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The Number e

Date: 6/3/96 at 1:5:14
From: Anonymous
Subject: The Number e 

Dear Dr.Math,

I have a project to do and I would like to know what "e" is. Is it 
a number like Pi? How does "e" relate to continued fractions?

Oliver Wong

P.S. I am a 1st year Algebra Student.

Date: 6/14/96 at 0:40:34
From: Doctor 
Subject: The Number e 

I'm not sure what you have or have not covered yet, so I won't assume 
too much. 

That's a pretty big question....  There are a lot of interesting
properties of "e", as there are with Pi. Some of them are pretty 
advanced, and some of them are easy to write down but hard to prove.
"e" is a number, just like Pi, and it has the value 
2.718281828459045235306.... So e is not the same as Pi, which is about 
3.14159265358979323846....  But while lots of people know Pi, not so 
many know about e.

Now, what exactly is so special about e?  Does it have a geometric 
meaning like Pi (which is the ratio of the circumference of a circle 
to its diameter)?  

Why does e=2.71828... instead of, say, 17.391526381?  Or -592?  
Or Pi/2 (which it isn't)?  Perhaps you've been told that the 
decimal digits of Pi never end, and they never repeat like            
22/7 = 3.142857142857142857.... Well, e is just like Pi in this 
respect; if you keep going, it never stops, and it never repeats.
What sets them apart from fractions like 3/4 and -65/11, is just this 
fact, which is more commonly known as irrationality. Fractions are 
*rational*, while numbers like the square root of 2, Pi, and e are

Although e is irrational, we can approximate it with rational numbers.  
If you're familiar with the concept of a continued fraction,

  e = 2+ ------------------------------
         1+ ---------------------------
            2+ ------------------------
               1+ ---------------------
                  1+ ------------------
                      4+ --------------
                         1+ -----------
                            1+ --------
                               6+ -----
                                   1+ ...

or, if you prefer,

 e = 2+1/(1+1/(2+1/(1+1/(1+1/(4+1/(1+1/(1+1/(6+1/(1+1/(1+1/(8+....

This method of computing e is very fast - try it by stopping the 
continued fraction somewhere, as I did here, and punch it into a 

Still, we haven't really talked about what e is useful for. To get to 
this, we need to talk about logarithms. Say you have 2^x = 64 
(2^x is my way of saying "2 raised to the x power"), and you want to 
solve for x. 

Well, we know that 
64 = 2*32 = 2*(2*16) = 2*2*2*8 = 2*2*2*2*2*2 = 2^6, so x = 6. 

How about 2^x = 5? There we're stuck, because 2^2 = 4 and 2^3 = 8 
so x should be somewhere between 2 and 3 to make 2^x = 5. 

There's another way of saying what x should be, and this is called the 
logarithm of 5 to the base 2. That is, x = log[2](5); (the 2 should be 
a subscript, like a power but typed a bit below the log, so it isn't 
log 10). In general, the solution to b^x = n for some given b and n is 
x = log[b](n). b is called the *base* of the logarithm. 

Logarithms are useful, but there is a particular kind of logarithm 
that is used the most often: the *natural* logarithm. This is just the 
logarithm to the base [e]. In fact, the natural logarithm is so useful 
that people often say "ln(n)" instead of log[e](n). Now, why is all 
this important? It's hard to say without going into a lot of details, 
but here's a little hint of the interesting things about e and ln(n):

Think about (1+1/n)^n for some value n. For n=1, this is 2. For n=2, 
this is 2.25. For n=5, this is 2.48832. For n=10, this is 2.5937.... 
For n=100, this is 2.7048.... For n=10000, this is 2.7169.... Can you 
guess what happens to (1+1/n)^n as n gets larger and larger? In fact, 
it becomes e. A way of expressing this in mathematical notation is

          lim      (1+1/n)^n = e.

(the "lim" stands for "limit"; we say "the limit as n goes to infinity 
of the quantity one plus one over n to the nth power is e.)

Another thing to think about: If you've graphed equations, look at the graph
of y=1/x. If you look at the region enclosed by y=1/x, the line y=0 (the
x-axis), and the lines x=1 and x=e, it looks like a rectangle but with one
curved side. What is the area of this shape? In fact, it is exactly 1.
Mathematically speaking, we say "the area under the curve y=1/x from 1 to e
is 1," or even better, "the *integral* of 1/x from 1 to e is 1." This is
because if we replaced the line x=e with some line x=b for some b>1, the
area of the region is the natural logarithm of b. Note the natural logarithm
of e is 1, because e^1 = e; that is, 1 is the exponent for which the base
(e) is equal to e.

And finally, for something I hope someone else will explain,

      e^(i*Pi) = -1, where i is the square root of -1.

-Doctor Pete,  The Math Forum
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