Until that is I came across this[1] wonderful article. That site is a treasure trove of very good insightful articles. Can't recommend enough.
[1] https://betterexplained.com/articles/an-intuitive-guide-to-e...
Eg. if an event has a 1/10 chance of occuring, how likely am I to encounter it at least once after 10 trials? What about a 1/1000 event after 1000 trials?
Well, if it has probability p, then there is 1-p of it NOT happening. So the chance of seeing a 1/p event after p trials is:
= 1 - (1-p)^p (looks a lot like e) = 1 - 1 / (1-p)^-p = 1 - 1/e (63.2%)
This comes up a lot in real-life, since it feels like if we do something with a 1% chance 100 times, it should occur, but there is really a more than 1/3 chance against it!
Visual Complex Analysis, by Tristan Needham, covers this lucidly.
Or you can think of the function exp() represented by the Taylor series and e just happens to be exp(1). But the particular number itself isn’t as important as the function.
To be fair, neither is important. You don't have to reach for a transcendental if you want infinite polynomials, you can manufacture one for integers, rationals, irrationals...the familiar example -
f(x) = 1 + 1/x + 1/x^2 + 1/x^3 + 1/x^4 + ... f(1) = 2
But its still a really interesting video!
I feel like the most clear statement from that article is "it's the base of exponential growth", but that (for a math inclined audience) the best way to show that is via equations like dy/dt = \alpha y.
e is the base of the exponential function, the solution of
y = y'
y'' = -yReplace dy/dx with ∆y/∆x, so
∆y = y(x+∆x) - y(x)
Once you solve that (use some induction and numerical methods), see what happens when ∆x appraches to 0 and x=1.
For example try with ∆x=1 and then generalize that case to ∆x=n.
Could you break it down further?