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EXCELLENCE OF "e",

DEFINING  “e”

     Let us attempt a practical definition of the constant e in this article.

FINANCIAL CRISIS:
    The following formula calculates the compound interest and is used by the banks.  The formula is so designed to calculate the interest n times the year and each time the interest is added to the principle enhancing it continuously.  This is called the compounding the interest.
S = P*(1+r//n)^n*t

   S--- Amount yielded.
   P --- Principle
   r--- Percentage rate of interest
   n ---number of times interest compounded in a year.
   t---- Number of years.

      First, let us take the principle P as one dollar, rate of interest r as 100% = 100/100= 1; time t as one year and number of times n = 1, the interest is calculated only once at the end of the year.  So what is the yield S?

S=1*(1+1/1)^1*1=1*(1+1)^1=2


So one dollar becomes 2 dollar at the end of a year for 100 % rate of interest. 

     Now let us compound the interest 3 times the year- i.e.- every four months.  So n = 3, retaining other quantities same, the yield will be

S=1*(1+1/3)^1*3=1*(1+1/3)^3=2.37

    $1  becomes 2.37 dollar in a year, when the interest is compounded every 4 months.

     Now, let us do it 6 times a year- i.e.- every 2 months.

S=1*(1+1/6)^1*6=1*(1+1/6)^6=2.52

     So 1 dollar yields 2.52 dollars, when the interest is compounded every 2 months

     Suppose we compound the interest every day, every hour, or every second, will the yield sky rocket?  Let us find out




No.of times a year-compounded. n
Amount yielded P
1
2
3
5
10
100
1000
10000
100000
1000000
.
.
.
2
2.25
2.37
2.488
2.5937
2.7048
2.7169
2.71814
2.718268
2.7182804.
.
.
.


     So the yield reaches a dead end.  The one dollar cannot be grown more than the magic number 2.7182804...., however the larger n may be.  Here we try to compound (grow) the interest as continuously as possible,  but one dollar only becomes 2.71.. dollar in one year in the seamless continuous growth.

NATURES IDEA:  – A CONSTANT FOR GROWTH

    The limit 2.7182804...  is known as euler's constant and is denoted by” e”. An important mathematical constant  on par with π.

DEFINITION
  ' The constant e = 2.7182804 … is the maximum yield when unit quantity grows continuously at unit rate in unit time'

  The above definition holds good for decay process also.  We can simply say e is the unit of natural growth or decay.
   Many natural processes of growth and decay does not take place step by step, but rather continuously.  Wherever continuous change takes place, e will show its face in some form.

Examples:  growth of population, growth of a baby, raise and fall of electric current, fall of temperature of hot cup of coffee, etc..

MATHEMATICAL PROOF:

    Let us start with original banker's formula

S = P*(1+r//n)^n*t

  for larger n, using binomial approximation we can write.

S = P*(1+1//n)^r*n*t
S = P*(1+1//n)^n*(r*t)

When n becomes larger and larger,- i.e- when we make growth as continuous as possible .

(1+1//n)^n approaches 2.7182804... = e

hence
S = P*e^(r*t)

     Now let us write the above formula in other notations.
   Let S = N, number of quantity yielded at the time t.
  P =N0, the initial number of quantity at the time t = 0
  r= rate of growth for unit time.  r  will be negative for decay.
  t= number of units of time
so     N = N0*e^(r*t)

   let N0 = 1 , initial unit quantity
   r= 1, unit rate of growth
   t=1, unit time 

therefore  N = e
    Hence our definition of e has been cross checked

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