sqrtn

Sqrtn refers to the square root function, most commonly written as sqrt(n). It assigns to a nonnegative number n the unique nonnegative real x such that x^2 = n. The value sqrt(n) is called the principal square root. For negative real numbers, the real-valued square root is not defined; in the complex plane, sqrt is multivalued, with a standard principal branch chosen by convention.

Domain and range: Over the real numbers, sqrt(n) is defined only for n ≥ 0 and yields sqrt(n) ≥ 0. Over the complex numbers, sqrt extends to all n with two values ±, though a principal value is typically selected to be the one with nonnegative real part.

Basic properties: sqrt(a^2) = |a| for any real a; sqrt(ab) = sqrt(a) sqrt(b) for nonnegative a and b; (sqrt(n))^2 = n for n ≥ 0; sqrt(n) is equal to n^(1/2). The function is increasing on [0, ∞) and satisfies sqrt(n) ≤ n for n ≥ 1.

Computation: In hardware and software, sqrt is computed by dedicated instructions or algorithms such as Newton's method, digit-by-digit computation, or binary search when dealing with integers. The integer square root returns the floor of sqrt(n).

Applications and examples: In geometry, sqrt appears in the Pythagorean theorem, circle equations, and when converting squared quantities back to linear dimensions. In computation, it is used in solving equations, normalizing vectors, statistical measures, and many numerical methods.

Notation and history: The radical symbol √ was introduced in the 16th century. The term sqrt(n) is widely used in mathematics, science, and programming to denote the square root function, with context clarifying whether a real, complex, or integer-valued version is intended.