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 the original estimate and its square. Furthermore, ("x" + "c") ≈ "x" + 2"xc" when "c" is close to 0, because the tangent line to the graph of "x" + 2"xc" + "c" at "c" = 0, as a function of "c" alone, is "y" = 2"xc" + "x". Thus, small adjustments to "x" can be planned out by setting 2"xc" to "a", or "c" = "a"/(2"x"). The most common iterative method of square root calculation by hand is known as the "Babylonian method" or "Heron's method" after the first-century Greek philosopher Heron of Alexandria, who first described it.