One of the most well known proofs is to argue by contradiction as follows: assume √ 2 is rational and write it as a/b where a and b are relatively prime positive rational integers. Then a = 2b. It follows that a is even. Write a = 2a′. From 2a′ = b one deduces that b also is even, contradicting the assumption that a and b were relatively prime. There are variants of this proof a number of them ...

In this paper we investigate the infinite convergent sum T = ∑∞ n=0 P (n) Q(n) , where P (x) ∈ Q[x], Q(x) ∈ Q[x] and Q(x) has only simple rational zeros. N. Saradha and R. Tijdeman have obtained sufficient and necessary conditions for the transcendence of T if the degree of Q(x) is 3. In this paper we give sufficient and necessary conditions for the transcendence of T if the degree of Q(x) is 4...

Consider the minimal β-shift containing the shift space generated by a given Sturmian word. In this paper we characterize such β and investigate their combinatorial, dynamical and measure-theoretical properties and prove that such β are transcendental numbers.

Throughout this paper L will be an algebraically closed field and K an algebraically closed subfield such that the transcendence rank of L over K is at least 3. In [4] we considered the pregeometry G(L/K) = (L,clK), where, for Y a subset of L, the closure clK( Y) is the algebraic closure in L of the subfield generated by K and Y. In this paper we shall be concerned mainly with the geometry G(L/...