contain all the real transcendental numbers and a subset of the algebraic numbers, including the quadratic irrationals and other forms of algebraic irrationals. Applying any non-constant single-variable algebraic function to a transcendental argument yields a transcendental value. For example, from knowing that is transcendental, it can be immediately deduced that numbers such as , and are transcendental as well. However, an algebraic function of several variables may yield an algebraic number when applied to transcendental numbers if these numbers are not algebraically independent. For example, and are both transcendental, but is obviously not. It is unknown whether , for example, is transcendental, though at least one of and must be transcendental. More generally, for any two transcendental numbers and , at least one of and must be transcendental. To see this, consider the polynomial . If and were both algebraic, then this would be a polynomial with algebraic coefficients. Because algebraic numbers form an algebraically closed field, this would imply that the roots of the polynomial, and , must be algebraic. But this is a contradiction, and thus it must be the case that at least one of the coefficients is transcendental. The non-computable numbers are a strict subset of the transcendental numbers. All Liouville numbers are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its continued fraction expansion. Using a counting argument one can show that there exist transcendental numbers which have bounded partial quotients and hence are not Liouville numbers. Using the explicit continued fraction expansion of , one can show that is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded). Kurt Mahler showed in 1953 that is also not a Liouville number. It is conjectured that all infinite continued fractions with bounded terms that are not eventually periodic are transcendental (eventually periodic continued fractions correspond to quadratic irrationals). Numbers proven to be transcendental Numbers proven to be transcendental: if is algebraic and nonzero (by the Lindemann–Weierstrass theorem). (by the Lindemann–Weierstrass theorem). , Gelfond's constant, as well as (by the Gelfond–Schneider theorem). where is algebraic but not 0 or 1, and is irrational algebraic (by the Gelfond–Schneider theorem), in particular: , the Gelfond–Schneider constant (or Hilbert number) , , , , , and , and their hyperbolic counterparts, for any nonzero algebraic number , expressed in radians (by the Lindemann–Weierstrass theorem). The fixed point of