It was not until Bernoulli ( Johann I, 1704 ) that the symbol e and the name exponential were first used. This arose because it was necessary to have a way of representing ex. The calculation of logarithms by infinite series was carried out by Gregory, Wallis and Halley beginning in the 1670's. It was not until much later ( 1742 ) that William Jones ( 1675 - 1749 ) used exponents to calculate logaritms and allowed present day methods to be developed. However it should be emphasised that Euler was the first to introduce this concept, and the symbol e, for the base of natural logarithms in an unpublished paper of 1728 ( he quoted e to 27 places ).

The number e is the base of Natural logarithms. Let us begin our description of e by writing down the first digits

e=2.7182818284590452353602874...

It is not known who was first determined:

It is well known that

for k>=0. This follows directly from the Taylor series expansion about x=0 for f(x)=e^x. From this we have for x=1

which has reasonably fast convergence and so may be used to calculate e. .

Continued Fraction Expansion

Here also is the regular continued fraction expansion for e:

There are several ways to compute e accurately. The Fibonacci numbers can be also used to compute e.

It is easy to verify that

where D is operator of differentation.

We shall now introduce the next operator :

As will be seen, we usually deal with the next operator`s equations for the Fibonacci and Lucas numbers :

and

By using this equations, we can write :

For the Fibonacci numbers we have:

For n=1 we have the next expansion for e:

        

        

See Also:  Archimedes` constant PI and the Golden Section   Archimedes` constant PI and the Fibonacci Numbers   Machin's Formula    Cassini formula for Fibonacci numbers    Archimedes` constant PI and the Square Root of 3    Fine Structure Constant and the Golden Section   

     

     

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