Edit made on November 20, 2008 by ColinWright at 21:33:53
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!! Definitions
So here are some characterisations of /e:/
 EQN:\frac{d}{dt}e^t=e^t 
 EQN:\frac{d}{dt}\log_e{t}=\frac{1}{t} 
 EQN:e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n 
 EQN:e=\sum_{n=0}^\infty\frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots 
 EQN:\int_{1}^{e}\frac{1}{t}\,dt={1} 
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The number /e/ is used to represent Euler's Number.
Its approximate value is 2.71828182845904523536...
/e/ turns up in several places, and has several definitions.
It is an irrational number  you can see the proof that e is irrational.

!! Motivation: Compound Interest
Suppose you have £1, and someone offers to pay you 100% interest, but only once a year.
At the end of the year you'll have £2.
Now suppose someone offers to pay 50% twice a year. How much will you end up with? After 6 months you get your first 50% of £1, giving you £1.50. Then at the end of the year you get an additional 50%, but this time it's 50% of £1.50. That makes a total of £2.25, which is clearly a better deal.
What about 20% paid quarterly? Is that bigger? What about 1% paid 100 times a year. Is that bigger?
The amount you get for /n/ payments each of EQN:\frac{1}{n}100% is given by this formula:
* EQN:(1+\frac{1}{n})^n
As EQN:n\rightarrow\infty this quantity approaches a limit. That limit is /e./
!! Differentiation
Another question we might ask is this.
If you take any point on a cubic equation, put a tangent, and ask what is the slope of that
tangent, the answer is given by the quadratic equation that is the derivative of the cubic.
Similarly, if you take any point on a quadratic equation, put a tangent, and ask what is the
slope of that tangent, the answer is given by the linear equation that is the derivative of
the quadratic.
Is there any function which, when you take its derivative, is the same as what you stated with?
Yes, and one of them is EQN:e^x. We can say even more, though. Any function /f/ that
satisfies the equation EQN:\frac{d}{dx}f(x)=f(x) is of the form EQN:f(x)=Ce^x where /C/
is a constant.
!! Other definitions
So here are some characterisations of /e:/
 EQN:\frac{d}{dt}e^t=e^t 
 EQN:\frac{d}{dt}\log_e{t}=\frac{1}{t} 
 EQN:e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n 
 EQN:e=\sum_{n=0}^\infty\frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots 
 EQN:\int_{1}^{e}\frac{1}{t}\,dt={1} 


!! Other sightings
Finally, there are places where /e/ turns up:
* Stirling's formula: EQN:n!\approx{\left(\frac{n}{e}\right)}^n\sqrt{2n\pi}
* If you rearrange /n/ objects at random, the probability that none end up where they started is approximately 1/e.
* If you rearrange /n/ objects at random, the probability that /*none*/ end up where they started is approximately 1/e.
* Euler's formula and Euler's identity.
* Continued fraction:
** EQN:e=2+\frac{1}{1+\frac{1}{{\mathbf{2}}+\frac{1}{1+\frac{1}{1+\frac{1}{{\mathbf{4}}+\frac{1}{\ddots}}}}}}
** More compactly: /e=[2;1,2,1,1,4,1,1,6,1,1,8,1,...,1,2n,1,...]/
*** The fact that this is an infinite continued fraction is another proof that e is irrational.
Many other series, sequence, continued fractions, and infinite product representations of /e/ have also been developed.

* http://www.google.com/search?q=euler's+number