What is the value of exponential e?

We know this value is varying between 2 and 3. But what is its actual value?

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In mathematics, the constant e has a number of important properties. The brief answer to your question is that it has a value of

e=2.7182818284 (to ten decimal places)

It is defined as the function f(x)=e^x whose derivative at the point x=0 is exactly 1. That is, the slope of the tangent line at x=0 is 1. For other numbers, the slope of the tangent line is not exactly 1. Another definition of e is that it is the number given by the limit as n approaches infinity of

(1+ 1/n)^n

Its usefulness lies in its properties. For one, the derivative of

f(x)=e^x

is

f'(x)=e^x

and for other exponential functions, the derivative of something like

f(x)=e^ax

is

f'(x)=a*e^ax

Another useful property is its relationship to complex numbers. We define Euler's formula as

e^ix = cos x + i sin x

giving rise to trigonometric representations on a graph in the complex plane.

**e** , like** pi**, is the most interesting constant in mathematics.

the definition of e according to Mathematical Analysis is given by:

(i)

e = limit x-->infinity of the finction (1+1/x)^x, aternatively,

(ii)

e = Limit x--> 0 (1+x)^(1/x), or

(iii)

e = 1+1+ 1/2!+1/3!+1/4!+1/5!+....+1n!+........................, a positive valued positive termed decreasing series.

Its value is clearly greater than 2 as 1st 2 terms itself add up to 2. And the series becomes bounded above as:

e=1+1/1!+ 1/2!+1/3!+1/4!+1/5!+....+1n!+........................ is

< 1+1+1/2+1/2^2+1/2^3+1/2^4+1/2^5, as each terms in the former series is < each term in the latter series, except the 1st 3terms which are equal. But the latter adds up in limit to 1+(1-1/2)^(-1) = 1+2 = 3. So , 2 <e <3.

So e has definite value.

We can calculate approximate values of e from (1+1/n)^n or the series form at (iii) .

e is not a rational number. It is an irrational number.

e is not the root of any rational polynomial. It is not a surd.In other words, it is a transcendental number.

It is Leonard Euler, the Swiss mathematician who started calling the number by the name **e**. And now it is Euler,s number or Euler's constant.

e^(i*pi) = -1 Or e^(i*pi)+1 = 1 is one of the wonderous equations in Mathematics as it connect two famous transcental numbers ** e** and ** pi** and relates with the rational real number **1 **or **-1** and also imaginary number **i** or **(-1)^(1/2)**. This equation is enlightened to us by Leonard Euler.

So the value of e is given by:

e = 1+1/1!+ 1/2!+1/3!+1/4!+1/5!+............ in infinite series form.

Also if you want in rational form the value of e is like:

e = 2. 7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274..

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