The Wizard, Euler's Number (e) part 2

In our previous lesson we studied the origin of the Euler number e and also took a glance at its mathematical background(focused mainly on the deduction of the number).. In this part we'll see its magical and profound implications in our daily experiences and will also try to unravel why it is as it is..

Part 2 (The Mysterious appearances of e ):-

From our previous lesson we came to know that wherever e appears in an equation, somehow some sort of growth or decay is lurking in the picture... For this property e ,it appears not only in Population Equation or Radioactive decay but also in Calculas, Probability, Statistics some other crucial Equations and functions like Hyperbolic Function, Logarithmic Function,  Gamma Function, Hardy-Ramanujan Partition Formula of Mathematics and even in Statistical Mechanics, Planck's Radiation Law etc...

Today we'll take into account some regular cases like Population Growth, Radioactive Decay and we will try to deduce their conventional equation from the scratch..
Make sure to take a look at the previous chapter as I shall not discuss about the derivation of equations which I had derived before..


Population Growth:-

# From a Rational as well as Reductionism Perspective 

Suppose that there are Nₒ number of rabbits in a rabbit Community.. Now if the factor of growth by which the population of the rabbit increases in a year is R ..Now if the rabbits mate once in a year, then after a year the total number of the rabbits will be

 And after mating t years the total Number of Rabbits will be

We know from our experiences that there is no specific time when babies are born.. Now if the the babies are born in n times in a year then the growth rate will in a period will be (R/n) and the total times of this recurring event of new births will be change into nt and by definition previous Equation will be changed into the following..

So, the population after each period of newbirths will total population will increase by some amount and if we plot this data on our graph, it would be like a exponential step function..And as we place more and more data this curve will look more and more like pure exponential function..

The newborns are given birth almost anytime in a year.. So, This whole event will be very rapid.. 

As the incidence takes place more and more frequently, our previous Graph will change into the exponential curve..

(**please take a note that in our graphs the curve is defined even if time is negative.. So, I would request you not to take account the outputs of the function where the time is Negative)..

# In the Context of Calculus 

Now here I shall take another approach to the problem, in terms of Calculus..Which is pretty straightforward but you have to know about the basics of Calculus (like what is limit, differntiation, Integration)..

This approach won't show any particular or different promise right now.. But in later contexts it would be super crucial..

by definition the change in the number of rabbit or the growth rate of Rabbits is R and the prior population was Nₒ and the population at an instant is N(t) which changes with time.. So, we can say the total number of members in the population is a function of time..

And it is quite plausible, that the rate of change of the total population is proportional to the size of population..as it is logical to say that the more the rabbits in the community the more the new babies are born..

Now this is an example of what we call Differential equation.. We can arrive at a definitive idea about how the population changes with time In technical terms we can find a absolute function of N(t)..

We, Can see that we have at the same result from both approach..


#Deviation from actual Data

Our Previous understanding would have fully comprised the actual occurence but it doesn't..There are two main reasons for this..

                 • Firstly, we presumed the all rabbits who ever was in the community will always be a part of the community..In other words we took into account that no rabbits will neither die nor leave the population as well as become infertile..

But we know that this is certainly not the case..Some Rabbits will always die, some will leave the population..No one can stop that..

                • Secondly, There is always an upper limit about the number of the rabbits of the population..Because, there is a finite amount of their domiciles... 

 So, we have to introduce these parameters in our calculations..


Here I will take the the path of calculus (We can evenachieve the results from the rational approach but it would be too complicated)..

Now in the first case we can just introduce another constant in our equation of rate of change to encapsulate the mortality rate..which doesn't effect much rather than decreasing the growth rate..

and our population function will be 

and for our second concern we can slightly change the relation between rate of growth rate and the size of population..The final relation will be

Here K represents the maximum capacity of their Habitation.. As the number N(t) approaches to K the growth rate decreases by some factor.. At N(t) = K , the growth rate ceases to zero..

By solving the differential equation, we get,

Now this prediction almost accurately predicts our observations..

That's an example about where e appears magically..

Cast a glance at the Population Curve 

Radioactive Decay:-

This phenomena is same as the Population.. But, Since Due to Radiation of an atom makes it non-Radioactive after Radiation, The Growth Rate is negative or Decay rate positive..Which Reduces the number of Radioactive atoms less and less from the Sample as time pass by..

Here, I am not elaborating the incidence as it is identical to the previous one..But I am providing the Relevant Graph here.. Click here 

Shravan Ghosh 

Click here to go to the first part