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:

1.  e^a  – where a is algebraic and non-zero
2.  phi
3.  e^phi 
4.  a^b  – where a is algebraic but not 0 or 1 and b is irrational algebraic.
5.  sina,  cosa,  tana and their multiplicative inverses for any nonzero algebraic number a.
6.  ln(a)  – where a is algebraic and not equal to 0 or 1, for any branch of the logarithm function
7.  W(a)  – where a is algebraic and nonzero, for any branch of the Lambert W function.
8.  ∑ _j = 1^∞β²ⁿ  – where  0 < detβ < 1  and  β  is algebraic.
9.  ∑ _j = 1^∞β^n!  – where  0 < detβ < 1  and  β  is algebraic.