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#1 2015-02-12 15:51:17

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,306

e, the letter and number...

The e - life..

In this world of e-mails,
e-ticket
e-paper
e-recharge
e-transfer &
e-governance...

e is the most eminent and prolific letter..

men or women don't exist e..

house or home can't be made without e...

bread or butter can't be possible without e...

e is the beginning of existance and end f trouble...

its not present in 'war' but twice in 'peace'...

its once in hell but twice in heaven...

emotional relations like father, mother, brother, sister, wife and friends have e...

e represents effort and energy...

but for e,  life, wife, friends or hope would be non-existant...

see, hear, smell, taste, touch..are indispensable for the letter e....

e signifies eternity, endurance and everlasting...

The words eulogy, euphoria, euphemism are incomplete bur for e...

e in Mathematics :

The number e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828, and is the limit of

as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series.


The constant can be defined in many ways. For example, e can be defined as the unique positive number a such that the graph of the function
has unit slope at x = 0. The function
is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.

The number e is of eminent importance in mathematics, alongside

. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant
, e is irrational: it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is

2.71828182845904523536028747135266249775724709369995....

Last edited by ganesh (2015-02-12 18:54:21)


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#2 2015-02-12 23:42:27

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,306

Re: e, the letter and number...

The e constant is defined as the limit:


Euler's formula

The complex number

has the identity:

i is the imaginary unit (the square root of -1).
is any real number.

The number e occurs naturally in connection with many problems involving asymptotics. A prominent example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers e and π enter:

.
A particular consequence of this is

.

As in the motivation, the exponential function

is important in part because it is the unique nontrivial function (up to multiplication by a constant) which is its own derivative


and therefore its own antiderivative as well:

The global maximum for the function


occurs at x = e. Similarly, x = 1/e is where the global minimum occurs for the function

occurs at x = e.

,
defined for positive x. More generally,
is where the global the function 
is
for any n > 0. The infinite tetration

  or

converges if and only if
(or approximately between 0.0660 and 1.4447), due to a theorem of Leonhard Euler.

Number theory

The real number e is irrational. Euler proved this by showing that its simple continued fraction expansion is infinite. (See also Fourier's proof that e is irrational.)

Furthermore, by the Lindemann–Weierstrass theorem, e is transcendental, meaning that it is not a solution of any non-constant polynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); the proof was given by Charles Hermite in 1873.

It is conjectured that e is normal, meaning that when e is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).

Complex numbers

The exponential function ex may be written as a Taylor series

Last edited by ganesh (2015-02-13 02:25:51)


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#3 2015-02-13 00:52:29

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,838
Website

Re: e, the letter and number...


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#4 2015-02-15 03:30:42

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,306

Re: e, the letter and number...

Hi Agnishom Chattopadhyay,

Excellent post!


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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