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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 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 is2.71828182845904523536028747135266249775724709369995....
Last edited by Jai Ganesh (2015-02-12 18:54:21)
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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The e constant is defined as the limit:
Euler's formula
The complex number
has the identity: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:
.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 derivativeThe global maximum for the function
occurs at x = e.
,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 Jai Ganesh (2015-02-13 02:25:51)
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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'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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Hi Agnishom Chattopadhyay,
Excellent post!
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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