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Every non-zero number x, real or complex, has n different complex number nth roots. (In the case x is real, this count includes any real nth roots.) The only complex root of 0 is 0. The nth roots of almost all numbers (all integers except the nth powers, and all rationals except the quotients of two nth powers) are irrational. For example,
Geometric representation of the 2nd to 6th root of a general complex number in polar form. For the nth root of unity, set r = 1 and φ = 0. The principal root is in black. An n th root of unity, where n is a positive integer, is a number z satisfying the equation [1] [2] =
Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds.
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (+) /, is an algebraic number, because it is a root of the polynomial x 2 − x − 1. That is, it is a value for x for which the polynomial evaluates to zero.
It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity (/ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is
Roots are the opposite of exponents. The nth root of a number x (written ) is a number r which when raised to the power n yields x. That is, = =, where n is the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root.
The two square roots of a negative number are both imaginary numbers, and the square root symbol refers to the principal square root, the one with a positive imaginary part. For the definition of the principal square root of other complex numbers, see Square root § Principal square root of a complex number.
The root separation is a fundamental parameter of the computational complexity of root-finding algorithms for polynomials. In fact, the root separation determines the precision of number representation that is needed for being certain of distinguishing distinct roots.