A related notion is whether there is a closed-form expression for a number, including exponentials and logarithms as well as algebraic operations.
[4] His original papers on the matter in the 1840s sketched out arguments using simple continued fractions to construct transcendental numbers.
The theorem is still not strong enough to detect all transcendental numbers, though, and many famous constants including e and π either are not or are not known to be very well approximable in the above sense.
[9] Fortunately other methods were pioneered in the nineteenth century to deal with the algebraic properties of e, and consequently of π through Euler's identity.
[10] His work was built upon by Ferdinand von Lindemann in the 1880s[11] in order to prove that eα is transcendental for nonzero algebraic numbers α.
In particular this proved that π is transcendental since eπi is algebraic, and thus answered in the negative the problem of antiquity as to whether it was possible to square the circle.
The seventh of these, and one of the hardest in Hilbert's estimation, asked about the transcendence of numbers of the form ab where a and b are algebraic, a is not zero or one, and b is irrational.
In the 1930s Alexander Gelfond[13] and Theodor Schneider[14] proved that all such numbers were indeed transcendental using a non-explicit auxiliary function whose existence was granted by Siegel's lemma.
The next big result in this field occurred in the 1960s, when Alan Baker made progress on a problem posed by Gelfond on linear forms in logarithms.
Gelfond himself had managed to find a non-trivial lower bound for the quantity where all four unknowns are algebraic, the αs being neither zero nor one and the βs being irrational.
Finding similar lower bounds for the sum of three or more logarithms had eluded Gelfond, though.
This work won Baker the Fields medal for its uses in solving Diophantine equations.
[17] While Cantor's result is often quoted as being purely existential and thus unusable for constructing a single transcendental number,[18][19] the proofs in both the aforementioned papers give methods to construct transcendental numbers.
In 2004, though, Boris Zilber published a paper that used model theoretic techniques to create a structure that behaves very much like the complex numbers equipped with the operations of addition, multiplication, and exponentiation.
For example, if one supposes that the number α is algebraic then can one show that it must have very high degree or a minimum polynomial with very large coefficients?
Ultimately if it is possible to show that no finite degree or size of coefficient is sufficient then the number must be transcendental.
Since a number α is transcendental if and only if P(α) ≠ 0 for every non-zero polynomial P with integer coefficients, this problem can be approached by trying to find lower bounds of the form where the right hand side is some positive function depending on some measure A of the size of the coefficients of P, and its degree d, and such that these lower bounds apply to all P ≠ 0.
In the 1960s the method of Alan Baker on linear forms in logarithms of algebraic numbers reanimated transcendence theory, with applications to numerous classical problems and diophantine equations.
[23] Definition of these classes draws on an extension of the idea of a Liouville number (cited above).
be the minimum non-zero absolute value these polynomials take and take: ω(x, 1) is often called the measure of irrationality of a real number x.
Roth's theorem says that irrational real algebraic numbers have measure of irrationality 1.
Two numbers x, y are called algebraically dependent if there is a non-zero polynomial P in two indeterminates with integer coefficients such that P(x, y) = 0.
There is a powerful theorem that two complex numbers that are algebraically dependent belong to the same Mahler class.
The symbol S probably stood for the name of Mahler's teacher Carl Ludwig Siegel, and T and U are just the next two letters.
Jurjen Koksma in 1939 proposed another classification based on approximation by algebraic numbers.
If the ω*(x, n) are all finite but unbounded, x is called a T*-number, Koksma's and Mahler's classifications are equivalent in that they divide the transcendental numbers into the same classes.
[27] Let It can be shown that the nth root of λ (a Liouville number) is a U-number of degree n.[33] 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.
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. The supremum of the sequence {ω(x, n)} is called the type.
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.