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 number, and quadratic irrational for a proof for all non-square natural numbers.) The square root function maps rational numbers into algebraic numbers (a superset of the rational numbers). For all real numbers "x", For all nonnegative real numbers "x" and "y", and The square root function is continuous for all nonnegative "x" and differentiable for all positive "x". If "f" denotes the square root function, its derivative is given by: The Taylor series of about "x" = 0 converges for ≤ 1 and is given by The square root of a nonnegative number is used in the definition of Euclidean