WHAT ARE CONTINUED FRACTIONS?
A number like 6/23 can be written as 1/(23/6) which can be
written as
1/(3+5/6)
and this in turn can be written as
1/(3+ [1/(6/5)] )
and continuing..
1/(3+ [1/ (1+1/5) ])
This gets to look rather messy so we have a convention for
writing continued fractions so that
6/23 = /3,1,5/
In a more familiar way:
1

6 1
 = 3 + 
23 1
1 + 
5
The expression "6/23 = /3,1,5/" is more compact.
The process of working out a continued fraction is called
"continued fraction expansion" whilst the method of evaluating
the fraction from the continued fraction is called "continued
fraction evaluation". It is quite easy to write computer
programs for both these operations.
Continued fractions (cf) have many interesting properties but the
most useful is that if we terminate a cf early and evaluate the
truncated first part, we get a good approximation to the original
fraction in smaller numbers. The resulting fraction is also
always in its lowest terms (i.e. no common factors).
A good introduction to continued fractions is contained in Donald
Knuth's excellent series of books "The Art of Computer
Programming". You need volume 2.
Strictly, these are called "regular continued fractions" because
they always have "1" as the numerator part. Other kinds of cf
are less interesting.
Contact David Broughton if you would like to know more of the
programming techniques for expansion and evaluation of continued
fractions.
