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

Hi, Clarissa.

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://jeff560.tripod.com/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/
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Associated Topics:
High School History/Biography
High School Transcendental Numbers
Middle School History/Biography