A*-numbers are the algebraic numbers. LeVeque's construction Let It can be shown that the nth root of λ (a Liouville number) is a U-number of degree n. This construction can be improved to create an uncountable family of U-numbers of degree n. Let Z be the set consisting of every other power of 10 in the series above for λ. The set of all subsets of Z is uncountable. Deleting any of the subsets of Z from the series for λ creates uncountably many distinct Liouville numbers, whose nth roots are U-numbers of degree n. Type The supremum of the sequence {ω(x, n)} is called the type. Almost all real numbers are S numbers of type 1, which is minimal for real S numbers. Almost all complex numbers are S numbers of type 1/2, which is also minimal. The claims of almost all numbers were conjectured by Mahler and in 1965 proved by Vladimir Sprindzhuk. Open problems While the Gelfond–Schneider theorem proved that a large class of numbers was transcendental, this class was still countable. Many well-known mathematical constants are still not known to be transcendental, and in some cases it is not even known whether they are rational or irrational. A partial list can be found here. A major problem in transcendence theory is showing that a particular set of numbers is algebraically independent rather than just showing that individual elements are transcendental. So while we know that e and π are transcendental that doesn't imply that e + π is transcendental, nor other combinations of the two (except eπ, Gelfond's constant, which is known to be transcendental). Another major problem is dealing with numbers that are not related to the exponential function. The main results in transcendence theory tend to revolve around e and the logarithm function, which means that wholly new methods tend to be required to deal with numbers that cannot be expressed in terms of these two objects in an elementary fashion. Schanuel's conjecture would solve the first of these problems somewhat as it deals with algebraic independence and would indeed confirm that e + π is transcendental. It still revolves around the exponential function, however, and so would not necessarily deal with numbers such as Apéry's constant or the Euler–Mascheroni constant. Another extremely difficult unsolved problem is the so-called constant or identity problem. Notes References Further reading Alan Baker and Gisbert Wüstholz, Logarithmic Forms