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 2 Since it is a convergent of the continued fraction representation of the square root of two, any better rational approximation has a denominator not less than 169, since (≈ 1.4142012) is the next convergent with an error of approx. . The rational approximation of the square root of two, , derived from the fourth step in the Babylonian algorithm starting with , is too large by approx. : its square is … This is a table of recent records in calculating digits of A short proof of the irrationality of can be obtained from the rational root theorem, that is,