Quote:
Originally Posted by evay
I know pi is used in measuring circles, and i is the square root of -1 (so the square root of -4 is 2i), but what's "e" in mathematical terms?
|
Scooter's correct, except about Caesar. (And he
is right about Caesar where cryptography is concerned. Caesar devised the Caesar shift, rot-3 in modern parlance, which is why I had the
Romulans use it.)
Anyway,
e is as deeply embedded in calculus as pi is in geometry.
y =
C × ex is the only function which is its own derivative. As such, it's the best base for logarithms (just as pi is the best basis for angle measurement), so logs to the base
e are called natural. Common logs, i.e. to base 10, are strictly for engineers and the occasional number theorist.
e also arises in accounting. If you have $1 in the bank at 100% simple interest, your balance will be $2 in a year. If the interest is compounded semiannually, that rises to $2.25. Compound it monthly and you'll end up with $2.61. It turns out that there's a limit to this process, representing a sort of continuous compounding of interest, and that limit is
e dollars.
A third way
e can be found is through infinite series.
e is the sum of the series 1/0! + 1/1! + 1/2! + 1/3! + ..., where the exclamation point is a factorial sign. (4! = 4 x 3 x 2 x 1, and so on. Before you ask, there
is a double factorial sign, and it's the only kind of double exclamation point I can stand.) Any of these three properties of
e can be, and has been, used as its definition; you can derive the other properties from the one you start with.
As you can see, it's easy to get me talking about this stuff. I did some quality math-related ranting in
this LJ post, for anyone interested.