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The positive 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. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an n th root is a root extraction.
The other roots of the equation are obtained either by changing of cube root or, equivalently, by multiplying the cube root by a primitive cube root of unity, that is . This formula for the roots is always correct except when p = q = 0 , with the proviso that if p = 0 , the square root is chosen so that C ≠ 0 .
The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of the square root of 2 in four sexagesimal figures, 𒐕 𒌋𒌋𒐼 𒐐𒐕 𒌋 = 1;24,51,10, [13] which is accurate to about six decimal digits, [14] and is the closest possible three-place sexagesimal representation of √ 2:
Since the root of unity is a root of the polynomial x n − 1, it is algebraic. Since the trigonometric number is the average of the root of unity and its complex conjugate, and algebraic numbers are closed under arithmetic operations, every trigonometric number is algebraic. [2]
The number of roots of a nonzero polynomial P, counted with their respective multiplicities, cannot exceed the degree of P, [25] and equals this degree if all complex roots are considered (this is a consequence of the fundamental theorem of algebra). The coefficients of a polynomial and its roots are related by Vieta's formulas.
The principal cube root is the cube root with the largest real part. In the case of negative real numbers, the largest real part is shared by the two nonreal cube roots, and the principal cube root is the one with positive imaginary part. So, for negative real numbers, the real cube root is not the principal cube root. For positive real numbers ...
This is a reference implementation, which can find routinely the roots of polynomials of degree larger than 1,000, with more than 1,000 significant decimal digits. The methods for computing all roots may be used for computing real roots. However, it may be difficult to decide whether a root with a small imaginary part is real or not.
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.