|Definition: The set of all algebraic numbers|
where is the set of all polynomial functions with rational coefficients.
For example, if we take , , and , we will find these numbers to be algebraic since:
Also, any rational number is also algebraic, since it is a root of .
Any number that is not algebraic, that is, not a root of any non-trivial polynomial function with rational roots, is defined to be transcendental.
In this way, constructing such a polynomial with a specific root proves that the given root is an algebraic number.
Algebraic numbers are either rational or irrational numbers, purely real or imaginary, or a complex combination. Essentially, most all complex numbers conceivable through simple algebraic relationships are algebraic, while the transcendentals exist between them and algebraically unrelated to them.