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A measure of intelligence of an approximation of a real number in a given model

Bakir Farhi

TL;DR

A way to measure the intelligence (or the interest) of an approximation of a given real number in a given model of approximation, based on the idea of the complexity of a number, which characterises the intelligence of that approximation.

Abstract

In this paper, we introduce a way to measure the intelligence (or relevance) of an approximation of a given real number in a given model of approximation. Based on the notion of complexity of a number, defined as the number of its digits (in a given base), we introduce a function noted $μ$ (called a measure of intelligence) that associates to any approximation $\mathbf{app}$ of a given real number in a given model a positive number $μ(\mathbf{app})$, which measures the quality of that approximation. More precisely, an approximation $\mathbf{app}$ is deemed intelligent if and only if $μ(\mathbf{app}) \geq 1$. We illustrate the theory with several numerical examples and apply it to the rational model. In this case, we show that it is consistent with the classical theory of rational Diophantine approximation. We conclude by stating an open problem, namely whether any real number can be intelligently approximated in a given model for which it is a limit point.

A measure of intelligence of an approximation of a real number in a given model

TL;DR

A way to measure the intelligence (or the interest) of an approximation of a given real number in a given model of approximation, based on the idea of the complexity of a number, which characterises the intelligence of that approximation.

Abstract

In this paper, we introduce a way to measure the intelligence (or relevance) of an approximation of a given real number in a given model of approximation. Based on the notion of complexity of a number, defined as the number of its digits (in a given base), we introduce a function noted (called a measure of intelligence) that associates to any approximation of a given real number in a given model a positive number , which measures the quality of that approximation. More precisely, an approximation is deemed intelligent if and only if . We illustrate the theory with several numerical examples and apply it to the rational model. In this case, we show that it is consistent with the classical theory of rational Diophantine approximation. We conclude by stating an open problem, namely whether any real number can be intelligently approximated in a given model for which it is a limit point.

Paper Structure

This paper contains 13 sections, 10 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Models of approximation
  3. The size and the logarithmic size of an approximation lying in a given model
  4. A measure of intelligence of an approximation lying in a given model
  5. Examples
  6. Approximations concerning the number ${\boldsymbol \pi}$
  7. Approximations concerning the number ${\boldsymbol e}$
  8. Other notable approximations
  9. Intelligent rational approximations
  10. Main results
  11. Applications of Theorems \ref{['t1']} and \ref{['t2']} to $\pi$, $e$, and various other real numbers
  12. Irrational numbers with bounded or unbounded intelligence measure in the rational model
  13. An important open problem

Key Result

Proposition 6.1

Let $\alpha$ be an irrational number and let $p , q \in {\mathbb Z}^*$ such that $|p q| \neq 1$. Then the rational approximation $\alpha \approx \frac{p}{q}$ is intelligent (in the rational model) if and only if one of the two following equivalent inequalities holds:

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Definition 4.1
  • Remark 4.2
  • Proposition 6.1
  • proof
  • Corollary 6.2
  • proof
  • ...and 17 more