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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 , ( 1 + 5 ) / 2 {\displaystyle (1+{\sqrt {5}})/2} , is an algebraic number, because it …

May 22, 2025 · Algebraic numbers are represented in the Wolfram Language as indexed polynomial roots by the symbol Root [f, n], where is a number from 1 to the degree of the polynomial (represented as a so-called "pure function") . Examples of some significant algebraic numbers and their degrees are summarized in the following table.

To be algebraic, a number must be a root of a non-zero polynomial equation with rational coefficients. So x is algebraic in this example: 2x 3 − 5x + 39 = 0

Definition 1.1. The number α ∈ C is said to be algebraic if it satisfies a polynomial equation x n+a 1x −1 +···+a n with rational coefficients a i ∈ Q. We denote the set of algebraic numbers by Q¯. Examples: 1. α = 1 2 √ 2 is algebraic, since it satisfies the equation x2 − 1 2 = 0. 2. α = 3 √ 2+1 is algebraic, since it ...

Aug 8, 2023 · An algebraic number is a fundamental concept of mathematics that comes from the understanding of algebraic equations and number theory. An algebraic number confines a vast range of numbers, containing rational numbers, specific types of irrational numbers, and integers.

Algebraic number, real number for which there exists a polynomial equation with integer coefficients such that the given real number is a solution. Algebraic numbers include all of the natural numbers, all rational numbers, some irrational numbers, and complex numbers of the form pi + q, where p.

Feb 14, 2020 · An algebraic number is called an algebraic integer if all the coefficients of its minimal polynomial are rational integers. For instance, $i$ and $1+\sqrt2$ are algebraic integers, being roots of the polynomials $x^2+1$ and $x^2-2x-1$.