Why is Euler's number $2.71828$ and not anything else?

$\sum\frac1{n!}$ is not that special.

$\lim_{n\to\infty}\left(1+\frac1n\right)^n$ is not really special.

$f'(x)=f(x)$ is a very simple differential equation, but unremarkable, really.

$\ln (x)$ is only marginally nicer than other logarithms, in that its derivative is $\frac1x$.

The fact that a single number connects all of these (and many, many others) as intimately as $e$ does is nothing short of a miracle. Oh, and also $e$ happens to have the decimal expansion $2.718\ldots$

We use $e$ because it a natural choice, as it yields a simple derivative:

$$(e^x)'=e^x.$$

For other bases, we have

$$(a^x)'=\ln a\,a^x$$ and the factor $\ln a$ is annoying.

For a very similar reason we use radians in the trigonometric functions:

$$(\sin x)'=\cos x.$$

With degrees, we would have

$$(\sin_d x)'=\frac\pi{180}\cos_d x,$$ once more an embarrassing factor.

As shown by Hyperion, the condition $(e^x)'=e^x$ induces the value

$$1+1+\frac12+\frac1{3!}+\frac1{4!}+\cdots$$

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Assume you wanted to find a number $b$ such that $(b^x)'=b^x$. Using the definition of the derivative, you could try to solve

$$\frac{b^{x+h}-b^x}h\approx b^x$$ where $h$ is a small increment.

Then $$\frac{b^{x+h}-b^x}h=b^x\frac{b^h-1}h\approx b^x$$ leads to

$$b^h\approx 1+h$$ or $$b\approx(1+h)^{1/h}.$$

It turns out that this expression has a limit for $h\to0$, which you can obtain using the generalized binomial theorem.

E.g.,

$$1.000001^{1000000}=2.718280469\cdots$$

Clearly, one answer is "because that's the value that the various definitions produce, and when we follow them $\sum_{n=0}^{\infty}\frac{1}{n!}$ pops out". But it's not a very satisfying answer (in fact I think you're asking for an underlying reason why that happens).

I can't give a definitive why, but my suggestion is that it's something to do with iterated processes like

• taking the next derivative
• dividing by the next integer
• choosing the next item in a permutation
• multiplying by the next bracketed expression

all of which are quite good at producing sequences of factorials.

But of course I've now got $e^{iπ}=-1$ nagging at me, and even though that can be explained in terms of "exponential growth sideways" and proved to be true, it doesn't in itself seem that related to any iterated process, and @Arthur's comment that it's "nothing short of miraculous" seems more accurate than any proof of the connection would be.

My suggested explanation, if true, just pushes the question back a level: "Why do iterated processes that produce the series for $e$ pop up all over the place?"

Typically if you ask Why? more than about four or maybe five times (following underlying reasons rather than a chain of trivial causal events or a string of theorems), you'll get to unanswerable philosophical questions—for instance "Why is it raining?" leads me after a few steps to "why is there such a thing as the laws of physics?" I suspect that pursuing the reasons why a particular number is as it is will have the same result.