# e – like everywhere

Everybody doing any kind of geometry, calculus or higher algebra will have stumbled over this number:

2.71828182845904523536028747135266249775724709369995…

More commonly it is known as $e$, Euler’s number. You may wonder, where does this number come from and why is it so important? Well… If you are like me that is: waiting for simulations to finish, having 20 minutes of spare time on your hands, and having a tenacious drive to understand things. So come along as I explain two different ways that result explain where this beautiful number comes from.

## Compound interest

Rather than diving into derivatives and integrals, let me begin with an explanation that may be more comprehensible and easily relates to reality: compound interest. Interest is a percentage by which the amount of your savings in a bank increase. Imagine you are lucky and receive an interest offer of 100% per annum. This effectively means that your savings will double after every year.

$f\text{ }:\text{ }n \to (1+100\%)n = 2n$

Now, what if the bank offers you 50%, but after 6 months. Would you take the deal? Quick napkin-maths will show that you should definitely take that deal. In fact, you would not only double your savings, but receive 1.25 times what you already have.

$f\text{ }:\text{ }n \to (1+50\%)(1+50\%)n = (1.5)(1.5)n = 2.25n$

Since no bank in the world would offer such extreme interest rates, let’s carry on with this slightly ludicrous thought experiment, Let’s decrease the interest rate to $\tiny{\frac{1}{12}}$, which is paid every month. The formula would now become quite long, yet the result is an increase by 1.613. Rather than writing out the full chain of products I will simply raise the brackets to the power of 12:

$f\text{ }:\text{ }n \to (1+\frac{100}{12}\%)^{12}n = 1.083^{12}n = 2.613n$

As you can see, despite reducing the interest rate, if the payout frequency is inversely proportional, you will always receive a tiny bit more. So you might think: so why not pay out every day?

$f\text{ }:\text{ }n \to (1+\frac{100}{365}\%)^{365}n = 1.00273^{365}n = 2.714n$

Or why not every second of the year?

$f\text{ }:\text{ }n \to (1+\frac{100}{31557600}\%)^{31557600}n = 1,0000000317^{31557600}n = 2.7182n$

Very cool!