I know how they got pi, i, but by what process did the arrive at the conclusion e=2.7828…?
e = 1 + 1 + 1/2 + 1/(23) + 1/(234) + 1/(2345) + 1/(23456) + 1/(23456*7)…
e = limit as i -> infinity of (1 + 1/i)^i
As you plug in larger and larger numbers for i, you get closer to 2.718281828459045…
(1 + 1)^1 = 2
(1 + 1/100)^100 = 2.70481382942152609311
(1 + 2^-30)^(2^30) = 2.71828182715555896830
etc.
There’s a wonderful book on this that came out a couple of years ago, apparently in imitation of Petr Beckmann’s book on “pi”. I think the title is something like " e – the biography of a number". My copy is at home, though.
I don’t know the details of the history of e, but I’ll give it a shot.
According to this page which I merely glanced at, e apparently first popped up implicitly when Napier first studied logarithms.
e occurs in a natural way in many cases. As far as I know, the first two (that led to the discovery of e, I think, but I could be wrong about that part) are as follows:
Trying to find the area under the curve f(x) = 1/x from x=1 to x=n. It turns out to be log(n).
Trying to find for which functions f(x) the value of the function is always equal to the instantaneous rate of change of the function with respect to x. All such functions are of the form f(x) = Ae[sup]x[/sup], where A is any constant.
easy, they just multiplied m by c squared.
The book that CalMeacham refers to is:
e: Story of a Number
by Eli Maor, 1994, Princeton University Press
I read that book a while back, but if I recall correctly, the first situation in which e is suspected to have been discovered is that of compound interest. If interest is compounded frequently enough, the amount of money you have approaches P * e ^ (r * t), where P is the principal, r is the interest rate, and t is the time. If I am remembering the book correctly, a very early work on logarithms contains a statement equivalent to the fact that the natural log of 10 is 2.30259. Because this was before the application of logarithms to hyperbolas was discovered, the only known use of this fact would be to calculate compound interest from a table of Naperian logarithms.