Saturday, 24 June 2017

Why 'e' is a Universal Constant - 2.718 ?

What does exponential mean?

Physical Interpretation of the Exponential Constant e

Exponential is a mathematical term,which indicates the variation of anything in terms of the exponential constant - e. Whenever something gets increased or gets reduced in terms of 2.718 times,we call it exponential

What is e?

e is a universal constant , having a value approximately 2.718

What does e represent? 

The e represent the scale of growth of such an exponential action.

In the physical world,we can see things grows and dies. Mostly living things,or running phenomenon's. Nature have a policy of growth. We don't know why; may be because that's easier,faster and comfortable. The term exponential refers to such a growth. 

Generally,everything grows by multiplying. The micro organisms like bacteria,Radio Active decaying etc

So How do we interpret e?

Before interpreting e and exponential growth,lets examine how things multiply.

How do nature multiplies things?

We know, the easier and simplest multiplication is doubling. Nature too ,depends on it.
1 become 2,2 become 4,4 become 8 and so on.
Mathematically we can represent this in terms of 2's power or 2's multiples. 
2^x is the general equation of multiplication. It continues as 2*2*2*2*2*2...

Then what Is the need of exponent?

We already said, exponential means growth/multiplying. So,lets consider this 2*2*2*2* the growing rate.

Growth = 2^x

But from Where did we got this 2?

Actually it is 1 in the beginning. 1 becomes 2,2 becomes 4...
So,in the beginning there was 1,it multiplied into 2. Which means,the multiplication was 100%.

A 100% multiplication means,doubling. Getting the equal amount again,means getting the whole again. The whole is 100%. Thus we can think, the 2 is actually the initial 1 and the multiplied 100%

2 = ( initial condition + finalising condition)

Initial condition is the whatever amount we had. If we wanna double; the finalizing condition is a 1. If we only wanted to increase the growth by a small percentage, the value of 1 reduces up to 0, or becomes larger when we want a big growth.

Generally big and low growths are artificial. Every growth (positive or negative) depends upon the current condition (current condition will be the initial condition of that particular instant)
A negative growth is nothing but decay. 

So,try drawing a flow chart. 1 becomes 2,2 becomes 4 and so on. 
The interesting twisting fact is that, at the beginning the responsibility of growth was with the 1. 
In the next step the growth,which is at a constant rate,became the responsibility of 2. 
This two divides the responsibility and creates another 2-the new 4. 
So totally the responsibility of growth resides now with 4.

To easily visualize this, consider a family. Parents give birth two 2 children. The responsibility of family now resides within the two children. Who will marry, give birth to 2 each. The total 4 children again marries and give birth to 8. The only difference here is no marriage is taking pace. 

Finding the value of exponential function
Flow Chart 

Now remember, initially the first grandfather had the responsibility to start his family. Only him can - 100%.
Next the two children had the equal responsibility to grow the family 50% and 50%
Generation by generation, it becomes - 25%,25%,25%,25%.
Then next generation will be 12.5%,12.5%,12.5%,12.5%,12.5%,12.5%,12.5% ,12.5% and so on.

Finding the value of exponential function
Multiplication / Growth
(The Green and Blue markings
 represents the indivitual growth,
which we finally adds)

Finding the value of exponential function
Individual Contribution

Finding the value of exponential function
nth Generation

Finding the value of exponential function
Generation Growth

Finding the value of exponential function
Finding the Limiting Value of e

The responsibility of each generation reduces,but the growth rate of family is equal ,if we consider the whole members of each generation.

Now we can consider our doubling equation. 
2 = (1 + 1) becomes, 2 = (1 + 0.5 + 0.5),It again becomes 2 = (1+ 0.25 + 0.25 + 0.25 + 0.25) 

When it undergoes x times growth, the final part,splits into  x equal parts. Where the individual contribution of a single one (the single family of nth generation) will be 1/n 

We take the 2^x equation,and substitute  (1 + 1/n) to get the individual growth sequence.

To get the total value of entire families of nth generation, we add them,n times. We don't know n, so we consider n is approaching infinity,and applies the techniques from calculus to solve. This will give us a value Approximately 2.71828... That's how e got born. This is not the exact mathematical approach,yet it is simpler than any textbook language. (I hope)

Every Process in nature,multiplies in this mode, at the rate of 2.718. It is a factor like Pi,which is 3.1428...
Pi is actually the ratio between circumference and diameter of a circle,while e is the constant growth factor of nature.

Now we know,what is e.
When we make a function,on its terms,which is causing the growth ,we call the growth is exponential,and function is - Exponential Function

Finding the value of exponential function

This is how,if we analyse the limit graphically
(image credit : math is fun)

I will be telling more about e,in my next article :
what is the application of e?
what is the physical meaning of e intuitively? 
what is the significance of e? 
why always e?
where all we can find e?
How is e related to logarithm?

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