ruutjuurt

Ruutjuur is the Estonian term for the mathematical square root. It assigns to a nonnegative real number x the nonnegative number y such that y squared equals x. In mathematical notation, the square root is written as √x or x^(1/2). The principal square root refers to the nonnegative solution. In Estonian, ruutjuur is the noun for the operation, while ruutjuurt can appear as a grammatical form of the word in a sentence.

For real numbers, the square root is defined only for x ≥ 0. If x < 0, the square root is not a real number; in the complex number system, √x equals i√|x|. Basic properties include √(ab) = √a · √b when a and b are nonnegative, and √(a^2) = |a|. The square root function is monotone on the nonnegative real numbers: larger inputs yield larger outputs. It can be expressed as x^(1/2), linking it to exponentiation.

Common methods for computing square roots range from exact factorization to numerical approximation. Factoring x into prime components allows simplification when x has perfect square factors, yielding simple radicals. For general values, iterative methods such as the Newton-Raphson formula r_{n+1} = (r_n + x/r_n)/2 provide approximate results. Historical algorithms include the Babylonian method, used since ancient times, and, in modern calculators, fast algorithms for real and complex inputs.

Applications of the ruutjuur appear across geometry, physics, engineering, computer science, and statistics, wherever lengths, distances, or radial quantities are involved. Related concepts include the radical sign, irrational numbers, and the square root function more broadly.