sqrt

Square root, often written sqrt(x), is the function that assigns to a nonnegative real number x the nonnegative number y such that y^2 = x. It is the principal square root and serves as the inverse of the squaring function on the domain of nonnegative real numbers. For any real x ≥ 0, sqrt(x) ≥ 0 and (sqrt(x))^2 = x. The equation y^2 = x has two solutions y = ±√x when x > 0, but the sqrt function returns the nonnegative root by convention. The domain of sqrt over the real numbers is x ≥ 0, and its range is y ≥ 0.

Properties: sqrt(a^2) = |a|. For nonnegative a,b, sqrt(ab) = sqrt(a) sqrt(b). For any nonnegative x, sqrt(x^n) = x^{n/2}. A common relation with exponents is sqrt(x) = x^{1/2}.

Extensions to complex numbers: sqrt is multi-valued; the principal value is defined by the branch cut along the negative real axis, with sqrt(re^{iθ}) = √r e^{iθ/2} for θ ∈ (-π, π].

Computation and algorithms: manual methods include the digit-by-digit calculation and Newton-Raphson iteration; modern calculators and computer languages implement efficient algorithms.

Irrationality and examples: sqrt(2) is irrational; in fact, sqrt(n) is irrational for any non-square integer n. Decimal expansions may be nonrepeating.

History and usage: the concept appears in ancient mathematics; the term square root and the radical symbol were popularized in early modern Europe; symbol √ was introduced by European mathematicians in the 16th century.

Applications: geometry, physics, statistics, engineering, computer science; appears in area calculations, normalization, distance formulas, and many numerical methods.