Why are $e$ and $\pi$ so common as results of seemingly unrelated fields?

I'm sure this gets asked all the time but I swear I googled with no useful result. What I'm looking for is a reasonably intuitive answer.

Those two constants have some pretty interesting properties. $\pi$ is often used in geometry while $e$ is for example used in statistics, yet $\int_{-\infty}^{+\infty}e^{-x^2}=\sqrt \pi$

Weird right? We'll it's weird to me.

$e$ itself is pretty unusual. It's both the $\sum_{n=0}^{\infty}\frac{1}{n!}$ and the limit of $(1+\frac{1}{n})^n$. Personally, I've done proofs that those two converge to the same number but I really can't say that I get why.

Then there's the famous $e^{i\pi}=-1$

This can't all be a coincidence, right? In fact, seeing how there's not only an infinity of irrational numbers but an uncountable one, this literally can't be a coincidence.

There must be some level of dissection on which you can say "See, there's e. If we add 1 to this we get $\pi$".

For example, say we turn the tables. $\cos(\pi)=-1$ and $\frac{1}{e}\sum_{n=0}^{\infty}\frac{1}{n!}=1$. I could say it's pretty weird how these unrelated things have those numbers as their results. $-1$ and $1$ pop up everywhere, right? The intuitive explanation is that I obviously chose those two things specifically because I know what their results are. As for what their connection is, they obviously come from the same "factory". They are both whole numbers that you get by slightly incrementing or decrementing zero.

In fact, I'd be willing to bet that whole numbers with their absolute values less than, say, 10 pop up pretty often and the explanation for that is the same as above.

So what's the deal with those irrationals? They aren't countable. You can't just add one to one of them to get the other. What's their "common factory"?

The gist of what I'm saying is, $e$ and $\pi$ show up all the time in unrelated circumstances and they even seem to know each other. Their connection, however, eludes me.

Solution 1:

$e$ is not important. What is important is the function $e^x$, which is an eigenvector of the differentiation operator $f \mapsto \frac{\partial f}{\partial x}$ with eigenvalue $1$. Differentiation is important, e.g. because it lets us pose differential equations, and eigenvectors of differentiation are important because they help us solve differential equations. Differential equations are important in many fields, although the connection is harder to see in some fields than others.

$\pi$ is important because it is closely relate to the period of $e^x$, regarded as an analytic function taking in complex values, which is important. More precisely, it's important because $e^{x + 2 \pi i} = e^x$. Essentially every appearance of $\pi$ in mathematics I know of, including its relevance to the circumference of a circle, can be traced back to this.

Solution 2:

Bounded sums, and bounded differences.

• $\pi$ is the constant of the circle, whose algebraic equation is $x^2+y^2=r^2$.

• e is the basis of the natural logarithm, whose derivative is the hyperbola, whose algebraic equation can be written as $x^2-y^2=r^2$, following a rotation of $45^\circ$.

So they both define geometric shapes of the form $x^2\pm y^2=r^2$. For imaginary values of y, the circle becomes a hyperbola, and vice-versa. Hence, by mixing e with imaginary exponents, we have a parameterization of the unit circle.