What is the best way to prove that e = 2.7182818...

What is the best way to prove that the continuously important natural logarithm is equal to 2.7182818... I got lucky and proved it in year 2002 while trying to prove something else. The technology applications are very Infinite. You never know how we might arrive at something.

What are the best applications (among the Infinite) of e?

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.

I just looked in my calculus textbook, and another way to define e is as follows:

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[edit on 30-8-2005 by DragonsDemesne]

Model a perfect mixing device. A minute quantity of New substance is introduced to Old material in the full mixer, totally mixed, and then the same quantity of now mixed Old and New material is vented.

After the introduction of exactly one complete volume of New material, introduced incrementally as described, the amount of New material actually in the mixer is 1/e.