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:

- Any rational number can be expressed as a finite simple
continued fraction, i.e. finite number of terms in the square brackets above.
- 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.
- 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`.