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Defining $ e $

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

Let us consider the formula 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 us 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} $

By careful analysis of this table, it can be seen that value of this formula seems to approach some number. The number that this sequence approaches is called $ e $. From this, we can extract a limit that can be used to express $ 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 using the symbol e in a paper he wrote in 1731. Surprisingly, 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.

Upon further study of this constant, 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

Haran, B. [Numberphile]. (2016, December 19). e (Euler's Number) - Numberphile [Video File]. Retrieved from https://youtu.be/AuA2EAgAegE
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
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 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