The Euler Number e

    We will discuss what the following means:

    as      ;

and in doing so, introduce the number e as discovered and defined by Jacob Bernoulli (1654-1705) and Leonard Euler (1707-1783), respectively.
    First we start off by mentioning Jacob Bernoulli (1654-1705) and his study of the calculus of exponential functions in 1697. His work can be found in his publication: Principia calculi exponentialium seu percurrentium where he investigates properties of exponential functions by using recently developed methods of calculus. It is interesting that he first recognized the importance of the number e, by studying compound interest problems. Here is what we are talking about, in modern terms:  

    Here are some values of the function

Definition (Euler's Number e) Euler's number is defined by the value of the limit:  

    Finally, we end this topic with five hundred digits of the number e.

2.71828182845904523536028747135266249775724709369995957496696762772407663035354759
45713821785251664274274663919320030599218174135966290435729003342952605956307381
32328627943490763233829880753195251019011573834187930702154089149934884167509244
76146066808226480016847741185374234544243710753907774499206955170276183860626133
13845830007520449338265602976067371132007093287091274437470472306969772093101416
92836819025515108657463772111252389784425056953696770785449969967946864454905987
9316368892300987931

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Cite this as:
The Euler Number E
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/the-euler-number-e.html