e is: the unique number such that for all real x, e^x >= x^e the x-coordinate of the maximum of the function x^(1/x); for all real x, x^(1/x) < e^(1/e) the r

e is: the unique number such that for all real x, e^x >= x^e the x-coordinate of the maximum of the function x^(1/x); for all real x, x^(1/x) < e^(1/e) the root of the hyperbolic logarithm ...hyperbolic logarithm? Well, 1/x -- the reciprocal function -- is a hyperbola, and the area under that function turns out to be: http://en.wikipedia.org/wiki/Natural_logarithm#Definitions http://en.wikipedia.org/wiki/Natural_logarithm#Definitions Unfortunately, it's kind of hard to try to make sense of e without going into at least some basic calculus (which betterexplained does, in the form of a limit, but glosses over). It is commonly stated that e was discovered by Bernoulli, but the first references are from Napier, who presented a table of natural logarithms without any reference to e the number itself. I'm not a terribly huge fan of the two most common explanations of e: e = lim (1+1/n)^n as n grows e defined such that d/dx e^x = e^x The first definition has no obvious connection with most of the interesting properties of e, so while it makes plenty of sense it doesn't actually clarify anything about exponential functions or natural logarithms. The second definition asks students to simply assume one of the most interesting properties of e, without any clarification as to the derivation of such a number. Worse yet, students are usually introduced to both long before they are capable of understanding the connection between these definitions (this requiring L'Hopital's rule to properly understand) which basically tells them that "e is some magic that you can't understand". My favorite treatment of the idea is the one I had the privilege of learning as a student (perhaps I'm biased), which ignores e at first and defines the natural logarithm as shown above, as a definite integral resulting from the function 1/x, and then the fundamental property that d/dx e^x = e^x is proven from the previous definition, which requires only a little basic calculus. In fact, the text did not even say it was a logarithm, merely "Consider the function L(x) such that L(x) = the integral of dt/t from 1 to x", and then went on to prove that this satisfies several properties of a logarithm, that e is the unique number such that L(e) = 1, and finally derives the fundamental property of the exponential function (dy/dx = y) and so forth. I think that the fear that surrounds calculus and the use of its methods does many students a disservice by rejecting a more complete explanation in favor of one that requires slightly fewer uses of the word "integral". Of course, I'm a cantankerous prick.