Algebraic number

An algebraic number is a number that is a root of a non-zero polynomial (of finite degree) 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 (complex) 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. Similarly, the set of algebraic (real) numbers is countably infinite and has Lebesgue measure zero as a subset of the real numbers, and in that sense, almost all real numbers are transcendental.