Origin of the number 'e' and the mysterious properties of the Exponential Function

The number 'e' appears very commonly in a lot of natural processes. In fact, it is bested only by 𝜋 as far as its number of times of appearance is concerned. But it is indeed quite mysterious as to why nature has such a preference for a number that is neither a whole number but also irrational. So in this article we are going to find about the discovery about this mysterious number.

Let's see how the number 'e' arises from calculating the interest of 1 unit of currency annually. Suppose we have 1 dollar and we get 100%  interest annually. We deposit 1 dollar on the New Year's Day. Then at the end of one year, the total amount stands to be 2 dollars. This is because of the 1 dollar initially deposited plus the interest obtained.

Pretty, simple. Now if we calculate the interest twice every year, then after six months, the interest calculated is 1 plus one-half of one i.e. 3/2(one-half because only half of the year has passed). After another six months the interest calculated is 3/2 plus one-half of 3/2. Finally we get 2.25 dollars at the end of the year. 

If it is not understood why 3/2 is used instead of 1 to calculate the interest for the last six months( I personally didn't understand initially), this is because after the first six months, the principal amount( the amount for which the interest is to be calculated) becomes 3/2 instead of 1 as the interest accumulated in the first six months should be added.

The reason for the seemingly useless mathematical manipulation will become apparent later on.

If the interest is calculated thrice every year i.e. once every four months then

If the interest is calculated four times every year i.e. once every three months then

We can see that in the last step, a pattern is noticeable. To calculate the net amount at the end of the year, we just have to calculate the reciprocal of the number of times we want to calculate the interest each year, add the reciprocal to one and the take the power of sum to the number of times we want to calculate the interest each year.

Therefore if we want to calculate the annual interest at the rate of 'n' times each year then the expression will be

Now, let us calculate the value of the net annual amount for a few increasingly larger values of 'n'

We can see that as the value of 'n' increases, the value of the given expression reaches a definite value. That definite value is defined as the Real Irrational number 'e'.

Mathematically, 'e' is defined as the value of the given expression as the value of 'n' tends to infinity.

As it turns out, this number occurs repeatedly in nature in a number of different cases. The value of 'e' calculated using a calculator turns out to be 2.7182818285.......If this number is special then its power function e^(x) must also be special. This function indeed is special. It turns out that the function e^(x) is the only function whose derivative is the function itself(i.e. invariant under differentiation). Let's see why this is the case.

To find this function, we use Power Series. This powerful method was discovered and used by Newton himself. In Power Series Expression, a function is defined as an infinite series of polynomials with constant coefficients.

Now to find the function that is invariant under differentiation, first we take the derivative of f(x)

and then equate it to the function f(x)

equating the corresponding coefficients

this implies

By this expression, all the coefficients of the function f(x) can be found.

Therefore the function f(x) is

 Now, for 'a sub 0' equal to 1 we get

Things get interesting when we put x equal to 1

Which is exactly equal to 'e' calculated initially!!
What is even more interesting that f(2) is exactly equal to e^(2)

From these examples, it can be safely( in this case at least) concluded that-

This formulation is quite convenient, 'x' is no longer required to be a whole number, it can be any real number and even a complex number. It also redefines exponentiation, it is no longer repeated multiplication. 
We have finally reached the conclusion of this article. We began with the origin of the mysterious number 'e' and as it turns out, it has a pretty humble origin. From there we went on to find the properties of exponentials and derive the expression of e^(x). The Power Series method is literally quite powerful and can be used in a wide range of applications. Please mention any advice for improvement in the comment section. Thank you!! 

(An exciting thing to try out- in the series, substitute ix instead of x and separate the real and the imaginary parts)

(This article is intended to be only for reference and to be an aid to proper understanding. In no way is this article intended to be a rigorous Mathematical Proof for the concepts and the theorems involved)