The Wizard, Euler's Number (e) part 1

- January 26, 2021

After Archimedes' constant (π) the number (e) or Euler's number is perhaps the second most famous number in all Mathematics and Physics..It appears numerous occasions in various phenomenons from simple population growth, spread of pandemics, to radioactive decay etc.. So, in General wherever there is growth or decrement based on previous amount, somehow e is embedded there..

I am going to break this topic on two parts..In part one I will try to bring forth the basic properties of e and in second part I would try to explain why does it appear in various real world aspects...

Part 1(The Mathematical Sorcery)

The journey of begin with the curiosity of one of the Bernoulli brothers (Jacob Bernoulli)

Jacob Barnouli

Jacob Bernoulli was interested in Special kind of interest, Compound Interest..It is a type of interest where each time the amount of interest is counted on the the first investment plus the previously accumulated interest..

As example if we invest (£P) in a bank and the rate of interest over the investment in a year is r, then after t year we get the interest (i).. We can intuitively prove the mentioned formula..

So, after a year we get total of

Next year our prior investment is

So, at the end of 2nd year we get the interest

and our total balance is

by some rearrangement we get,

Now in general total Principle after t years looks like this,

Suppose that the bank reduce the rate of interest by 1/n but will give interest n times in a year.. In this case our previous formula will look like this,

Now, you might ask which one is better, whether if the bank gives interest by the rate of r over a year or by the rate of r/n, over n times in a year..For this let's take an example

If the bank follows the first method, then if we invest £1 with the rate of interest 100%..then after a year we get

£[1+{1×(100/100)}]= £2

Now if the bank follow the second procedure and give the interest 2 times in a year with the reduced rate of interest of 50% then after a year, we get

£1×[1+(50/100)]²= £2.25

which is certainly better than the previous one...

So, the Question that Jacob Barnouli asked himself, "what happen if I make the time period infinitesimally small(in other words we get the interest every instances) with the reduced interest rates.."

For our analogy the situation will look something like this,

This answer was given more or less 50 years later after Bernoulli posed the problem by another Mathematical Monster Leonard Euler..He quantitatively proved that this sum does blows to infinity but converges at a finite number and also proved that this number is irrational(which means that this number can't be expressed as the ratio of two integers)..

Leonard Euler

[***The proof I am going to show here might not be Euler's original proof.. I devised the proof from my own arguments(which I hope is correct)..]

If we expand the expression following the Binomial Theorem, we get

Now, if we work out the sum we get
e = 2.71828...
So, we can see from the above calculation that though the interest period goes to infinity, we won't get infinite amount of money.. The sum will converge at a finite number..

Now if we ask in general what happens if the the rate of increment here rate of interest is x or the total time of the increment is increased by a factor of x..

For small values of n (number of times interest applied per time period) this two conditions are different but as we increase the number of times of the interest applied per unit time, this two aspects become identical.. which means if number of times interest applied per time period is small enough then the total amount we'll get by the increment factor by x will be same as if we invest our money with the enhancement factor of 1 for x years...