Algebraic number

In mathematics, 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 ${\displaystyle (1+{\sqrt {5}})/2}$ is an algebraic number, because it is a root of the polynomial ${\displaystyle X^{2}-X-1}$, i.e., a solution of the equation ${\displaystyle x^{2}-x-1=0}$, and the complex number ${\displaystyle 1+i}$ is algebraic as a root of ${\displaystyle X^{4}+4}$. Algebraic numbers include all integers, rational numbers, and n-th roots of integers.

Algebraic complex numbers are closed under addition, subtraction, multiplication and division, and hence form a field, denoted ${\displaystyle {\overline {\mathbb {Q} }}}$. The set of algebraic real numbers ${\displaystyle {\overline {\mathbb {Q} }}\cap \mathbb {R} }$ is also a field.

Numbers which are not algebraic are called transcendental and include π and e. There are countably many algebraic numbers, hence almost all real (or complex) numbers (in the sense of Lebesgue measure) are transcendental.