b2 b3
fraction of the form a1 + -- -- ... is called a _continued fraction
a2 + a3 +
b2 b3 b4
of the first class_; a continued fraction of the form -- -- -- ...
a2 - a3 - a4 -
called a _continued fraction of the second class_.
1 1 1
A continued fraction of the form a1 + -- -- -- ..., where
a2 + a3 + a4 +
a1, a2, a3, a4 ... are all _positive integers_, is called a _simple
continued fraction_. In the case of this fraction a1, a2, a3, a4 ... are
called the successive _partial quotients_. It is evident that, in this
case,
p1, p2, p3 ..., q1, q2, q3 ...,
are two series of positive integers increasing without limit if the
fraction does not terminate.
b2 b3 b4
The general continued fraction a1 + -- -- -- ... is evidently
a2 + a3 + a4 +
equal, convergent by convergent, to the continued fraction
[lambda]2b2 [lambda]2[lambda]3b3 [lambda]3[lambda]4b4
a1 + ----------- -------------------- -------------------- ...,
[lambda]2a2 + [lambda]3a3 + [lambda]4a4 +
where [lambda]2, [lambda]3, [lambda]4, ... are any quantities whatever,
so that by choosing [lambda]2b2 = 1, [lambda]2[lambda]3b3 = 1, &c., it
can be reduced to any equivalent continued fraction of the form
1 1 1
a1 + -- -- -- ...
d2 + d3 + d4 +
_Simple Continued Fractions._
1. The simple continued fraction is both the most interesting and
important kind of continued fraction.
Any quantity, commensurable or incommensurable, can be expressed
uniquely as a simple continued fraction, terminating in the case of a
commensurable quantity, non-terminating in the case of an
incommensurable quantity. A non-terminating simple continued fraction
must be incommensurable.
In the case of a terminating simple continued fraction the number of
partial quotients may be odd or even as we please by writing the last
1
partial quotient, a_n as a_n - 1 + --.
1
The numerators and denominators of the successive convergents obey the
law p_{n}q_{n-1} - p_{n-1}q_n = (-1)^n, from which it follows at once
that every convergent is in its lowest terms. The other principal
properties of the convergents are:--
The odd convergents form an increasing series of rational fractions
continually approaching to the value of the whole continued fraction;
the even convergents form a decreasing series having the same property.
Every even convergent is greater than every odd convergent; every odd
convergent is less than, and every even convergent greater than, any
following convergent.
Every convergent is nearer to the value of the whole fraction than any
preceding convergent.
Every convergent is a nearer approximation to the value of the whole
fraction than any fraction whose denominator is less than that of the
convergent.