b2
a1 +- -----------
b3
a2 +- -----------
b4
a3 +- ----------
b5
a4 +- ----
a5 +- ...,
where a1, a2, a3, ... and b2, b3, b4, ... are any quantities whatever,
positive or negative, is called a "continued fraction." The quantities
a1 ..., b2 ... may follow any law whatsoever. If the continued fraction
terminates, it is said to be a terminating continued fraction; if the
number of the quantities a1 ..., b2 ... is infinite it is said to be a
_non-terminating_ or _infinite_ continued fraction. If b2/a2, b3/a3 ...,
the _component fractions_, as they are called, recur, either from the
commencement or from some fixed term, the continued fraction is said to
be _recurring_ or _periodic_. It is obvious that every terminating
continued fraction reduces to a commensurable number.
The notation employed by English writers for the general continued
fraction is
b2 b3 b4
a1 +- -- -- -- ...
a2 +- a3 +- a4 +-