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Does any generalization for "famous" math constants (like $\pi$ or $e$) exist.

I know how those constants are useful, but I do not know what property makes them useful.

Is there any definition of those numbers which can include some additional numbers. (Those additional number can be part any super-set of $R$ or $C$).

What property of those constants is making them "special"?

Vasoli
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    $\pi$ and $e$ are irrational, so a little bit special. – jupiter_jazz Nov 17 '17 at 21:38
  • just google the definitions of $e$ and $\pi$ – jupiter_jazz Nov 17 '17 at 21:40
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    $3$ is a famous constant. What makes it special? – user4894 Nov 17 '17 at 21:40
  • @Kirill "irrationality" is rather common ($\sqrt{2}$ is irrational) and does not distinguish them from the "vulgus pecus"; transcendentality is much less common... – Jean Marie Nov 17 '17 at 21:40
  • I don't understand what you mean by generalizing a constant. They are fixed and so how could you generalize them? – Qudit Nov 17 '17 at 21:43
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    They are like the Kardashians. They are useful precisely because they are famous. – Will Jagy Nov 17 '17 at 21:52
  • @Kirill I was googling and I did not found answer to my question. – Vasoli Nov 17 '17 at 21:57
  • @Qudit I cannot exactly answer to your question, because it is a part of my question. In mathematics we are doing all kinds of generalization. Like extending $sin$ function from right triangle to unit circle definition. Addition of complex number is some kind of generalization of addition of real numbers. So with every "generalization" we are somehow extending a scope. So are those "special" number part of some bigger set of numbers which all share same characteristic as those numbers. – Vasoli Nov 17 '17 at 21:57
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    @JeanMarie Might depend on what you mean by "common". In the sense of measure theory, almost every real number is transcendental. In the sense of Baire category, a generic real number is transcendental. On the other hand, being proved transcendental is somewhat special. – Robert Israel Nov 17 '17 at 22:04
  • so, @Vasoli, what are possible definitions of $e$ and $\pi$? – jupiter_jazz Nov 17 '17 at 22:10
  • @Vasoli As other have mentioned, they are irrational and transcendental. I don't think much more can be said beyond that in terms generalizing them. – Qudit Nov 17 '17 at 22:11
  • @Qudit There might be some generalization of $\pi$ and some other generalization of $e$. I do not expect to have same generalization of both of those numbers. We can generalize that 1 and 0 as neutral elements in $(R, *, +)$ field. So we can observe other neutral elements, and neutral elements are some kind of generalization of 0 and 1. Maybe there is some generalization of $\pi$ and $e$. I do not know, that is why I am asking. – Vasoli Nov 17 '17 at 22:20
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    @Robert Israel You are right: It is "proved transcendentality" which isn't common. – Jean Marie Nov 17 '17 at 22:30
  • Relevant: https://mathoverflow.net/questions/341470/is-there-any-deep-philosophy-or-intuition-behind-the-similarity-between-pi-4 – Anixx Oct 31 '23 at 13:50

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Perhaps you may be interested in ring of periods, recently developed by Kontsevich and Zagier, generalizing the constants you mentioned. More details in the Wikipedia article and its references. According to Kontsevich and Zagier, "all classical constants are periods in the appropriate sense".

Somos
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  • I voted up your answer (Also Adams). I will keep question open. Maybe we get some additional interesting answer. – Vasoli Nov 17 '17 at 23:12
  • How many periods are there? One reason the transcendental numbers do not seem like a good generalization is because almost every real number is transcendental. – Qudit Nov 18 '17 at 07:30
  • I have not read much about "periods" but they seem more like algebraic numbers which are countable. – Somos Nov 18 '17 at 13:35
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Perhaps you are looking for non-algebraic ( transcendental ) numbers? Certainly, something shared with pi and e.

http://mathworld.wolfram.com/AlgebraicNumber.html

Adam
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  • That might be one answer. But I do not see that transcendental have that much in common with those 2 except being transcendental. I will leave question unanswered because maybe someone have some better answer then that, if that answer exist. – Vasoli Nov 17 '17 at 22:48
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Well, we can calculate $\pi$ by $$\pi = \lim_\limits{n\to \infty} \frac{1}{2} * n * \sin{(\frac{n}{360})}$$ And $e$ by using $$e = \sum_{n=0}^\infty (\frac{1}{n!})$$ Those two definitions are somewhat the proof that they can't actually be calculated. However, since they do occure in nature, they are very speciall. Also, the beautiful equation $$e^{i * \pi} + 1 = 0$$ States that they are relative to each other and $\sqrt{-1}$ (an imaginary number), which, once again, makes both of these very special.

SuperSjoerdie
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    This celebrated relationship should not be interpreted as meaning a special link between $e$ and $\pi$ ! You could find other constants with other "close relationships". – Jean Marie Nov 17 '17 at 22:35
  • I knew all of this, but this does not answer my question. This is not "generalizing" them in any way. I not saying that there is a generalization, I am asking if there is one. – Vasoli Nov 17 '17 at 22:39
  • You question was "what property of these two constants is making them special?" Which is that they both can be defined by either using $\infty$ or each other and $i$, and they still are to be found in nature. That's what makes these two special. Either this or redefine the last line of the question. – SuperSjoerdie Nov 17 '17 at 22:48
  • And if you want to know what makes them useful: $\pi$ is used to calculate circle-related equations, and $e$ is used to calculate air pressure by height, if I'm not mistaken. – SuperSjoerdie Nov 17 '17 at 22:56
  • You can probably say the same thing for 2 and 3 and they are not "special". That they can be calculated in terms of each other and i.

    I know how they can be used, how they can be calculated and lots of their "fine" properties. What I do not know is how I can distinguish them from other numbers. With precise mathematical definition. I have that property for 1 and 0. They are neutral elements. But I do not see such a property with pi, e or i.

     – Vasoli Nov 17 '17 at 22:59
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    Ach so, well $e$ and $/pi$ also are irrational (unable to write it in a fraction) unlike 2 and 3. Also, 2 and 3 don't have a non-recursive relation to $\infty$, which $e$ and $\pi$ do – SuperSjoerdie Nov 17 '17 at 23:11
  • @SuperSjoerdie I like your "Ach so", somewhat not translatable in English... – Jean Marie Nov 19 '17 at 01:17