Imagine you put $1 in the bank, and it earns 100% interest over the course of the year... cha-ching! $2.
But if the interest is compounded monthly, then you get to earn "interest on the interest" and get a little more:
If you compounded every day, you'd get more. Every second would get you even more, but what about every instant? Infinite money?
Nope: `lim_(x->infty)(1+1/x)^x` = $2.71828...
The number e is important in calculus, for determining the rate at which exponential functions increase. It has the critical property that
(There's no other number like this).
This allows us to find the rate that any exponential function increases:
`y = 3^x = (e^(ln3))^x=e^(ln3*x)`
Since ln3 is just a constant, then chain rule gives us `y'=e^(ln3*x)*ln3` .
There is a long history as to how the number e evolved as given in the referred link and qouted below:
The first time the number e appears in its own right is in 1690. ......the notation e made its first appearance in a letter Euler wrote to Goldbach in 1731. He made various discoveries regarding e in the following years, but it was not until 1748 when Euler published Introductio in Analysin infinitorum that he gave a full treatment of the ideas surrounding e. He showed that
e = 1 + 1/1! +1/2! +1/3! + ...
and that e is the limit of (1 + 1/n)n as n tends to infinity. Euler gave an approximation for e to 18 decimal places,
e = 2.718281828459045235
For further details, go to the link