible --- this is a bit harsh...
What about i?-- madness, Jun 06 2007 Interesting. If you search cosh i pi in Google, it will calculate the answer (lowercase only). But it doesn't seem to do the gamma function.-- ldischler, Jun 06 2007 [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.-- Cosh i Pi, Jun 06 2007 i am sqrt cosh i pi
whoami-- madness, Jun 06 2007 [madness] The implication of that is that I am u². Also that u r imaginary, and I am less than nothing.-- Cosh i Pi, Jun 06 2007 //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, Jun 06 2007 [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.-- Cosh i Pi, Jun 06 2007 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-- Dub, Jun 06 2007 [cosh i pi] Indeed, Athough it is more fun to say that u are imaginary and I am less than nothing... (or is that the other way around?)-- madness, Jun 06 2007 [madness] Good point. Edited.-- Cosh i Pi, Jun 06 2007 <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, Jun 06 2007 [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".-- Cosh i Pi, Jun 07 2007 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?
i.e
1) what is x!?
x*(x-1)*(x-2)*...3*2*1
2)what is x!??
x
3) what is x!?!?
4) what is x!?!??
x-- bleh, Jun 07 2007 //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.-- MaxwellBuchanan, Jun 07 2007 ¿How about using an inverted exclamation mark ¡ instead?-- xaviergisz, Jun 07 2007 I like that better, less confusing.-- bleh, Jun 07 2007 i imagine inestimable interference in inverting ! instead of ?-- half, Jun 07 2007 You just took a time warp back to the time of caveman, everybody on the bakery is now stupider for having read these words.
8*4=8*(23+1)?=8(1+1*6?)-- quantum_flux, Dec 19 2008 random, halfbakery