Natural Numbers (e) for Dummies

In this piece of writing, I assume you already have sufficient knowledge of what natural number is. I am also sure many of you are familiar with the constant e in middle, or even high school. The value of e itself was discovered by Leonhard Euler and is defined by an approximation as follows:

e = 2.718281828.....

I have no idea why it is named e, instead of any other letters, but I do know that e is not derived from the initials of Euler's name. This constant is very well known in the world of mathematics and plays an important role in calculus and other fields of mathematics.

Where does e come from, again?

In 1683, someone named Jacob Bernoulli studied compound interest in banks. The initial problem was this: I had $1, then deposited it in a bank that (somehow, seems a bit unrealistic) paid 100% interest once annually. With that, by the end of the year, the account should have $2.

Then, what if the interest was 50%, but was given every 6 months or twice a year? Would the amount of money in the end be different?

With a simple calculation, on the first interest payment, the amount of money becomes:

$1 x 50% + $1 = $1.5

In the second payment, the amount of money is:

$1.5 x 50% + $1.5 = $2.25

or

$1 x 1.5 x 1.5 = $2.25

Now, if the bank were to pay interest four times a year, 25% each for every 3 months, the result would be:

$1 x 1.25 x 1.25 x 1.25 x 1.25 = $2.4414...

If the interest was paid daily, then:

The calculations above converges to a specific value as we increase the amount of payment per year indefinitely, or mathematically speaking, as x goes towards infinity:

This is how e was found.

Unique Properties of e

In everyday life, the natural number e is often synonymous with the exponential growth rate. For example, population growth, radioactive decay, and economic income growth. The question is, why e?

In mathematics, the exponential function with the number e has the following graph:

Consider the green graph. It can be observed that as x or the independent variable gets bigger, the value of y will increase faster. This is quite in line with population growth, where the population continues to increase rapidly, instead of in a linear trend. So, the exponential graph can be the right approach to find the rate of population growth.

This still does not solve the bigger question, if we wanted to use a constant number as a base, why has it to be e, instead of some other arbitrary numbers like 2 or 3?

We will see how the constant e greatly simplifies calculus. The derivative of the e to the x is the function itself:

And the integral of the function is the function itself, plus the constant C.

Another thing that makes an exponential function with the number e is that the value of the gradient and the value of y at a point, as well as the area from a point to negative infinity, are always the same for all real x values! Please prove it yourself.

When we use other numbers, other constants will appear which complicate further calculations. For example:

This has shown that e is a special number, just like 0, 1, phi (3.1415...), and i (an imaginary number defined as the root of minus 1). When we see math as something that means a lot beyond just numbers, math becomes a lot more fun. As a bonus, there's one simple formula from Euler (Eulerian identity) that combines the constants elegantly in my opinion: