Facts about e

Date: 11/09/98 at 17:22:19
From: clarissa
Subject: Algebra II - the meaning of "e"

We have been talking about the number "e" a lot in class lately. I was 
just wondering where it came from, and what uses it has in the real 
world.

Date: 11/10/98 at 11:53:04
From: Doctor Rick
Subject: Re: Algebra II - the meaning of "e"

Hi, Clarissa.

For some interesting information about e, try this page from our 
archives. In some of the answers there, you will find links to other
stuff on the Web that you won't want to miss:

  Transcendental Numbers
   http://mathforum.org/dr.math/tocs/transcendental.high.html   

Here is a good reference on the origin of the use of the letter "e" 
for the base of natural logarithms:

  Earliest Uses of Symbols for Constants (Jeff Miller)
   http://members.aol.com/jeff570/constants.html   

Some of the materials in the archives say that e stands for Euler's 
Number, but it isn't that simple. Euler first called it e, but it's 
very doubtful that he named it after himself. (Most things that are 
named for people were named by other people.)

Most of the uses of e in the real world start with a special property 
of the function e^x (e to the xth power). If you draw the graph of e^x, 
draw a tangent to the curve at some point (x, e^x), and measure the 
slope of the tangent line, it is exactly e^x. This property makes e^x 
very useful in calculus, and physics (the mathematical description of 
the real world) uses calculus a lot.

The number e popped up in an interesting way in my work recently. I did 
some work with Mercator maps. It turns out that the formula for 
computing distances on a Mercator map has a constant in it, and the 
constant is 180/(pi log e). It's easy to understand that the pi is 
there because the earth is round and pi comes from circles. The e is 
there as well, because it takes calculus to compute the distance, and 
it turns out that (you don't need to understand this) the integral of a 
secant function is the natural logarithm of a tangent function, and to 
convert from a natural log to a common log you need to divide by log e.

So, e is a very important and useful number. It's a nice number, too. 
When I was in high school I memorized the first 52 decimal places of e 
- a lot more than I learned of pi or the square root of 2:

   2.7182818284590452353602874713526624977572470936999595

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/