How was the number 'e' 2.718 originated?

[PrakashPhy]

Since past two years I have been using the mathematical constant 'e' over and again in mathematics (calculus and logarithm). I wonder how this number first originated. Who first used this number , this particular number e=2.718281828459045... .

Thank you for your help in advance.

<<<sorry if this thread already exists in the forum; i searched in the forum but couldn't find it>>>

 

[wukunlin]

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier.[4] However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli, who attempted to find the value of the following expression (which is in fact e):

$$e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n$$

The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler started to use the letter e for the constant in 1727 or 1728,[5] and the first use of e in a publication was Euler's Mechanica (1736). While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard.

http://en.wikipedia.org/wiki/Euler's_number#History

 

[PrakashPhy]

The discovery of the constant itself is credited to Jacob Bernoulli, who attempted to find the value of the following expression (which is in fact e):

e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n

How at once did he encounter this limit? There must have been some earlier equations or problems or something that led to this equation.

Like pi =3.14... is the ratio of circumference of a circle to its diameter which is a simplest way to understand pi and probably which is the origin of of pi ; I wish to get such simpler and similar information on origin of e

 

[gb7nash]

And with a little more digging...

Bernoulli discovered the constant e by studying a question about compound interest which required him to find the value of the following expression (which is in fact e):

$$e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n$$

http://en.wikipedia.org/wiki/Jacob_Bernoulli

That's how it was discovered anyway. As far as what uses it has, the more time that passed, the more people that found a particular use for e. e^x is the only function who's derivative is equal to itself. e is a key component of Euler's identity. etc.

 

[Dr. Seafood]

How at once did he encounter this limit? There must have been some earlier equations or problems or something that led to this equation.

Check http://summer-time-nerd.tumblr.com/post/8628678169" out (it's my blog . Apparently Bernoulli (one of them :p) explored this finance problem, where he encountered this limit in this theoretical situation.

 

[JDoolin]

Another useful definition of e can be found by taking the taylor series (more specifically, the Maclaurin series) of e^x.

$$\begin{align*} e^{x} &= \sum_{n=0}^\infty \frac{( x)^n }{n !} \\ e^1 &= \frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!} +...\\ &=1+1 +\frac{1}{2}+\frac{1}{6}+\frac{1}{24}+\frac{1}{120}+... \end{align*}$$

If you do the same for sine and cosine, and e^{(i x)}
$$e^{ix} = \sum_n \frac{(i x)^n }{n !}$$

you'll find a very clear relationship between the exponential functions and the trigonometric functions, using complex numbers.