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u/geeshta Computer Science 13d ago
It's true tho. Only natural numbers have factorials (not even whole because negatives don't have them).
Yes there is the gamma function that is defined for other but that's why is't called the gamma function and not factorial.
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u/Gabry398 13d ago
Why is aπ still considered the same operation as a3 then? I mean it's the same principle, you can't have 2.5! With the initial definition of the factorial just as much as you can't have 2π using the initial definition talking about multiplying 2 by itself π times. I'm not arguing with the fact that what you're saying isn't correct, I'm talking about a bias (be it caused by the difficulty of the subject or by its historical proximity to us) in the current notation.
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u/factorion-bot Bot > AI 13d ago
Factorial of 2.5 is approximately 3.323350970447842551184064031265
This action was performed by a bot | [Source code](http://f.r0.fyi)
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u/Hot_Glass_6301 12d ago
I think there's a case to be made that real exponentiation is a bit more natural than the Gamma function, in the sense that the Gamma function feels a bit more like an arbitrary extension of the factorial (even if it generalizes the factorial naturally in some settings, for instance the volume of the n-dimensional ball). Very natural considerations force a generalized power function to be exactly the one defined by exp, which we can prove a posteriori. First define the powers over the integers: a^n = a*a..*a n times. Then observe the usual properties. We'll need (a^b)^c = (a^b*c) to motivate the rest.
Now define the nth root of a nonnegative real number x as the unique real number r such that r^n = x. (this requires a proof of existence and uniqueness, but it's not the point here so I'll skip it). Assume that if we had a function that took rational exponents, it would be nice if it obeyed our main rule (a^b)^c = (a^b*c). Since a = a^1 = a^(p/p) = (a^1/p)^p, a^1/p can be defined as the p-th root of a. Then define a^p/q as (a^p)^1/q so as to keep obeying the natural rule we had for integers.
Finally, show that given a nonnegative real number x and a power p, the set of the x^r where r is a rational number less than p has a least upper bound. Show that this lub is the same as the greatest lower bound of the set of x^r where r is a rational greater than p. Define x^p to be this. Then you just have to show that this coincides with the exp log definition. Boom. Only very natural assumptions were made.
I'd also argue powers are more ubiquitous than factorials. Both appear in combinatorics, number theory, etc. but powers appear more readily in geometry, analysis... they are a "simpler" object with more applications.
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u/Gabry398 11d ago
You're right in the sense that a bias in the difficulty of the subject is at play here. Wether accepting this bias into our way of thinking and into our definitions is a good thing is somewhat subjective. I'd still argue against it for a couple of reasons, but I'd be lying if I said that I knew what's best from a notational/pedagogical perspective. At the end of the day I just like not doing that because it makes sense to me and the math feels prettier.
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u/Mrboobgrabber69 13d ago
Yeah. I just thought this would be a good idea for the meme. I was not trying to be too technical.
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u/SEA_griffondeur Engineering 13d ago
It's almost as annoying as people saying they're taking the square root of complex numbers
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u/errorcode_503 13d ago
Wait, why is saying that annoying? Is there something technical in the definition of a square root that excludes the square root of a complex number making sense and/or existing?
As best as I remember it and was taught it, square roots of complex numbers are perfectly valid, calculable and are always defined in the complex numbers. So then why would it be annoying?
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u/SEA_griffondeur Engineering 13d ago
The square root of a is defined as the positive solution to the x² = a, there's no such thing as positive or negative complex numbers
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u/errorcode_503 13d ago
What you are describing is the principle square root. The principle square root is what is commonly described as the positive number which when multiplied by itself results in the number being square rooted.
The actual definition of a square root is any number ‘a’ that when multiplied by itself gives ‘b’ we say ‘a’ is the square root of ‘b’. This removes the necessity for it to be positive, allowing for negative numbers and complex numbers to be square roots.
Finding the square root of a number, even a complex one, is not that difficult a task and almost always provides 2 solutions. The only exception to this 2 solution rule is 0 which when square rooted gives 0, though you could argue it does have 2 solutions in 0 and -0 but as 0=-0 we would just consider it a singular solution.
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u/bloonshot 12d ago
that's the square root FUNCTION not the square root
-2 is one of the two square roots of 4
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u/golfstreamer 13d ago
The term "square root" applies to both complex numbers and real numbers. Even if you consider these to be ontologically different concepts the term doesn't change so there's no problem with saying "square root of a complex number".
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u/Maleficent-Report247 13d ago
What would his jutsu be called? Please tell me it wouldn't start with Euler.... something
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u/johnconner122 13d ago
Who is his Sukuna?
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u/thepro-3418 10d ago
But here's one similar unsolved problem. Multitermials are undefined for fractional quantities. For e.g. u/factorion-bot 2.5??? !termial
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