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Continued Fractions
A continued fraction is anything that has the form

(2.1) 
where each a_{i}a partial denominatorand b_{i}a
a partial numeratorare objects which can be added,
multiplied, or divided. In other words, a_{i} and b_{i} are
elements of some subset of a field, for all i. Often the partial
numerators and denominators are restricted to the set of integers,
which is a subset of many fields. The whole continued fraction
itself is an element of the field that is a superset of the set
containing the partial numerators and denominators. Thus,
abstractly, a continued fraction is a number ,
where Fis an arbitrary field, such that x has the form
where
,
with
,
for all .
Since 0 has no inverse, there is the further condition that
for all .
If after some n the expansion
() terminates, then the continued fraction is
called finite, otherwise it is infinite. If a_{0} is
an integer, a_{i} is a positive integer for all ,
and
b_{i} = 1 for all ,
then the continued fraction is
called simple, and each a_{i} is called a partial
quotient. For the remainder of the report, every continued
fraction discussed is assumed to be simple. Below are some
theorems regarding this type of continued fraction.
Suppose that () is simple and terminated after
a_{n}, so that a continued fraction x has the form

(2.2) 
In standard notation, the continued fraction
() is denoted
,
and if () is
infinite, then it is denoted
.
The kth
convergent of x is the continued fraction
,
where the partial quotients
are truncated from the continued fraction of x.
Since
x = P_{n}/Q_{n}, one would suspect that an investigation of
convergents would lead to some insight on the nature of continued
fractions. Indeed, convergents are the most important part of the
theory of continued fractions. The following theorem is the
foundation for almost the whole theory of simple continued
fractions.
Theorem 1
For continued fractions of the form (
),
convergents satisfy the
fundamental recurrence relation

(2.3) 
where
P_{2} = 0,
P_{1} = 1,
Q_{2} = 1, and
Q_{1} = 0.
Another theorem from the theory of simple continued
fractions is needed for the development of the rest of the report,
which is the following theorem.
Theorem 2
For all
,

(2.4) 
These two theorems are the only ones from the theory of finite
continued fractions, although they apply to infinite continued
fractions as well, that will be needed for the study of families
of continued fractions. For more information on finite continued
fractions see [2], [6], [7], and
[8].
Finding the continued fraction for the square root of some
positive, nonsquare integer is an easy process. It proceeds by
first adding and subtracting the greatest integer in the square
root from the square root, then taking the reciprocal of the
reciprocal of the square root minus the greatest integer in it,
and then rationalizing the denominator of the resulting fraction.
The same process is applied to the new quadratic irrational, and
it continues until the resulting surd is the same as one of the
previous surds. For example,
Since
the continued fraction for
is
which is
in the standard notation,
where the bar covers the repeating partial quotients. This is the
most simple example, but continued fraction for any surd can be
found using this method. A method which takes less time and paper
is given in [6]. The continued fractions of pure
quadratic irrationals all have the same structure, which is given
in the following theorem.
Theorem 3
If
d is a positive, nonsquare integer, then
,
where
each partial quotient is a positive integer.
There are many patterns among the continued fractions expansions
of surds, some of them being more evident than others. In just
the short list of continued fractions below,
n 

n 

1 
1 
26 

2 

27 

3 

28 

4 
2 
29 

5 

30 

6 

31 

7 

32 

8 

33 

9 
3 
34 

10 

35 

11 

36 
6 
12 

37 

13 

38 

14 

39 

15 

40 

16 
4 
41 

17 

42 

18 

43 

19 

44 

20 

45 

21 

46 

22 

47 

23 

48 

24 

49 
7 
25 
5 
50 

one can see many patterns, such as:
,
,
,
and more
generally,
.
Groups of
continued fractions whose periods are predictable and static
abound, but the more interesting groups whose periods grow are
rare and harder to detect.
Next: Families of Continued Fractions
Up: Families of Continued Fractions
Previous: Introduction
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