h a l f b a k e r yNot so much a thought experiment as a single neuron misfire.
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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.
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...12eb79cabeb39f955e3
Scroll down to see a Python solution. [kwhitefoot, Jan 08 2014]
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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. |
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Just nod and smile [htj], just nod and smile. |
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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. |
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I really like this!? = I really like this |
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"It would really clear up mathematics,
especially for the novice."
Sure about that, now? |
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3!=6
2!=2
1!=1
0!=1 (by definition) |
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Careful, now. Not all functions have
easy and exact inverses. Now that's an
important
lesson to learn early on in maths. |
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2^2 = 4
1^2 = 1
0^2 = 0
-1^2 = 1
2^2 = 4 |
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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. |
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Would this allow me to divide by zero without fear of the ground opening up and swallowing me whole? |
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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). |
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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. |
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Mathematician: "The answer is 9."
Student [incredulously]: "Did you say 9???!!!!?!?!?!?"
Mathematician: "That's right" |
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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. |
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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. |
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Maybe there should be new classes of numbers for <non-integer>! and <non-factorial>?
I propose 'contrary' and 'pretend' respectively. |
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[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. |
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Cool link, Steve. It seems to work, except it goes to zero for anything less than .8957. (Probably where the function becomes two valued.) |
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Jutta, you're right. I was misremembering how the ! function works. |
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God but you people make my head hurt sometimes.
Keep up the good work. |
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9?! = 9 is not valid. In order to perform the ? function, the operand must be a valid factorial. |
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All full factorials above 1 are even numbers. 9 is not even, therefore it is not a valid full factorial. |
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9!? = 9, however, would be valid. |
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[Freefall], wouldn't
a!? = a? !
just be defined as an identity? |
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a!? = a?! only where a? is defined. |
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What about ? of odd numbers or negative numbers (or ! of negative numbers which my calculator claims to be ERROR 2)? |
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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. |
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sp. aesthetics, forsooth, [forsooth]. |
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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. |
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[phundug]
//Mathematician: "The answer is 9." |
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Student [incredulously]: "Did you say 9???!!!!?!?!?!?" |
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Mathematician: "That's right"// |
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(Explains why Spock often had one brow-up and the other down) |
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//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. |
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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. |
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That said, over the real domain [1,infinity], the Gamma function is single valued, and its inverse is therefore also a function. |
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// a!? = a?! only where a? is defined [freefall] |
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By defined you mean reducible --- this is a bit harsh... |
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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. |
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[ldischler] I didn't know Google did that (cosh i pi). At least it gets it correct! 8~) |
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[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. |
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[madness] The implication of that is that I am u². Also that u r imaginary, and I am less than nothing. |
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//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. |
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[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.) |
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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. |
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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 |
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[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?) |
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[madness] Good point. Edited. |
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<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. |
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In this case, shirley, x? would always
have a value for positive real x -
wouldn't it? |
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[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". |
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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? |
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//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. |
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¿How about using an inverted exclamation mark ¡ instead? |
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I like that better, less confusing. |
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i imagine inestimable interference in inverting ! instead of ? |
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You just took a time warp back to the time of caveman, everybody on the bakery is now stupider for having read these words. |
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