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science:mathematics

# Inverse Factorial '?'

The inverse of the factorial operation should be '?'   (+25, -3) [vote for, against]

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

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...12eb79cabeb39f955e3
Scroll 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

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)

 — 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...

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] 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?)

 — Cosh i Pi, Jun 06 2007

 ! 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!?!?

 x*(x-1)*(x-2)*...3*2*1

 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

 — 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

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