How was $e$ first calculated? I understand how $\pi$ is calculated, but I am interested in references that explain when and how the natural exponent $e$ was developed.  What mathematical principles are behind the value of $e$?
 A: Here are four different examples to peruse.


*

*Book: "e: The Story of a Number", Eli Maor 

*Website: The Number e

*Paper: e The Exponential - the Magic Number of Growth

*Journal: J L Coolidge, The number e, Amer. Math. Monthly 57 (1950), 591-602

A: My favorite explanation involves adding interest to an investment.
Let's say you invest some money and the bank gives you interest yearly. At the end of the first year you would have $I(1+r)$, where $I$ is the investment and $r$ is the interest rate as a decimal, i.e. 5% = 0.05.
Now let's say you're impatient and you want interest more often. The bank might offer to apply half the interest rate twice in a year. So instead of 5% once, you get 2.5% twice: $I(1+\tfrac{r}{2})^2$.
Now let's say you're still not happy and you want interest more often. The bank might offer to apply quarter of the interest rate four times in a year. So instead of 5% once, you get 1.25% fourtimes: $I(1+\tfrac{r}{4})^4$.
If the bank applies the interest $n$ times in a year then you would have $I(1+\tfrac{r}{n})^n$ after a year.
What happens as $n$ gets larger and larger? Say the bank start paying interest not monthly, not weekly, not daily, not even hourly. What if the bank pay continuous compound interest? Well:
$$\lim_{n \to \infty}\left(1+\frac{r}{n}\right)^n = \operatorname{e}^r$$
This is why the exponential turns up in population dynamics and nuclear decay. Someone is being born almost every instant which gives a tiny continuous increase to the population, just like the bank adding tiny amounts of interest continuously.
If you want to work out $\operatorname{e}$ then just note that
$$\lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n = \operatorname{e}$$
For a very good approximation, just pick a very large $n$. For example:
$$\left(1+\frac{1}{10,000}\right)^{10,000} \approx 2.71815$$
Of course, a better way to find an approximation would be to use its power series:
$$\operatorname{e}^x \sim 1 + x + \tfrac{x^2}{2!} + \tfrac{x^3}{3!} + \cdots$$
This is much easier to calculate, and it gets very close very quickly:
$$1 + 1 + \tfrac{1}{2!} + \cdots +\tfrac{1}{10!} \approx 2.71828$$
