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Defining $ e $

There are a number of constants in mathematics that are defined by geometry in some way. For example, Archimedes' constant $ \pi $ is defined as the ratio between the circumference and diameter of a circle. Pythagoras' constant $ \sqrt2 $ is defined as the length of the diagonal of a square with sides of 1. Unlike these other constants, $ e $ is more easily defined using formulas.

The formula we are concerned with is the one for compounding interest:

                $ \begin{align} P' = P\left(1+\frac rn\right)^t \end{align} $

Where $ P $ is initial principle, $ P $ is final principle, $ r $ is the interest rate, and $ n $ is the number of times the interest is compounded. To make analysis simpler, we'll set $ P = r = 1 $ and $ t = n $.

Let's observe what happens to the value of $ P' $ when different values of $ n $ are used:

                $ \begin{array}{|c|c|}\hline n & P'\\\hline 5 & 2.48832\\\hline 50 & 2.69159\\\hline 100 & 2.70481\\\hline 100,000 & 2.71827\\\hline 1,000,000 & 2.71828\\\hline \end{array} $

As can be seen from the table, the value of the formula seems to approach some number. The number that this sequence approaches is called $ e $. We can extract a limit to define $ e $ from the formula for P':

                $ \begin{align} e := \lim_{n\to\infty} \left(1+\frac1n\right)^n \end{align} $

Leonhard Euler was the one who defined this constant with the symbol e in one of his papers in 1731. The agreed upon theory is that he named the number e not after himself but because it was the next letter in the variables he was using.

In his study of the number, Euler found the following fractional expansion for $ e $:

                $ \begin{align} e = 2 + \frac1{1 + \frac1{2 + \frac1{1+\frac1{1+\frac1{4 + \ddots}}}}} \end{align} $

With coefficients $ 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \cdots, 1, 2n, 1, \cdots $. He used this expansion to show that the number is irrational.

He was later able to find the first 18 digits of $ e $ using a different formula he came up with to define the number. This particular formula was based on the Taylor Series of $ e^x $:

                $ \begin{align} e^x = \sum^{\infty}_{n=0}{\frac{x^n}{n!}} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots \end{align} $

Obviously, evaluating this series at $ x = 1 $ yields approximations of $ e $. This method of calculating $ e $ is much easier to work out by hand than the initial formula based on compounding interest as it does not require taking large powers of already long fractions.

References
Maor, E. (1994). E: The Story of a Number. Princeton, NJ: Princeton University Press. Retrieved from http://webwork.utleon.edu.mx/Paginas/Libros/E%20-%20The%20Story%20of%20a%20Number.pdf
Haran, B. [Numberphile]. (2016, December 19). e (Euler's Number) - Numberphile [Video File]. Retrieved from https://youtu.be/AuA2EAgAegE
Sýkora, S. (2008, March 31). Mathematical Constants and Sequences. In Mathematical Constants and Sequences. Retrieved from http://www.ebyte.it/library/educards/constants/MathConstants.html
O'Connor, J. J., & Robertson, E. F. (2001, September). The number e. Retrieved from http://www-history.mcs.st-and.ac.uk/HistTopics/e.html
Sandifer, E. (2006, February). How Euler Did It. Retrieved from http://eulerarchive.maa.org/hedi/HEDI-2006-02.pdf