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Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate.

I know there are many ways of calculating (or should I say "ending up at") the constant e. How would you explain e concisely?

It's a rather beautiful number, but when friends have asked me "what is e?" I'm usually at a loss for words -- I always figured the math explains it, but I would really like to know how others conceptualize it, especially in common-language (say, "English").


related but not the same: Could you explain why $\frac{d}{dx} e^x = e^x$ "intuitively"?

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    Can you justify why this is not the same question? My answer to this question is the same as my answer to that question.2011-03-09
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    There are any number of "semi-natural" ways to arrive at $e$ (what you call "ending up at $e$"), such as compound interest, a function that is equal to its own rate of growth, etc. The compound interest one is particularly succint ("$e$ is the amount of interest you would have at the end of one year if you deposit one dollar, at 100% annual compound interest, compounded each and every instant."), though it may take some motivation to explain to the lay public (who has enough trouble grasping financial matters, it would seem), why the frequency of compounding matters.2011-03-09
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    @Qiaochu: IMHO the questions are not at all the same; they are complementary and so it is natural that they share the same basis for an answer.2011-03-09
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    @Arturo Magidin: The number $e$ is what everybody wants to have (at least on his/her bankaccount). Imagine a world where the banks would pay interest continously...2011-03-09
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    @Fabian: Well, keep in mind that they would likely also *charge* interest continuously...2011-03-09
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    @Arturo Magidin: at least I would then know how much I owe in a year. Now we have to use those ugly looking [formulas](http://en.wikipedia.org/wiki/Compound_interest) which I keep forgetting.2011-03-09
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    I like your "ending up at" phrase so much that I am up-voting your question just because of that:-)2011-07-29
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    @Mike: haha, awesome. A lot of math seems to be: given a set of equations, can you reproduce the insight that lead to them? :)2011-08-03
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    @sova : on the topic of 'questions that are a lot like this one', what about http://math.stackexchange.com/questions/387746/pi-from-the-unit-circle-sqrt-2-from-the-unit-square-but-what-about-e/387764#387764 ? That seems to be asking for almost exactly the same thing...2013-06-19
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    Is it safe to say "e" comes into play when there is some sort of infinitely small compounding or growth? ie: Is the continuously compounding interest example using A=Pert just one random example of using e, or does it truly reflect the underlying essence of e? As in, infinitely small exponential instantaneous growth intervals, which can apply to many things besides calculating interest.2013-01-29

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