average of these two numbers is a better approximation than either of them. However, the inequality of arithmetic and geometric means shows this average is always an overestimate of the square root (as noted below), and so it can serve as a new overestimate with which to repeat the process, which converges as a consequence of the successive overestimates and underestimates being closer to each other after each iteration. To find x: Start with an arbitrary positive start value x. The closer to the square root of a, the fewer the iterations that will be needed to achieve the desired precision. Replace x by the average (x + a/x) / 2 between x and a/x. Repeat from step 2, using this average as the new value of x. That is, if an arbitrary guess for is x0, and , then each xn is an approximation of which is better for large n than for small n. If a is positive, the convergence is quadratic, which means that in approaching the limit, the number of correct digits roughly doubles in each next iteration. If , the convergence is only linear. Using the identity the computation of the square root of a positive number can be reduced to that of a number in the range . This simplifies finding a start value for the iterative method that is close to the square root, for which a polynomial or piecewise-linear approximation can be used. The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers. Another useful method for calculating the square root is the shifting nth root algorithm, applied for . The name of the square root function varies from programming language to programming language, with sqrt (often pronounced "squirt" ) being common, used in C, C++, and derived languages like JavaScript, PHP, and Python. Square roots of negative and complex numbers The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work with a more inclusive set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by i (sometimes by j, especially in the context of electricity where "i" traditionally