Properties of Continued Fractions

    • A continued fraction is finite if and only if the number it represents is a rational number.
    • A continued fraction is infinite if and only if the number it represents is an irrational number.
    • Continued fraction representations for "simple" rational numbers are short.
    • A continued fraction representation of an irrational number is unique.
    • A continued fraction representation of a rational number is almost unique (there are exactly two representations -- one representation always ends in one).

Most irrational numbers do not have any periodic or regular patterns in their continued fraction representations. However, some irrational numbers have very nice patterns:

Continued fraction form of squareroot2 : [1; 2, 2, 2, 2, ...]

Continued fraction form of e : [2;1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, ...]

Continued fraction form of phi : [1; 1, 1, 1, 1, 1, ...]

Phi has the distinct honor of being the slowest to converge of all continued fractions; taking 26 iterations to converge to 10 decimal places. On the other hand, the continued fraction form of pi has no discernable pattern, but it can be computed to 10 decimal places of accuracy in just 7 iterations of the continued fraction (Herkommer, 2003).

pi = [ 3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, ... ]

References

Herkommer, M. (2003). Continued fractions. [Online] http://www.petrospec-technologies.com/ Herkommer/contfrac.htm