A number of expansions to the geometric proof of the irrationality of the square root of two have been put forward in the paper “Irrationality From The Book” by Steven J. Miller and David Montague. There are a number of other conceptually simple expansions which can be attempted. This paper shows some basic work towards proving the irrationality of √ p, where p is a prime, and of 3 √ 2. Althoug...

We show how one can use Hermite-Padé approximation and little q-Jacobi polynomials to construct rational approximants for ζq(2). These numbers are qanalogues of the well known ζ(2). Here q = 1 p , with p an integer greater than one. These approximants are good enough to show the irrationality of ζq(2) and they allow us to calculate an upper bound for its measure of irrationality: μ (ζq(2)) ≤ 10...

Based on Rational Emotive Behavior Therapy (REBT) we tested the hitherto unexplored assumption that irrationality as conceptualized by REBT (demandingness, self evaluation, low frustration tolerance), is associated with erroneous statistical reasoning. We assessed trait irrationality of 216 respondents and individual estimates of future winning probabilities in the context of the Wortman (1975)...

We present a game-theoretic account of irrational agent behavior and define conditions under which irrational behavior may be considered quasi-rational. To do so, we use a 2-player, zero-sum strategic game, parameterize the reward structure and study how the value of the game changes with this parameter. We argue that for any “underdog” agent, there is a point at which the asymmetry of the game...

1. Irrationality Measures An irrationality measure of x ∈ R \Q is a number μ such that ∀ > 0,∃C > 0,∀(p, q) ∈ Z, ∣∣∣∣x− pq ∣∣∣∣ ≥ C qμ+ . This is a way to measure how well the number x can be approximated by rational numbers. The measure is effective when C( ) is known. We denote inf {μ | μ is an irrationality measure of x } by μ(x), and we call it the irrationality measure of x. By definition,...

In this paper we give irrationality results for numbers of the form ∑∞ n=1 an n! where the numbers an behave like a geometric progression for a while. The method is elementary, not using differentiation or integration. In particular, we derive elementary proofs of the irrationality of π and em for Gaussian integers m 6= 0.

We describe an unexpected connection between bounded height in families of finitely generated subgroups tori problems and irrationality. Our method allow us to recover effective irrationality measures for some values algebraic functions

There are not many new results concerning the linear independence of numbers. Exceptions in the last decade are, e.g., the result of Sorokin [8] which proves the linear independence of logarithmus of special rational numbers, or that of Bezivin [2] which proves linear independence of roots of special functional equations. The algebraic independence of numbers can be considered as a generalizati...