Realsqrt

Realsqrt refers to the real-valued square root function. It is commonly denoted sqrt(x) and, in the context of real numbers, returns the nonnegative number y such that y^2 = x. The function is defined for nonnegative inputs and yields nonnegative outputs.

The domain of Realsqrt is x ≥ 0, and its range is y ≥ 0. It is the inverse of the squaring function restricted to nonnegative inputs, meaning that sqrt(x^2) = |x| and, for x ≥ 0, sqrt(x) = x. The function is continuous and increases monotonically on [0, ∞). It is differentiable for all x > 0 with derivative 1/(2 sqrt(x)); at x = 0 the derivative is not finite.

In terms of properties, Realsqrt satisfies sqrt(0) = 0 and sqrt(1) = 1. It can be used to simplify geometric and physical calculations, such as computing lengths, distances, and standard deviations in statistics. Since negative real numbers have no real square root, Realsqrt is not defined over the negative real axis in the real-number setting. In broader contexts, the square root extends to complex numbers, where each nonzero complex number has two square roots; a principal branch is often chosen to be single-valued with a cut along the negative real axis.

Computationally, Realsqrt is implemented in most math libraries and calculators. Common algorithms include Newton’s method (Heron’s method) and table-driven or digit-extraction methods for efficient computation.

See also: square root, principal square root, complex square root.