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What is the value of exponential e? We know this value is varying between 2 and 3. But...

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amitparmar | Student, Undergraduate | (Level 3) eNoter

Posted February 24, 2010 at 5:37 AM via web

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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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maadhav19 | College Teacher | (Level 2) Assistant Educator

Posted February 24, 2010 at 8:24 AM (Answer #1)

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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.

 

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neela | High School Teacher | (Level 3) Valedictorian

Posted February 24, 2010 at 12:48 PM (Answer #2)

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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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maria-vivanco | Student, Grade 11 | (Level 1) Valedictorian

Posted July 30, 2014 at 10:06 PM (Answer #3)

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e=2.7182818284 

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