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### Continued Fractions

```
Date: 10/07/97 at 16:15:35
From: Sarah
Subject: Continued fractions

Dear Dr. Math,

I was recently assigned a project in math class in which I must
present to the class a lesson on continued fractions. Up until this
point I have not been able to figure out exactly what "continued
fractions" are. I have searched the Net and also a few math textbooks
and I have not found even the simplest definition of the fractions.
I would appreciate you sharing ANY knowlegde that you have on
continued fractions.

Thank you,

Sarah
```

```
Date: 10/10/97 at 11:53:05
From: Doctor Luis
Subject: Re: Continued fractions

A continued fraction is an expression of the form,

A = a_0 + 1/c_0

where

c_0 = a_1 + 1/c_1

c_1 = a_2 + 1/c_2

.
.                   (0 denominators are not allowed, of course)
.

If the process stopped at, say,

c_n = a_(n+1) + 1/c_(n+1)

then we would have a finite continued fraction.

Now, the initial question is whether the process converges in the
infinite case.

If a_0 is an integer and the rest of the a's are positive integers,
the process converges to a real number; and all real numbers can be
represented in this way. Simple (or ``regular'', as some authors
write) continued fractions are those with these requirements on
a_0, a_1, a_2, ....

It is standard to use [a_0,a_1,a_2,a_3,...] to represent

a =  a_0 +      1
--------------
a_1 +    1
---------
a_2 +  1
----
a_3 + ...

There's an interesting way you can find a continued fraction
representation of 1 + the square root of 2.

x   = 1 + sqrt(2)

x - 1 = sqrt(2)

x^2 - 2x + 1 = 2

x^2  = 2x + 1

1
or     x    = 2 + ---
x

"Substituting" for x in the lefthand side

1
x   =  2 + --------
1
2 + ---
x

Repeating this process an infinite number of times you'll
get the infinite continued fraction,

1
x  =  2 + -------------------
1
2 + ----------------
1
2 + -----------
1
2 + -------
2 + ...

Using the notation I introduced above, we have

1+sqrt(2) = [2,2,2,2,2,....]

Note that this gives us the continued fraction representation
of the square root of 2, namely

sqrt(2) = [1,2,2,2,2,....]

Which is interesting because, if you remember, the sqrt(2)
has a nonterminating and nonrepeating (since it's irrational)
decimal representation 1.414213562373...... BUT it has the
repeating continued fraction representation [1,2,2,2,...]

(Remember that rational numbers have repeating decimal
representations, like:

1/3 = 0.3333333.....

2/7 = 0.285714285714285714.......

1/1 = 0.99999999999......

and so on )

Why don't you try to solve the following?

Problem: Show that the continued fraction for sqrt(n(n+1)) is
[n,2,2n,2,2n,2,2n...].

You should be able to find the theory of continued fractions in
(almost) any elementary or standard number theory textbook. I
subject.

Here are a couple of links I found that you might find interesting:

Continued Fraction Representations involving e

http://pauillac.inria.fr/algo/bsolve/constant/e/cntfrc.html

Continued Fractions Involving Pi

http://www.mathsoft.com/asolve/constant/pi/frc.html

I hope this helped :)

-Doctor Luis,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory
High School Sequences, Series

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