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
(iii) It provides an opportunity to introduce the subject of group theory via
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] .
1. −2 = [−1.5]
2. 3 = [ ]
For any real x with n = [x] the number u = x − n falls in the unit interval I
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
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
that procedes one step at a time. Given x one begins with the mod one
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
If > 0, then the reciprocal 1/ of satisfies 1/ > 1 since is in I
< 1. In this case the second step in the recursive determination of the
expansion of x is to apply the mod one decomposition to 1/. One writes
with discriminant 60. Since y > 2, one of the two roots of the quadratic
equation cannot be
y, and, therefore,
The idea of the calculation above leads to the conclusion that any continued
that eventually repeats is the solution of a quadratic equation with positive
integer coefficients. The converse of this statement is also true, but a proof
The process of finding the continued fraction expansion of a rational number is
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
= 0 is the case where x is an integer. Otherwise > 0, and the mod one
of 1/ gives
This shows that , where
is the remainder for division of b by
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
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
fraction represents a rational number. That statement will be demonstrated in
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