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.