Square root any nonnegative number, then the principal square root of −"x" is The right side (as well as its negative) is indeed a square root of −"x", since For every non-zero complex number "z" there exist precisely two numbers "w" such that : the principal square root of "z" (defined below), and its negative. There are two complex numbers that square to a given arbitrary non-zero imaginary number formula_11 with real formula_12 where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since , the above is also identical to the following generalized continued fractions: The
Square root of 5 of five can be used. Despite having a denominator of only 72, it differs from the correct value by less than (approx. ). As of December 2013, its numerical value in decimal has been computed to at least ten billion digits. 1. This irrationality proof for the square root of 5 uses Fermat's method of infinite descent: 2. This irrationality proof is also a proof by contradiction: It can be expressed as the continued fraction The convergents and semiconvergents of this continued fraction are as follows (the black terms are the semiconvergents): Convergents of the continued fraction are colored red;