E

Today we are going to discuss the origins and applications of e. No, not the letter, rather the numerical constant: Euler’s Number. Just like pi, e is an irrational number meaning that it “goes on“ forever. If one had to approximate e, they could write 2.7182818, which is about equal to e. Even though this number is used in many equations, its seemingly obscure vale makes it hard to understand where if came from. The simplest way to derive e is to look at problems with continuously compounding interest. If you put money into a bank and it compounded your money at a rate of 5% every month, that money would be compounded 12 times a year. When something is compounded continuously, ever split second, it would grow by a certain rate. Lets use some examples to find the relationships between the different compound rates.

Lets say say you put $1 into an acount. The formula for compounding anything is A = P(1 + r/n)^nt where;

P = Original value

r = rate

n = number of times compounded per period (usually years)

t = amount of periods elapsed (usually years) Our first example will compound at %100 every year.

Whith this you get the equation:

1 + (1+1)¹ = $2

The next rate would compund quarterly or 4 times a year. Because the same %100 is needed, you can find the rate at 1.00/4

1 + (1+ 1/4)⁴ = $2.44

Continuing this same pattern, let's say it is compounded every day or 1.00/365.

1+ (1 + 1/365)³⁶⁵ = $2.71

The more and more precise that you get with how frequently an initial value is “compounded,“ the close the value will get to the true value of e. The output also switches between over and underestimating, narrowing on a more precise value of Euler's number. The way above is not the only way to find the value of e. Another method discovered by Euler also proves that e is irrational. That method is pictured below:

So what is this value useful for? Well, the first and most obvious is using it to determine continuously compounded values. This is useful not only with money, but also with bacteria, chemical reaction, ect. This constant is also used in physics for the same type of exponential problems.