I wanted to bring to greater prominence something that might have got lost in comments. Elke Stangl, author of the Theory And Practice Of Trying To Combine Just Anything blog, noticed that among the Richard Feynman Lectures on Physics, and available online, is his derivation of how to discover e — the base of natural logarithms — from playing around.
e is an important number, certainly, but it’s tricky to explain why it’s important; it hasn’t got a catchy definition like pi has, and even the description that most efficiently says why it’s interesting (“the base of the natural logarithm”) sounds perilously close to technobabble. As an explanation for why e should be interesting Feynman’s text isn’t economical — I make it out as something around two thousand words — but it’s a really good explanation since it starts from a good starting point.
That point is: it’s easy to understand what you mean by raising a number, say 10, to a positive integer: 104, for example, is four tens multiplied together. And it doesn’t take much work to extend that to negative numbers: 10-4 is one divided by the product of four tens multiplied together. Fractions aren’t too bad either: 101/2 would be the number which, multiplied by itself, gives you 10. 103/2 would be 101/2 times 101/2 times 101/2; or if you think this is easier (it might be!), the number which, multiplied by itself, gives you 103. But what about the number 
There’s a pretty good, natural way to go about writing that and as Feynman shows you find there’s something special about some particular number pretty close to 2.71828 by doing so.
nebusresearch 
So there is not an easy way to boil down this section to a very short post, is it? I suppose you would need to resort to Taylor’s expansions, I guess? But Feynman tried to do without explaining too much theoretical concepts upfront – so that’s probably why it takes more lines and one complete example…
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Taylor series do rather depend on calculus and the fact that d/dx(exp(x)) = exp(x), and he wanted to do it all without calculus.
You can define e as the solution to the equation
ln(x)=1, and with a proper definition of the natural log ( which needs the ideas of calculus) this will work.
Check my recent post on this:
http://howardat58.wordpress.com/
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Yes, and I love definitions of e which come about from more natural methods like that.
(For those reading this long after publication date http://howardat58.wordpress.com/2014/10/15/calculus-without-limits-5-log-and-exp/ is the post of reference.)
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I don’t know that the section couldn’t be boiled down to something short, actually; I didn’t think to try. Probably it would be possible to get to the conclusion more quickly, but I think at the cost of giving up Feynman’s fairly clear intention to bring the reader there by a series of leading investigatory questions, of getting there the playful way.
It’s a good writing exercise to consider, though, and I might give it a try.
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Joseph, I read the log bit and the complex number bit. The latter is excellent and should be plastered on the wall of every high school math classroom. Thanks.
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Oh, you’re quite welcome. Glad you could draw something from it.
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