squareroot

The square root of a number x is the nonnegative number y such that y squared equals x, written y^2 = x. For nonnegative real x, sqrt(x) denotes this principal square root. If x is negative, a real square root does not exist; in the complex numbers every nonzero x has two square roots, ±z, with z^2 = x.

In notation, sqrt is the radical sign and the value under the sign is called the radicand. The square root function is defined for all x ≥ 0 on the real number line. It satisfies sqrt(x^2) = |x| and, for nonnegative a and b, sqrt(ab) = sqrt(a) sqrt(b). It also relates to exponents by sqrt(x) = x^(1/2).

Examples include sqrt(4) = 2, sqrt(2) ≈ 1.41421356, sqrt(9) = 3, and sqrt(0) = 0. For any positive integer n that is not a perfect square, sqrt(n) is irrational. Geometrically, sqrt(x) can be interpreted as the side length of a square with area x.

Computationally, square roots can be found through methods such as Newton–Raphson iteration or digit-by-digit algorithms. In mathematics, the concept extends to complex numbers, where each nonzero x has exactly two roots, differing in sign, and to higher roots via roots of unity. The term “square root” remains standard for the principal nonnegative root in the real domain.