Transcendental Number - Numbers Proved To Be Transcendental

Numbers Proved To Be Transcendental

Numbers proved to be transcendental:

• ea if a is algebraic and nonzero (by the Lindemann–Weierstrass theorem).
• π (by the Lindemann–Weierstrass theorem).
• eπ, Gelfond's constant, as well as e-π/2=i i (by the Gelfond–Schneider theorem).
• ab where a is algebraic but not 0 or 1, and b is irrational algebraic (by the Gelfond–Schneider theorem), in particular:
  • , the Gelfond–Schneider constant (Hilbert number).
• sin(a), cos(a) and tan(a), and their multiplicative inverses csc(a), sec(a) and cot(a), for any nonzero algebraic number a (by the Lindemann–Weierstrass theorem).
• ln(a) if a is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).
• W(a) if a is algebraic and nonzero, for any branch of the Lambert W Function (by the Lindemann–Weierstrass theorem).
• Γ(1/3), Γ(1/4), and Γ(1/6).
• 0.12345678910111213141516..., the Champernowne constant.
• Ω, Chaitin's constant (since it is a non-computable number).
• The Fredholm number ; more generally, any number of the form with algebraic.
• The Prouhet–Thue–Morse constant.
• Any number for which the digits with respect to some fixed base form a Sturmian word.
• where and is the floor function.