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It seems like most natural situations (that at least I've come across) where Euler's number shows up, there's usually a more general (yet still natural) version of it that contains the exponential function. Even its usual definition as: $$e = \lim_n\left(1+\frac{1}{n}\right)^n$$ Can be extended to: $$e^x = \lim_n\left(1+\frac{x}{n}\right)^n$$

This seems, at least to me, somewhat unique. Rarely do I find results with other constants like $\pi$ where I can naturally extend them to a consistent family of values like how I can with $e \to e^x$.

Thus the question, does Euler's number have any significance that can't be naturally extended to other values of the exponential? Or alternatively, is there any significance to Euler's number other than being equal to $\exp(1)$ or the base of the exponential function?

Note: I realize of course that this isn't really a mathematical question, and that this might not be the appropriate place for this sort of question, if that's indeed the case I will remove this post.

TC159
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  • $|e^{ix}|=1$ for any real x. $(e^x)'=e^x$. – herb steinberg Sep 15 '22 at 22:18
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    The continued fraction for $e$ follows a very simple pattern and I can't see how that naturally extends to arbitrary powers of $e$. (Correction: actually, it does extend in a way see [https://en.wikipedia.org/wiki/Euler%27s_continued_fraction_formula].) – Rob Arthan Sep 15 '22 at 22:21
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    An alternative definition of $e$ underlies its especially nice derivative formula. The number is $e$ is the unique base $b \in \mathbb{R}{>0}$ such that
    $$ \lim    {x \to 0} \frac{b^x - 1}{x} = 1. $$     – Sammy Black Sep 15 '22 at 22:22
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    @herbsteinberg I mean, as I wrote in my post I'd call that a property of the exponential function rather than Euler's number itself. The only real importance to Euler's number in those cases is that it can be used as the base of the exponential function under usual notation, take the notation away and those properties disappear. – TC159 Sep 15 '22 at 22:27
  • @SammyBlack while I can't really argue with that exactly. I'd also like to point out that you can also give it a meaning in the exponential that is:

    The exponential function is the unique function of the form $b^x$ such that $b^x-1$ is asymptotically equal to $x$ as $x \to 0$. In fact this statement is much more useful than the original meaning.

     – TC159 Sep 15 '22 at 22:41
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    $\frac{1}{e}$ comes up in some probability questions. For example the Secretary problem (See here https://en.wikipedia.org/wiki/Secretary_problem) – Fishbane Sep 15 '22 at 23:36
  • @Fishbane that counts as an answer doesn't it? – Suzu Hirose Sep 16 '22 at 01:32
  • @SuzuHirose If you want to add some more info and write it as an answer, that is fine, I probably won't be. It's mostly up to the original asker to decide if something works. – Fishbane Sep 16 '22 at 02:17
  • Other than its name, what are you questioning? – herb steinberg Sep 18 '22 at 21:47

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