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A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of the "square" of a mathematical object is defined. These include function spaces and square matrices, among other mathematical structures.
Newton's method is one of many known methods of computing square roots. Given a positive number a, the problem of finding a number x such that x 2 = a is equivalent to finding a root of the function f(x) = x 2 − a. The Newton iteration defined by this function is given by
A negative real number −x has no real-valued square roots, but when x is treated as a complex number it has two imaginary square roots, + and , where i is the imaginary unit. In general, any non-zero complex number has n distinct complex-valued n th roots, equally distributed around a complex circle of constant absolute value .
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
Thus the square roots of A are given by VD 1/2 V −1, where D 1/2 is any square root matrix of D, which, for distinct eigenvalues, must be diagonal with diagonal elements equal to square roots of the diagonal elements of D; since there are two possible choices for a square root of each diagonal element of D, there are 2 n choices for the ...
The most common algorithm for this, which is used as a basis in many computers and calculators, is the Babylonian method [9] for computing square roots, an example of Newton's method for computing roots of arbitrary functions. It goes as follows:
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