About Euler’s Number (Exponential Constant )

About Euler’s  Number (Exponential Constant )

By Samar Ch. Dey

In the discussion of mathematical constant the number ‘e”  is the base of the natural logarithm. It is the Unique number whose logarithm is equal to 1. The numerical value of  e is

e = 2.71828182845904523536028747135266240097754…

Sometimes, it is known as Euler’s Number after the Swiss mathematician Leonhard Euler. Simply  it is called Euler’s constant. The number ‘e’ is also known as Napier’s constant . The constant was discovered by the Swiss mathematician Jacob Bernoulli in 1683 while he was studying compound Interest. This content ‘e’ can be written by the limit

Which gives a pretty nested radical result when x is taken to the power of both side.

Calculus :

                The function f(x) = e^x  is called the (Nataural)exponential function , and is the uniqueexponential function equal to its derivative

[ i e . ( e^x )= e^x ] . At x=1 , f(x)= e

= 2.71828   ( Approximetely )

and therefore its own antiderivative or Integration as well:  = e^x +c.  The natural logarithm or logarithm to the base e is the inverse function to thenatural exponential function.‘e’ is the unique number with theproperty that the area of the region bounded ley the hyperbola y =  .

ie x = t , y =  ,

The x- axis and the vertical

line x=1 and x=e is 1.

In other word  = lne=1

The derivative of logarithm with base e is

equal to   where x is a independent

variable. Therefore   (log _e ^x    ) =     , (x

it behaves well under differentiation

since there is now undetermined limit

to carry through the calculation .

Characteristic

The number e plays important and recurring roles across mathematics like the Archimedesconstant  , e is irrational number.  it can not be expressed as the form   were p , q  I and  q  . It is also a    transcendental number, it is not a root of any non – zero polynomial with rational coefficients.

The name “ Transcendental” comes from Leibnitz in his 1682 paper . Euler was probably the firstman to define transcendental Number in the modern sense. Charles Hermite proved that ‘ e ‘ is a transcendental number in 1873 . It is also the least transcendental possible with irrationality measure  .Mathematician Joseph Liouvill first proved the existence of transcendental Number in 1844 .He expressed first the decimal examples such as the Liovill constant in 1851 .