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Methods of computing square roots are algorithms for approximating the non-negative square root of a positive real number . Since all square roots of natural numbers, other than of perfect squares, are irrational, [1] square roots can usually only be computed to some finite precision: these methods typically construct a series of increasingly accurate approximations.
In mathematics, an nth root of a number x is a number r (the root) which, when raised to the power of the positive integer n, yields x: The integer n is called the index or degree, and the number x of which the root is taken is the radicand. A root of degree 2 is called a square root and a root of degree 3, a cube root.
The square root function maps rational numbers into algebraic numbers, the latter being a superset of the rational numbers). For all real numbers x, (see absolute value). For all nonnegative real numbers x and y, and. The square root function is continuous for all nonnegative x, and differentiable for all positive x.
Quadratic formula. The roots of the quadratic function y = 1 2 x2 − 3x + 5 2 are the places where the graph intersects the x -axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
Newton's method is a powerful technique—in general the convergence is quadratic: as the method converges on the root, the difference between the root and the approximation is squared (the number of accurate digits roughly doubles) at each step. However, there are some difficulties with the method.
Polynomial root-finding algorithms Finding polynomial roots is a long-standing problem that has been the object of much research throughout history. A testament to this is that up until the 19th century, algebra meant essentially theory of polynomial equations.
The square root of two is the frequency ratio of a tritone interval in twelve-tone equal temperament music. The square root of two forms the relationship of f-stops in photographic lenses, which in turn means that the ratio of areas between two successive apertures is 2.
Some matrices have no square root. An example is the matrix While the square root of a nonnegative integer is either again an integer or an irrational number, in contrast an integer matrix can have a square root whose entries are rational, yet non-integral, as in examples above.
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