+ b3a1a4 + b4a1a2 + b2b4
------------------------------------------, ...
a2a3a4 + a4b3 + a2b4

are called the successive convergents to the general continued fraction.

Their numerators are denoted by p1, p2, p3, p4...; their
denominators by q1, q2, q3, q4....

We have the relations

p_n = a_{n}p_{n-1} + b_{n}p_{n-2}, q_n = a_{n}q_{n-1} + b_{n}q_{n-2}.

b2 b3 b4
In the case of the fraction a1 - -- -- -- ..., we have the
a2 - a3 - a4 -

relations

p_n = a_{n}p_{n-1} - b_{n}p_{n-2}, q_n= a_{n}q_{n-1} - b_{n}q_{n-2}.

Taking the quantities a1 ..., b2 ... to be all positive, a continued

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._