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The term (or number) whose square root is being considered is known as the radicand.The radicand is the number or expression underneath the radical sign, in this example 9.  It is exactly the length of the diagonal of a square with side length 1.The real part of the principal value is always nonnegative.In the following, the complex z and w may be expressed as: The third equality cannot be justified (see invalid proof).In the Chinese mathematical work Writings on Reckoning, written between 202 BC and 186 BC during the early Han Dynasty, the square root is approximated by using an "excess and deficiency" method, which says to "...combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend." According to historian of mathematics D. Smith, Aryabhata's method for finding the square root was first introduced in Europe by Cataneo in 1546. Oaks, Arabs used the letter jīm/ĝīm () over a number to indicate its square root.The letter jīm resembles the present square root shape.Square roots frequently appear in mathematical formulas elsewhere, as well as in many physical laws. Computer spreadsheets and other software are also frequently used to calculate square roots.

Pocket calculators typically implement efficient routines, such as the Newton's method (frequently with an initial guess of 1), to compute the square root of a positive real number..

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. 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.

The principal square root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it isn't even continuous).

The above Taylor series for where the sign of the imaginary part of the root is taken to be the same as the sign of the imaginary part of the original number, or positive when zero.

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.