Algebraic numbers

Algebraic numbers, are real numbers which are roots of some finite degree polynomial with integer coefficients. Examples of algebraic numbers are 2, 1/3, and ${\sqrt {2}}$.

In contrast, transcendental numbers are real numbers which are not roots of any finite degree polynomial with integer coefficients. The most well-known transcendental numbers are e and $\pi$ (pi), though this must be proven using advanced calculus or number theory.

The set of algebraic numbers is countably infinite. This follows from the fact that the set of finite degree polynomials with integer coefficients is countably infinite. Conversely, the set of transcendental numbers is uncountable because for any finite degree polynomial with integer coefficients there are uncountably many numbers which are not solutions to it.

The algebraic numbers are closed under addition, subtraction, multiplication, division, square roots, radicals, and exponentiation. These properties imply that the algebraic numbers are an algebraically closed field.

If a is an algebraic number and t is a transcendental number, then a+t, a-t, at, a/t, a^t, and t^a are all transcendental.

A random real number is probably transcendental, though the exact value of this probability has baffled statisticians for almost 100 years.

Algebraic integers

An algebraic integer is an algebraic number which is also an integer, occurring in a subring of the algebraic number field.