Origin of e=2.7181...
Jeff560
>Similarly, what is the origin of e=2.7182..., the base of natural
>logarithms? I would also like to know any interesting properties of
>this constant.
CD, e is widely used in calculus, but here are some facts about it.
1. If the expression (1 + 1/x)^x is evaluated for very large numbers,
it approaches the value of e. (The ^ symbol means "to the power
of.")
2. If P dollars are invested at an interest rate of i compounded
continuously, the amount of money after t years will be P times
e^(it).
3. If a large number of letters are written and envelopes addressed,
and if the letters are randomly inserted into the envelopes, the
probability that every letter will wind up in a wrong envelope is
about 37% or, more precisely, 1/e.
4. The graph of y = x^x has a minimum (that is, bottoms out) when x
is 1/e.
5. The first few digits of e are: 2.7 1828 1828 45 90 45 2 35 36.
A convenient way to memorize these digits is as follows:
Andrew Jackson, Andrew Jackson (year he was first elected)
angles of an isosceles right triangle
number of acute angles in this triangle
how old I am (not really)
how old I'll be next year
6. The most commonly used letter in English is e. (Off the subject,
but I thought I'd mention it....)
A fact about e that I'd like to know is how the letter e was chosen
for this constant. Some books say it was chosen to honor the Swiss
mathematician Leonhard Euler, but actually it was he who first used
the letter for this constant and I don't believe he would name the
number for himself. I have been told that when Euler introduced the
letter e, he had already used a, b, c, and d in the paper, so he used
e because it was next in the alphabet. Also, e could stand for
"exponential." There is a new book with the title (I think) "E - The
Story of a Number." The book is supposed to address this question,
but I haven't seen the book yet.
Jeff Miller
New Port Richey, FL
jef...@aol.com
rosullivan on BIX
Recently dhan...@iastate.edu (C Dhanwada) wrote:
> Similarly, what is the origin of e=2.7182...,
I like to think of e as the unique constant that defines the 'fixed
point' for the differential operator, namely e^x. This places e in a
unique position amongst numbers.
Rick O'Sullivan
US Army Research Laboratory
Chris Delanoy
J> 4. The graph of y = x^x has a minimum (that is, bottoms out) when x
J> is 1/e.
Here's one more very similar to this one:
The graph of y = x^(1/x) (or the xth root of x, as some say), then the maximum value of y is calculated when x = e.
- Chris J. Delanoy
Bill Tihen
Commonly used in finace, but applies to many many areas.
(1+1/n)^n essentialy indicates that change occurs at discrete intervals.
e on the other hand indicates that change occurs continually, this is because
e = (1+1/n)^n when n=infinity. (I'm sorry I'm not using the proper notation
for limits). It looks like this post may have been around a while so I
appologize if this is old news.
_____________
Bill Tihen <Bi...@Tihen.dublin.mv.com>
Dublin School Dublin, NH USA <Tihen!Bi...@dublin.mv.com>