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NORTHERN ILLINOIS UNIVERSITY
Continued Fraction Sums and Products
A Thesis Submitted to the
University Honors Program
In Partial Fulfillment of the
Requirements of the Baccalaureate Degree
With Upper Division Honors
Department Of
Mathematics
By
Erik Rodgers
DeKalb, Illinois
May 2011
University Honors Program
Capstone Approval Page
Capstone Title (print or type)
Continued Fractions Sums and Products
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Date of Approval (print or type)
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HONORS THESIS ABSTRACT
THESIS SUBMISSION FORM
AUTHOR: Erik Rodgers
THESIS TITLE: Continued Fractions Sums and Products
ADVISOR: Richard Blecksmith
ADVISOR'S DEPARTMENT: Mathematics
DISCIPLINE: Computational Mathematics
YEAR: 2011 (~l\i6\)
PAGE LENGTH: 18
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Abstract
The focus of this project is to study the mathematical relationship of the continued fraction sum
and the continuedfraction product. Any real number can be written as a continued fraction, so the
addition and multiplication of any two continued fractions is the same as the real numbers they
represent. However, the continued fraction sum and continued fraction product each result in another
continued fraction. The problem is the determining the relationship that this new continued fraction
has with the original two. This will be achieved by observing the characteristics ofthe 3D graphs
contained within the unit cube. This topic is interesting because it is relatively unexplored. Whether or
not these relationships will lead to anything significant is yet to be determined.
Section 1
An Introduction to Continued Fractions
We begin by defining the general form of a continued fraction. It is the expression of the form
where each aj and b, is a real or complex number.
Definition 1: An continued fraction is called "simple" if all of the following are true:
(i) b, = 1 for i ~ 1
(ii) a, is a positive integer for i ~ 2
(iii) a, is an integer.
The expression for the general form of a simple continued fraction is given by
I
ao+----------------------a,+------------------a2+
+-----1
an-1+-an+···
Definition 2: Each a, of a continued fraction is called the partial quotient.
2
Definition 3: If a continued fraction is simple and has finitely many partial quotients, then it is called a
finite simple continuedfraction.
The expression for the general form a finite simple continued fraction is given by
1
ao+-----------------------1
a,+-------------------a2+
1
+---------
The finite simple continued fraction is be denoted
[ao;a"
... , aJ
and the infinite simple continued
fraction is be denoted
[ao; a" ... J .
Note: The continued fractions used in the following sections will be containded in the unit cube, where
ao= 0 . The ao term will be dropped and the continued fractions written as [a" ... ,aJ
Theorem 4: Any rational number can be expressed as a finite simple continued fraction.
Proof: Let
be any rational number where b > O. Using Euclid's algorithm, we get
~
a=b v cu+r,
b=r.w ac+r,
for
for
for
r,=r2*a3+r3
r 3=r,,_2*a,,_,+r
rn_2=rn_,*a
l1_
O<r, <b
O<r2<r,
O<r3<r2
for
l1_,
o <r,,_, <r..,
l1
Thus
a
b
-=a
'
r,
b
1
b
r2
r,
r,
r,
r3
l'
bl r,
+--
+-=a
-=a2+-=a2+----
1
r Jr,
1
-= a3 +r,
r,
=«, +---r21r3
r ,,-3
--=a
rn-2
rn-2
+ r n-' = a _, +
I1
n
rn-2
rn-2 rn-1
l1_,
=r=o;
rn_,
.
Using successive substitution,
it follows that
3
Section 2
Addition of Continued Fractions
Definition 5: Let a=[aI, a2, ••• J and P=[bI, b2, ••• J be simple infinite continued fractions.
Then the continued fraction sum of these two continued fractions is defined to be
a ffi p=[al
Definition 6: Let
Ifm
=
a = [a I' a2, ... , a J
and
+bI, a2+b2, ... J
P = [b I' b2, ... , b mJ
n, then the continued fraction sum ofthese two continued fractions is defined as
affip=[a1+bI,a2+b2,···,an+bJ
Ifm>
be simple finite continued fractions.
n, then the continued fraction sum is
a ffi P
.
=I al +b I' a2 +b2,
... , all +bll, bll+I, ... , b,J
Now the sum of two continued fractions is not equal to the sums of the numbers they represent. So the
question remains, is there a relationship between the continued fraction sum and the addition ofthe
numbers they represent?
We will contemplate this question first by getting a graphical representation
product. This will be achieved by using the program Mathematica.
of the continued fraction
The graph will be contained within the unit cube. Since writing zero as a continued fraction
poses problems we will make the starting point a random number very close to zero. Mathematica will
not perform continued fraction addition when the number of partial quotients differ. To compensate for
this, the random number will have a large number of decimal digits to ensure at least 20 partial
quotients in the continued fraction. The intervals between plot points will be
prime, which will help with the randomization
2 ~1 ,since 211 is
of plot points.
The following pictures are all different graphs, specifically different initial points, which are viewed
from different perspectives. The flat plane is the graph of x+ y for the numbers that the continued
fractions represent.
4
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'--
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9
These graphs are symmetric fractal patterns, and they can be explained by the partial quotients
of the continued fraction. When looking along the x or y axis we observe that there are discontinuities,
('jumps'), in the graph of x EB y . Going along either axis from 1 to we see that these jumps appear
111
to occur at "2 ' 3" ' "4 , ... etc.
°
Let a = [0, a2, ••• , azo 1 , P = [0, bz, ... , b20l be representative of the plot points in the graph, where
a's are the point on the x-axis and Ws are the points along the y-axis, let y =a EB P . The partial
quotients
a I and b, are zero because the plot points are contained within the unit square and
therefore never exceed a value of 1.
Now, consider only the first, nonzero partial quotient of each continued fraction. So
1
1
a=[0,a2l,
P=[0,b2l
andthefirstpartialapproximatesare
a=and
P=-b . Then the
a2
1
continued fraction sum is
decreases.
a2+b2
So, if we fix
bz= 1
. Now we know that as
and increment
a2
a2
+ b2
increases that
1
--1
n+
c2=k
} for a2 = {I ,2,... , n}. In general,
Y""
first approximate is
1
k::s-<k+l
for
={
a +b
we see that
2
1
2
y
1
"2 '
1
aZ+b2
1
3 , ... ,
2
; r.e.,
1
--<Y::S-k
k+l'
1
and the
1
a2
+b
2
Thus the most prominent behavior of the graph is explained, but what of the more subtle behavior?
Recall that the continued fraction plot points used 20 partial quotients. Now increase the number of
partial quotients that are considered.
So, fix c 2= k where k is some constant, so we can describe the behav ior ofthe graph between
discontinuities of the first approximate. As a result, all the continued fractions in this interval can be
1
y=--
written as
increases from
c2=1 ,for
k+~'
r3
k!
where
1 to
l::Sr2<1+1
~
1 ::Sr3<00 and
c2=[r3l
.
As
rz
increases from 1 to
,thus taking all values in the given interval.
00
,
Y
So, in general,
,I.e.,
1
I
--::Sy<--1
1
k+,
k+ 1+ 1
From this we can see that the second approximate of the sum of two continued fractions is
1
Y""-----
1
az+
a,+b'+--b-
2
The results for the successive approximations are similar, and we see that for each approximation,
behavior on the fixed interval is the same as the behavior of the whole graph.
the
10
Section 3
Multiplication of Continued Fractions
Definition 7: Let a = [a l' a2, ••• J and ,B = [b l' b2 ' ••• J be simple infinite continued fractions.
the continued fraction product of these two continued fractions is defined to be
a ® ,B =I a 1*b l' a2* b2, ••• J
Definition 8: Let
with m ~ n.
If m
Ifm>
=
a=[a1,a2,···,a"J
and
,B=[b1,b2,···,bmJ
Then
be simple finite continued fractions
n, then the continued fraction product of these two continued fractions is defined to be
a®,B=[al*bl,a2*b2,···,all*bJ
.
n, then the continuedfractionproduct
is
a ® ,B=[a1*b1, a2+b2, ... , an+bJ
Like the sum, the product of two continued fractions is not equivalent to the products of the numbers
they represent. So is there a relationship?
Once again, we'll start by looking at graphical representations
of the continued fraction products.
The parameters for these graphs will be the same as those used for the sum, but will be repeated
again for ease of reference. These graphs will be contained within the unit cube. Since writing zero as
a continued fraction poses problems we will make the starting point a random number very close to
zero. Mathematica will not perform continued fraction addition when the number of partial quotients
differ. To compensate for this, the random number will have a large number of decimal digits to
ensure at least 20 partial quotients in the continued fraction. The intervals between plot points will be
2 ~1 ,since 211 is prime, which will help with the randomization
of plot points.
The following pictures are all different graphs, specifically different initial points, which are viewed
from different perspectives. The flat plane is the graph of x* y for the numbers that the continued
fractions represent.
II
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15
16
As with addition, these graphs are symmetric fractal patterns, and they can be explained by the partial
quotients of the continued fraction, and once again when looking along the x or y axis we observe the
1
1
presence of discontinuities in the graph of x ® y . These jumps still appear to occur at
2 ' 3 '
1
4 , ... etc.
So, let a=[O,G2, ... , G20] and P=[O,b2, ••• , b20] be representative of the plot points in the graph,
where a's are the point on the x-axis and Ws are the points along the y-axis. Again, the partial quotients
a, and b, are zero because the plot points are contained within the unit square and therefore never
exceed a value of 1.
Again, consider only the first, nonzero partial quotient of each continued fraction.
a=[O,G2]
,
P=[O,b2]
1
a=-
and the first partial approximates are
and
z
1
I
P=-b
G
1
-*b
continued fraction sum is
bz=l
Gz
,and as
and increment a2 we see that
Gz*bz increases then -*b
G2
2
--b-
*
G2
= {
1
1 , -2 , ~ , ... ,
2
.)
decreases.
z
So
. Then the
So, fix
2
.ln } for a2 = {1,2,3,
... ,
n}. This explains the most prominent behavior of the graph.
The further behavior is explained in the same way as the material in chapter 2. However, the resulting
1
approximations are Y"""
b
for the first approximation and
-*
GI
for the second, etc.
I
17
Section 4
Multiplication Families of Continued Fractions
An interesting concept to consider is the behavior of the multiplication
when one of them is fixed.
Let 0. and fJ both be infinite simple continued fractions. Now set
such that fJ is the continued fraction of .J-;; .
of continued fractions
0.=12=[1,2]
fJ=Vn
and
Then as we increment n, we get the following:
12*)3=[1,2,4]=/6-1
12*V4=[2,O]=2
12*.J5=[2,8]=m-2
12*{6=[2,4,8]=m-2
.fi *.J7= [2,2,2,2,8]=
t ffs - 2
12 *.J8=[2,2,4]= 2/5-2
12*.J9=[3,O]=3
.J2* 110 = [3, 1,2]=137-3
.J2*fU =[3, 6,12]=ill-3
.J2 *m=
[3, 4J2]=.J39-9
1
2 9.J34597 - ~~
.fi*JT3=[3,2,2,2,2,1,2]=
So, we see a pattern that arises, specifically for fJ that can be written as
With these fJ the results of 0. fJ , for 0.= 12 , are given by
fJ = ~m2 - 1 .
*
.J2 *~m
2
b,
*
-1 = ~4 b~ + m -1 - n - b 1 where n is the greatest perfect square less than
2
is the first partial quotient of B. Other families for a of the form
0. =
rn
~ m2
-
I and
yield similar results.
18
Works Cited
Hensley, Doug. Continued Fractions. Hackensack, NJ: World Scientific, 2006. Print.
Khinchin, A. IA~ Continued Fractions. Chicago: University of Chicago, 1964. Print.
Liberman, Harry. Simple Continued Fractions: an Elementary to Research Level Approach.
Montreal: SMD Stock Analysts, 2003. Print.
Olds, C. D. Continued Fractions. [New York]: Random House, 1963. Print.
Schweiger, Fritz. Multidimensional
Continued Fractions. Oxford: Oxford UP, 2000. Print.