For simplicity, we follow the rules: k, n, p, K, N are natural numbers, x, y, e1 are real numbers, s1, s2, s3 are sequences of real numbers, and s4 is a finite sequence of elements of R. Let us consider x. We introduce x is irrational as an antonym of x is rational. Let us consider x, y. We introduce x as a synonym of x. One can prove the following two propositions: (1) If p is prime, then √ p ...

This paper is devoted to the rational approximation of automatic real numbers, that is, real numbers whose expansion in an integer base can be generated by a finite automaton. We derive upper bounds for the irrationality exponent of famous automatic real numbers associated with the Thue–Morse, Rudin–Shapiro, paperfolding and Baum–Sweet sequences. These upper bounds arise from the construction o...

Inspired by the epistemic game theory framework, I elicit subjects’ preferences over outcomes, beliefs about strategies, and beliefs about beliefs in a variety of simple games. I find that the prisoners’ dilemma and the traditional centipede game are both Bayesian games, with many non-selfish types. Many players choose strategies that are clearly inconsistent with their elicited beliefs and pre...

We apply the Padé technique to find rational approximations to h±(q1, q2) = ∞ ∑ k=1 q 1 1± q 2 , 0 < q1, q2 < 1, q1 ∈ Q, q2 = 1/p2, p2 ∈ N \ {1}. A separate section is dedicated to the special case qi = q ri , ri ∈ N, q = 1/p, p ∈ N \ {1}. In this construction we make use of little q-Jacobi polynomials. Our rational approximations are good enough to prove the irrationality of h(q1, q2) and give...

In 2006, Jonathan Sondow gave a nice geometric proof that e is irrational. Moreover, he said that a generalization of his construction may be used to prove the Cantor’s theorem. But, he didn’t do it in his paper, see [1]. So, this work will give a geometric proof to Cantor’s theorem using Sondow’s construction. After, it is given an irrationality measure to some Cantor series, for that, we gene...

This study explored how religious fundamentalism related to irrational beliefs and primitive defense mechanisms. We also explored how the personality factors of openness to experience and neuroticism moderated these relations. Participants (N = 120) were recruited in an urban area from a Northeastern university, a psychotherapy center, and through Internet advertising. The results demonstrated ...

0. In 1978, Apéry showed the irrationality of ζ(3) = ∑∞ n=1 1 n3 by giving the approximants `n = unζ(3) − vn ∈ Qζ(3) + Q, un, dnvn ∈ Z, dn = l.c.m.(1, 2, . . . , n), with the property |`n| → ( √ 2 − 1) < 1/e as n → ∞. A similar approach was put forward to show the irrationality of ζ(2) (which is π/6, hence transcendental thanks to Lindemann) but I will concentrate on the case of ζ(3). A few mon...

Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler’s constant γ. The proof is by reduction to known irrationality criteria for γ involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, γ, and logari...