Algebraic Number
An algebraic number is a complex number that is a root of a non-zero polynomial with rational coefficients. Formally, a number α is algebraic if there exists a polynomial equation of the form a₀ + a₁α + a₂α² + … + aₙαⁿ = 0, where the coefficients a₀, a₁, …, aₙ are rational numbers and at least one coefficient is non-zero. This definition encompasses all rational numbers (which satisfy linear equations) and certain irrational numbers like √2 (which satisfies α² − 2 = 0).
Key Properties
Algebraic numbers form a countable set, meaning they can be enumerated despite being infinite. The degree of an algebraic number is the lowest degree of any polynomial with rational coefficients that has it as a root. Algebraic numbers are closed under addition, subtraction, multiplication, and division (except by zero), making them a field. Every algebraic number is computable and can be represented exactly in symbolic form, distinguishing them from transcendental numbers like π and e.
Relationship to Other Numbers
The algebraic numbers form a proper subset of the complex numbers. Conversely, complex numbers that are not algebraic are called transcendental numbers. While there are countably many algebraic numbers, there are uncountably many transcendental numbers, meaning most real numbers are in fact transcendental. The field of algebraic numbers is fundamental to algebraic number theory, which studies properties such as factorization and divisibility within algebraic integers (a special subset of algebraic numbers).