# why the exponential value is considered as 2.718281828whats the history behind exponential funtion..?????

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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:

`(1+1/12)^12=$2.613...`

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

`d/(dx)e^x=e^x`

(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

eappears in its own right is in 1690. ......the notationemade its first appearance in a letter Euler wrote to Goldbach in 1731. He made various discoveries regardingein the following years, but it was not until 1748 when Euler publishedIntroductio in Analysin infinitorumthat he gave a full treatment of the ideas surroundinge. He showed that

e= 1 + 1/1! +1/2! +1/3! + ...and that

eis the limit of (1 + 1/n)nasntends to infinity. Euler gave an approximation foreto 18 decimal places,

e= 2.718281828459045235

For further details, go to the link

http://www.gap-system.org/~history/HistTopics/e.html