TITLE: What exactly ARE $\pi$ and $e$?
QUESTION [8 upvotes]: First of all, apologies if this is a bad question. I don't really know how to phrase it.
I first got introduced to  $\pi$ in elementary school, where it was presented as a ratio for a circle's area. I thought it was some special number that had to do with circles. You can imagine my confusion when I found it popping up in stuff that seemingly had nothing to do with circles, like in the Wallis formula or the Basel Problem. Same with $e$. I thought it was this financial growth thing, so I got pretty confused when it showed up in stuff like the Probability Distribution, and Euler's identity. Right now, they seem like two magic numbers with magical properties - same thing with $\cos$ and $\sin$ - magical equations that give you ratios of lengths. 
Could someone explain what the heck exactly is the significance of $e$ and  $\pi$? Beyond circles, beyond geometry. Because they certainly seem to be more than just that.

REPLY [10 votes]: This may surprise you, but every appearance of $\pi$, no matter how crazy-looking it is, traces to the involvement of a circle somewhere.
For example, where does the Wallis product come from? From evaluating $\frac{\sin x}{x}=\prod_{k\ge 1}(1-\frac{x^2}{k^2})$ at $x=\frac{\pi}{2}$ (i.e. a quarter-turn from $x=0$). And though we teach trigonometry to students with right-angled triangles, you can embed those triangles in a circle of which the hypotenuse is a radius, and generalise the cosine and sine beyond acute angles so that, if in the circle $x^2+y^2=1$ we consider a radius $(1,\,0)$ and another forming an anticlockwise angle $\theta$ with it, the latter meets the circle at $(\cos\theta,\,\sin\theta)$. So ultimately, these functions are defined with circles.
What about the Basel problem? We can prove $\sum_{k\ge 1}k^{-2}=\frac{\pi^2}{6}$ by using the fact that acute $x$ satisfy $\cot^2x\le x^{-2}\le\csc^2 x=1+\cot^2 x$, which we sum over $x\in\{\frac{k\pi}{2m+1}|1\le k\le m\}$. (The sum of the squared cotangents can be obtained from a polynomial's coefficients.) So once again, it leads back to trigonometry, hence circles.
Similarly, every appearance of $e$, no matter how crazy-looking it is, traces to either of its equivalent definitions (1) $e:=\lim_{n\to\infty}(1+\frac{1}{n})^n$ or (2) $\frac{d}{dx}e^x=e^x,\,e>0$. (1) implies $(1+\frac{x}{n})^n\approx e^x$ for $x\ll n$, from which we can deduce the central limit theorem using characteristic functions. But this proof also uses $e^{ix}=\cos x+i\sin x$, which can be proven by combining (2) with a proof that $\cos x,\,\sin x$ satisfy $y''=-y$.
Why does $\pi$ keep coming up? Because multivariable symmetries frequently concern tracing over the surface of a circle, sphere or hypersphere, which traces back to circles. Why does $e$ keep coming up? Because so many problems hinge on rates of change, which automatically include exponential functions if the differential equations involved take a certain form.