An algebraic number is a complex number that is a possible root of a non-zero polynomial function with rational coefficients and natural exponents.
Definition: The set of all algebraic numbers |
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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.