karekök

Karekök, or square root, is the mathematical operation that assigns to a nonnegative number x the unique nonnegative number y that satisfies y^2 = x. It is the inverse of the squaring function on the nonnegative reals. In real arithmetic, karekök x is defined only for x ≥ 0; for negative x, the square root is not a real number, although complex numbers provide two roots ±i√|x|. The standard notations are the radical symbol √x and the function name sqrt(x); in many Turkish texts it is written as karekök(x). In exponent form, y = x^(1/2).

Properties of the karekök include the rule sqrt(ab) = sqrt(a) sqrt(b) for nonnegative a and b, and sqrt(a^2) = |a|. It is a nonnegative function on [0, ∞), continuous and strictly increasing, and serves as the inverse of the squaring function restricted to nonnegative inputs. The operation is related to powers as x^(1/2).

Computation methods range from manual to algorithmic. The Babylonian (or Heron) method gives successive approximations via y_(n+1) = (y_n + x/y_n)/2 starting from an initial guess. A digit-by-digit extraction method allows decimal square roots to be found by hand. In computing, Newton-Raphson iterations, table-driven methods, and specialized hardware routines are common for efficiency and accuracy.

Applications of karekök span many fields. In geometry it relates to lengths and areas; in physics and engineering it appears in formulas for motion, energy, and oscillations; in statistics and finance it appears in measures such as standard deviation and volatility. The concept is foundational in algebra and analysis and appears in numerous scientific and practical contexts.