continued fraction

Any real number x can be represented as a continued fraction as:

                b
                 0
x = a + ----------------------
     0            b
                   1
         a  + ----------------
          1           b
                       2
               a  + ----------
                2
                     a  + ...
                      3

Typesetters hate this form (quite understandably), and prefer to write continued fractions as:

      b    b    b
       0    1    2
a  + ---- ---- ---- ....
 0   a  + a  + a  +
      1    2    3

A simple continued fraction is where all of the b_n are all identically equal to 1.

Continued fractions can be used to find best rational estimates for irrational numbers. Functions may also be written as continued fractions, giving successively better rational approximations.

Continued Fractions

p_k+1 = a_k+1p_k + p_k-1, and
q_k+1 = a_k+1q_k + q_k-1

p_-2 = q_-1 = 0 and
p_-1 = q_-2 = 1

Leveque, James. Fundamentals of Number Theory. See http://www.amazon.com/exec/obidos/tg/detail/-/0486689069/
qid=1074367205//ref=sr_8_xs_ap_i0_xgl14/002-8476672-9958465?v=glance&s=books&n=507846.

Davenport, H. The Higher Arithmetic. See http://www.amazon.com/exec/obidos/tg/detail/-/0486689069/
qid=1074367205//ref=sr_8_xs_ap_i0_xgl14/002-8476672-9958465?v=glance&s=books&n=507846.

Most information in the writeup comes from personal knowledge gleaned from various sources over the years. However, the above references are either very helpful, well-written, highly-recommended, or several of these things.