simple continued fraction (thing)

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.