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Also, the use of principal cube root may give a wrong result if the coefficients are non-real complex numbers. Moreover, if the coefficients belong to another field, the principal cube root is not defined in general. The second way for making Cardano's formula always correct, is to remark that the product of the two cube roots must be –p / 3.
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. 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.
Cube root In mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots.
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.
In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example 23 = 8 or (x + 1)3. The cube is also the number multiplied by its square:
Constructible number. The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1 and is therefore a constructible number. In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length can be constructed with compass and ...
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.
Examples Adjoining the real cube root of 2 to the rational numbers gives the cubic field Q ( 2 3 ) {\displaystyle \mathbf {Q} ( {\sqrt [ {3}] {2}})} This is an example of a pure cubic field, and hence of a complex cubic field. In fact, of all pure cubic fields, it has the smallest discriminant (in absolute value), namely −108. [2]
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