What Is E?

MATH TERM PAPER

What is e ?

Yield to the unique and extremely important constant , the constant e is the next important and irreplaceable number in math world. The constant e has a wide range of applications, in which made it so considerably important that many scientific foundations can not exist without the existence of e.

"The value of e is found in many formulas such as those describing a nonlinear increase or decrease such as growth or decay (including compound interest), the statistical 'bell curve,' the shape of a hanging cable or a standing arch. e also shows up in some problems of probability, something counting problems, and even the study of distribution of prime numbers. In the field of nondestructive evaluation it is found in formulas such as those used to describe ultrasound attenuation in a material ..."

The use of this constant involves in many different application and of limits and derivatives. Many mathematicians have been successfully calculating the constant to amazingly complex number. For instance, we have a computation of e to 22 decimal places:

e = 2.7182818284590452353602875 ...

More incredibly, a scientist name Sebastian Wedeniwski has been calculated to 869,894,101 decimal places and with his success, we are able to understand more about the mystery of this constant. The constant e was first studied by the Swiss mathematician Leonhard Euler in the 1720s. His work is said to be an implied work of John Napier, who invented logarithms, in 1614. In 1744, Euler proved e to be irrational. In 1873, Charles Hermite proved e transcendental.

It is usually shown in calculus that

and

Euler also gave an expression for e using continued fraction:

F= [2, 1, 2, 1, 1, . . . , 1, 2n, 1, . . . ].

Our first goal is to show prove show that e is irrational. Using the formula we can immediately conclude number e is irrational, because the sequence is added up by infinite series and each single piece yields to different value. For if the contrary is true, that is, if where p and q are integers, we can certainly choose n larger than q. Then must be integer. On the other hand, , and since , we must have . Hence the integer = the integer plus a non-vanishing proper fraction, which is impossible.

A different approach to the formation formula

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