information:extraction
Periodic continued fractions

The numbers with periodic continued fraction expansion are precisely the irrational solutions of quadratic equations with rational coefficients; rational solutions have finite continued fraction expansions as previously stated. The simplest examples are the golden ratio φ = and √ 2 =, while √ 14 = and √ 42 =. All irrational square roots of integers have a special form for the period; a symmetrical string, like the empty string (for √ 2) or 1,2,1 (for √ 14), followed by the double of the leading integer.