1 Introduction
Continued fractions offer a means of concrete representation for arbitrary real
numbers . The
continued fraction expansion of a real number is an alternative to the
representation of such a
number as a (possibly infinite) decimal .
The reasons for including this topic in the course on Classical Algebra are :
(i) The subject provides many applications of the method of recursion .
(ii) It is closely related to the Euclidean algorithm and, in particular, to
“Bezout’s Identity”.
(iii) It provides an opportunity to introduce the subject of group theory via
the 2-dimensional
unimodular group 
.
2 The continued fraction expansion of a real number
Every real number x is represented by a point on the real line and , as such,
falls between two
integers. For example, if n is an integer and
n ≤ x < n + 1 ,
x falls between n and n + 1, and there is one and only one such integer n for
any given real x.
In the case where x itself is an integer, one has n = x. The integer n is
sometimes called the
floor of x, and one often introduces a notation for the floor of x such as
n = [x] .
Examples:
1. 
−2 = [−1.5]
2. 3 = [ ]
For any real x with n = [x] the number u = x − n falls in the unit interval I
consisting of
all real numbers u for which 0 ≤ u < 1.
Thus, for given real x there is a unique decomposition
x = n + u
where n is an integer and u is in the unit interval. Moreover, u = 0 if and only
if x is an
integer. This decomposition is sometimes called the mod one decomposition of a
real number.
It is the first step in the process of expanding x as a continued fraction.
The process of finding the continued fraction expansion of a real number is a
recursive process
that procedes one step at a time. Given x one begins with the mod one
decomposition
where 
is an integer and 
.
If 
= 0, which happens if and only if x is an
integer, the recursive process terminates with
this first step. The idea is to obtain a sequence of integers that give a
precise determination
of x.
If > 0, then the reciprocal 1/ of satisfies 1/
> 1 since is in I
and, therefore,
< 1. In this case the second step in the recursive determination of the
continued fraction
expansion of x is to apply the mod one decomposition to 1/. One writes
where 
is an integer and 0 ≤
< 1. Combining the equations that represent the
first two
steps, one may write
Either = 0, in which case the process ends with the
expansion
or > 0. In the latter case one does to what had just been done to above
under the
assumption > 0. One writes
where 
is an integer and 0 ≤
< 1. Then combining the equations that represent
the first
three steps, one may write
After k steps, if the process has gone that far, one has integers
and real numbers
that are members of the unit interval I with
all positive. One
may write
Alternatively, one may write
If 
= 0, the process ends after k steps. Otherwise, the process continues at
least one more
step with
In this way one associates with any real number x a sequence, which could be
either finite or
infinite, 
of integers. This sequence is called the continued
fraction expansion of x.
Convention. When 
is called a continued fraction, it is understood
that all of the
numbers 
are integers and that
≥1 for j ≥ 2.
3 First examples
Let
In this case one finds that
where
Further reflection shows that the continued fraction structure for y is
self-similar:
This simplifies to
and leads to the quadratic equation
3y2 − 6y − 2 = 0
with discriminant 60. Since y > 2, one of the two roots of the quadratic
equation cannot be
y, and, therefore,
Finally,
The idea of the calculation above leads to the conclusion that any continued
fraction
that eventually repeats is the solution of a quadratic equation with positive
discriminant and
integer coefficients. The converse of this statement is also true, but a proof
requires further
consideration.
4 The case of a rational number
The process of finding the continued fraction expansion of a rational number is
essentially
identical to the process of applying the Euclidean algorithm to the pair of
integers given by its
numerator and denominator .
Let x = a/b, b > 0, be a representation of a rational number x as a quotient of
integers a and
b. The mod one decomposition
shows that 
, where
is the remainder for division of a by b. The case
where
= 0 is the case where x is an integer. Otherwise > 0, and the mod one
decomposition
of 1/ gives
This shows that 
, where
is the remainder for division of b by
.
Thus, the
successive quotients in Euclid’s algorithm are the integers
occurring in the continued
fraction. Euclid’s algorithm terminates after a finite number of steps with the
appearance of a
zero remainder . Hence, the continued fraction expansion of every rational number
is finite.
Theorem 1. The continued fraction expansion of a real number is finite if and
only if the real
number is rational.
Proof. It has just been shown that if x is rational, then the continued fraction
expansion of x is
finite because its calculation is given by application of the Euclidean
algorithm to the numerator
and denominator of x . The converse statement is the statement that every finite
continued
fraction represents a rational number. That statement will be demonstrated in
the following
section.
