ruutujuuren

Ruutujuuri is the mathematical concept that assigns to a nonnegative real number x its nonnegative square root y, such that y^2 = x. The symbol √ is widely used to denote this value, and the term principal square root refers to the nonnegative root. For negative real numbers, a real square root does not exist; in the complex numbers, every number has two square roots.

The square root of x is often written as sqrt(x) or x^(1/2). Key identities include sqrt(xy) = sqrt(x) sqrt(y) for x ≥ 0 and y ≥ 0, and sqrt(x^2) = |x|. In the real numbers, the domain is x ≥ 0.

Irrationality: For integers n that are not perfect squares, sqrt(n) is irrational. If n is a perfect square k^2, sqrt(n) = k is an integer.

Calculation: Square roots can be found by estimation or by iterative methods such as the Newton–Raphson method: r_{n+1} = (r_n + x/r_n) / 2, with r_0 a positive initial guess.

Applications include geometry (areas, the Pythagorean theorem), physics, computer science, and numerical analysis. Historically, the radical symbol and the concept appeared in early modern mathematics, and the √ notation was popularized during that period.

In Finnish usage, ruutujuuri denotes the square root, and ruutujuuren is its genitive form, used in phrases like “ruutujuuren arvo” (the value of the square root).

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