Rational approximations to two irrational numbers

Series of Error Terms for Rational Approximations of Irrational Numbers

Let pn/qn be the n-th convergent of a real irrational number α, and let εn = αqn−pn. In this paper we investigate various sums of the type ∑ m εm, ∑ m |εm|, and ∑ m εmx m. The main subject of the paper is bounds for these sums. In particular, we investigate the behaviour of such sums when α is a quadratic surd. The most significant properties of the error sums depend essentially on Fibonacci nu...

Rational and Irrational Bubbles

Rational Approximations to n

Using an IBM 1130 computer, we have generated the first 20,000 partial quotients in the ordinary continued-fraction representation of x.

The rational approach to the irrational.

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Rational Approximations to Rational Models of Categorization

Can statistical machine learning theories and algorithms help explain human learning? Broadly speaking, machine learning studies the fundamental laws that govern all learning processes, including both artificial systems (e.g., computers) and natural systems (e.g., humans). It has long been understood that theories and algorithms from machine learning are relevant to understanding aspects of hum...