Transcendental Numbers

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. ea – where a is algebraic and non-zero
  2. phi
  3. ephi
  4. ab – where a is algebraic but not 0 or 1 and b is irrational algebraic.
  5.  sina,  cosa,  tanaand 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 = 1β2n – where  0 < detβ < 1  and β is algebraic.
  9.  ∑ j = 1βn! – where  0 < detβ < 1  and β is algebraic.