Algebraic number

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 x² − x − 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number ${\displaystyle 1+i}$ is algebraic because it is a root of x⁴ + 4.

All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as π and e, are called transcendental numbers.

The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental.