e

[ydoaPs]

[math]\lim_{n\to\infty}(1+\frac{1}{n})^n=e[/math]

how was this discovered? where did e originally come from?

[Martin]

... where did e originally come from?

 

e goes back at least to newton

because he discovered the series called Mercator series

(which was also discovered by Mercator and published in 1668)

 

 

http://mathworld.wolfram.com/MercatorSeries.html

[kotake]

Newton first introduced the value, as an infinite sequence.

[Martin]

the invention of logarithms (base 10, though) is traced back to

a pamphlet published in 1619 by John Napier.

 

 

logarithms caught on, everybody wanted to use them because they didnt have calculators and they help do stuff like take square root and cube root etc, so they had to prepare log tables.

 

but it takes a significant amount of work to prepare logarithm tables when you are doing everything by hand.

 

It saves work to use "natural" base logarithms instead of base-10. there are fewer steps involved in calculating the logarithm base e. You can use the mercator, for instance.

 

Or you can use that limit you mentioned.

 

So after 1619 people were bound to stumble on natural logs and the base of the natural logs, which is e.

 

I dont know the first moment anyone glimpsed the number e, but I guess it was between 1619 and 1668.

 

Could well have been Newton.

 

Yes! look here

http://mathworld.wolfram.com/e.html

 

it says that newton published in 1669 the series that defines e

(the sum of the reciprocals of the factorials)

 

it also says that the letter e was chosen to honor Lenny Euler who proved that e is transcendental but that was much later I guess around 1810

[Martin]

Newton first introduced the value, as an infinite sequence.

 

I agree with kotake

hello kotake, I didnt see your post here until after I put up mine

but what you say is right. he published the infinite series for e, that you

mention, in 1669.

 

but even earlier than that, it seems, he discovered an infinite series

for calculating log(1+x)

 

this would be useful in preparing log tables (natural base logarithms)

 

[math]log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} +...[/math]

[srh]

"The notation e made its first appearance in a letter Euler wrote to Goldbach in 1731. He made various discoveries regarding e in the following years, but it was not until 1748 when Euler published Introductio in Analysin infinitorum that he gave a full treatment of the ideas surrounding e."

 

My maths lecturer showed the class a book the other day that had, among other things, Eulers letter to Goldbach, where he defines and uses the term e, officially for the first time. From memory, he related it to a flying spherical object, which was because he was doing military research at the time, i.e, a canonball? That could be wrong.

 

The following link has a bit more about the letter, halfway down the page:

http://www.gadsdenst.cc.al.us/math/Jesse's%20Papers/michael_shifflet.htm

[ydoaPs]

i wish i had a background in calculus so i could understand all of the series and other calculus-esque thingys. the only background i have is daves two lessons that basically amount to: [math]\lim_{h\to0}\frac{f(x+h)-f(x)}{h}[/math] and [math]y=ax^n\rightarrow\frac{dy}{dx}=anx^{n-1}[/math].

[john5746]

Googled this

 

History of e