Transcendental numbers are complex numbers that cannot be written as the zeros of a non-trivial, non-zero polynomial with rational coefficients and integer exponents. That is, a transcendental number is a number that is not algebraic.
Definition: Set of all transcendental numbers |
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The set of all transcendental numbers is defined as:
where is the set of all algebraic numbers. |
The set of all transcendental numbers is a subset of the set of all complex numbers. Transcendental numbers and algebraic numbers are mutually exclusive subsets of the complex numbers, themselves encompassing all complex numbers.
A few of the most well known transcendental numbers are represented as constants
- The mathematical constant pi (π)
- The mathematical constant known as Euler's number: e
Real transcendental numbers are also irrational numbers, but not all irrational numbers are transcendental, as many are algebraic. All rational numbers are algebraic, but not all algebraic numbers are rational.
These numbers are usually arrived at, mathematically, through calculus, transcendental functions, limit analysis, infinite summations, and concept definitions.