A simple continued fraction is one which has unity in each numerator. That is, it can be written as
                                   1
               x = a + -------------------------
                                       1  
                         b  +  -----------------
                                          1
                               c + -------------
                                             1
                                      d + ------
                                           .
                                            .
                                             .
A simplified notation for simple continued fractions is
               x = [a;b,c,d,...]
Note that the 'a' is set off by a semi-colon, implying that x > 1. If a = 0, one could just write x = [b,c,d,...] implying that x < 1.

A few interesting facts regarding simple continued fractions:

  1. Any rational number can be expressed as a finite simple continued fraction, i.e. finite number of terms in the square brackets above.
  2. Any quadratic irrationality, that is any irrational solution to a quadratic equation, can be expressed as an infinite simple continued fraction with periodic or repeating sequence of numbers in the square brackets.
  3. If a number x can be written as a simple continued fraction, that expression is a unique one. There is no other simple continued fraction which is equal to x.
It should also be noticed that the following infinite fraction :
                                 1
               x = a + ------------------------
                                       1  
                         b  +  ----------------
                                         1
                               c + ------------
                                           1
                                      d + -----
                                          ... 
can be synthesized by the following electrical ladder network :
           ____           ____          ____
          /    \         /    \        /    \
O---------  a   ---------   c  --------   e  --------- . . .
          \____/   _|_   \____/  _|_   \____/  _|_
                  /   \         /   \         /   \ 
                  |   |         |   |         |   |
                  | b |         | d |         | f |
                  |   |         |   |         |   |
                  \___/         \___/         \___/
                    |             |             |
O----------------------------------------------------- . . .

in case it x represents an impedance and by the following electrical ladder network :
                    ____          ____
                   /    \        /    \
O------------------   b  --------   d  --------------- . . .
             _|_   \____/  _|_   \____/  _|_
            /   \         /   \         /   \ 
            |   |         |   |         |   |
            | a |         | c |         | e |
            |   |         |   |         |   |
            \___/         \___/         \___/
              |             |             |
O----------------------------------------------------- . . .

in case it x represents an admittance.

An infinite simple continued fraction that has a repeating pattern is the root of a quadratic equation.
For instance, X^2-2 = 0 has the root X = √2 which can be expressed as X = [1;2,2,2,...]
The golden mean is the positive root of X^2-X-1 and can be expressed as X = [1;1,1,1,...]

HAKMEM has an interesting section on simple continued fractions, including algorithms for performing arithmetic operations on them which may be viewed at
http://www.inwap.com/pdp10/hbaker/hakmem/cf.html

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