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