Definition
Transcendental numbers are defined as those numbers which are not algebraic. In other words the numbers that are not the root of a non-zero polynomial equation with rational coefficients.
Corollary 1: Since all real, rational numbers are algebraic, all real, transcendental numbers are irrational.
Proven Transcendental numbers:
- e^{a} – where a is algebraic and non-zero
- phi
- e^{p}hi
- a^{b} – where a is algebraic but not 0 or 1 and b is irrational algebraic.
- sina, cosa, tana and their multiplicative inverses for any nonzero algebraic number a.
- ln(a) – where a is algebraic and not equal to 0 or 1, for any branch of the logarithm function
- W(a) – where a is algebraic and nonzero, for any branch of the Lambert W function.
- ∑ _{j = 1}^{∞}β^{2n} – where 0 < detβ < 1 and β is algebraic.
- ∑ _{j = 1}^{∞}β^{n!} – where 0 < detβ < 1 and β is algebraic.