Continued Fractions
A Presentation by Jeffrey Sachs, Bill
Gottesman, and Tim Mossey
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:
We can substitute for x:
Our best approximation of this is 3
Our best approximation of this is 3 and 1/3 =
3.333333
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.
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:
67 – 29(2) = 9
The
The 2 is known as the partial quotient. Thus,
our continued fraction begins as:
can be also represented as:
⇒
The
goes into the Euclidean Algorithm
as such:
This process continues, now with the
29 – 9(3) = 2:
9 – 2(4) = 1:
2 – 1(2) = 0
=
will be our final faction because it will have
no remainder in the Euclidean Algorithm.
And so, our finite continued fraction of
is:
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
.
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.
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?
It’s not
because that isn’t the best approximation of pi.
3.1 whereas
is 3.14. It is a simpler rational number.
Is only
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.
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:
The third convergent:
The fourth convergent:
=
=
= 3.141509…
= 3.1415929…
Thank you to
number theorist
Professor John
Voight
for helping us with
this topic.
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/sequences/DUNNE/TEMPERAMENT.HTML
(Applause)
To be continued…
Homework
Consider the improper fraction:
Construct its “terminating” continued fraction.