I know of a couple of ways to do this. The easiest is to use an infinite series. (This has the added bonus of being easy to calculate with a
computer program, too, you just need a loop with about 2 lines of code)
First, let me explain what a 'factorial' is in math, if you don't know. "!" is the factorial symbol. 2! = 1x2 = 2, 3! = 1x2x3 = 6, 4! =
1x2x3x4 = 24, and so on. Most scientific calculators will have a button that says "n!". Also, 0!=1 and 1!=1.
Now that that's been said...
e = 1/1! + 1/2! + 1/3! + 1/4! + .... + 1/10000! + 1/10001! + .... et cetera.
(Strictly speaking, that isn't a proof, that's just a statement, but it is true)
I've seen another way using calculus, limits, and the squeeze theorem, but it is more complicated to understand, more work for the same thing, and
really hard to explain on a forum. If you really want it, I'll draw it in MSPaint and send it to you. (This would actually be a proof, not a
statement)
Applications: e is really important in calculus. e to the power of x, or e^x, is the only function that is its own derivative. (I suppose the number
zero is also, but knowing that is totally useless) Because of this property of e^x, it ends up being the solution to MANY physical equations, so it
shows up in zillions of physics and math formulas. You can also take sine, cosine, and all the other trig functions, and express them in terms of e
to various powers. (For example cosh = 0.5 & [e^x + e^-x] ) Terms such as e^x or similar things also show up in some mathematical modeling, like
population growth models, because those models are often simple differential equations, that typically have solutions in terms of e to some power.
Since differential equations often have solutions that look something like Ae^Bx + Ce^Dx, or that sort of thing, and so many scientific equations are
differential equations, e shows up everywhere from F=ma (a differential equation in disguise) to the Schroedinger wave equation in quantum physics.
(yecch! I had to solve that one in school for a nanotech class, took 7 pages
)
I can't think of any 'best' application of e. If you get down to the fundamentals, I think it's safe to say that virtually all our technology,
once boiled down to equations on paper, requires e to do the calculations.