Continued Fractions

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Continued Fractions

• A Presentation by Jeffrey Sachs, Bill Gottesman,
  and Tim Mossey

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Introduction

• Continued Fractions provide insight into many
  mathematical problems, particularly into the
  nature of numbers.
• Consider the quadratic
  0
• By rearranging this, and dividing by x we can
  produce
•  
• 
  
• 
  
• 
  

This can be represented as
Our best approximation of this is 3
Our best approximation of this is 3 and 1/3
3.333333
We can substitute for x
Our best approximation of this is 3 and 3/10 3.3
Our best approximation of this is 3 and 10/33
3.30303
Using the quadratic formula, x
3.30278
Our approximations become more and more precise
with each expansion of the continued fraction.
This one will continue on forever. These types
of continued fractions are called infinite
continued fractions.
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Creating Continued Fractions



Consider the fraction
We want to turn this into a continued fraction,
but how?
Using the Euclidean algorithm we can start to
form our continued fraction
The 2 is known as the partial quotient. Thus,
our continued fraction begins as
67 29(2) 9
The can be also represented as
?
The goes into the Euclidean Algorithm as
such
This process continues, now with the

29 9(3) 2
will be our final faction because it will
have no remainder in the Euclidean Algorithm.
9 2(4) 1
2 1(2) 0
And so, our finite continued fraction of
is
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Unwinding a Continued Fraction
Consider the continued fraction from before
To find the initial rational number we must start
from the bottom.
Begin with
This equals
We now have
We now have
Thus, the continued fraction
represents the rational number .
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Continued Fraction Notation
A continued fraction is an expression of the
form
This is actually known as a simple continued
fraction.
is usually a negative or positive integer, and
all subsequent are positive integers.
A finite continued fraction is called a
terminating continued fraction.
Convenient ways to write continued fractions
include
This is the set of partial quotients of the
simple continued fraction.
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Infinite Continued Fractions
Infinite continued fractions never terminate, but
they converge.
A good example is pi.
pi 3.141592653589
What is the first good rational approximation of
pi?
What is the second good rational approximation of
pi?
Its not because that isnt the best
approximation of pi. Is only 3.1 whereas
is 3.14. It is a simpler rational number.
This all can be found with continued fractions
with what is called convergents.
Convergents are successive rational
representations of the continued fraction. Each
successive convergent is always a better
approximation of the original number than the one
before it.
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Convergents of pi
Using the Euclidean algorithm we can begin to
construct the infinite continued fraction of pi.
From this, the construction of the continued
fraction looks like this
3.141509

The third convergent
3.1415929
The fourth convergent


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Thank you to number theorist Professor John
Voightfor helping us with this topic.
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Questions?
Bibliography Khinchin, Aleksandr.  Continued
Fractions 1964, Chicago University Press. 
English translation edited by Herbert
Eagle. Olds, Carl Douglas. Continued fractions.
1963, Random house.Euler, Leonard. 
Introduction to analysis of the infinite, Book 1.
1742 English translation by J. D. Blanton 1988,
Springer. Pianos and Continued Fractions.
Edward G. Dunne, www.research.att.com/njas/sequen
ces/DUNNE/TEMPERAMENT.HTML
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(Applause)
To be continued
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Homework
Consider the improper fraction
Construct its terminating continued fraction.