I'm very eager to know and understand the definition of $e$. Textbooks define $e$ as follows
$$ e = \lim_{p\to\infty} \left[1+\frac1{p}\right]^p \approx 2.71828 $$
Is there an "easy to understand" proof of this? I'm really looking for a derivation of this which is very intuitive and easy to comprehend.
By the way I'm watching this video lecture.
Answer
$\pi$ is the name of the constant relating the diameter and circumferance of a circle. It's a definition, a particular constant that we thought deserved a name. $e$ happens to be the name of a constant from a particular limit. Like $\pi$, we named it because we thought it would be useful.
In Physics, many, many constants have names. Gravitational constants, expansional constants, electrical constants, etc. Each is useful, but somewhat arbitrarily chosen.
But I suspect you won't like the apparently nature of this. So I come up with something related: We call 2 the number s.t. 1 + 1 = 2. Why? What is its derivation? There is no derivation - we defined it, gave it a symbol, and a name.
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