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

## TL;DR

This paper proposes a new measure called ppa to evaluate the quality of approximations of real numbers within a model, based on number complexity, and explores its properties and applications.

## Contribution

It introduces a novel measure of approximation quality using complexity, linking it to classical Diophantine approximation and opening questions about universal approximation.

## Key findings

- ppa effectively quantifies approximation quality
- The measure aligns with classical rational approximation theory
- Open problem on universal approximation in limit models

## 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 $\mu$ (called a measure of intelligence) that associates to any approximation $\mathbf{app}$ of a given real number in a given model a positive number $\mu(\mathbf{app})$, which measures the quality of that approximation. More precisely, an approximation $\mathbf{app}$ is deemed intelligent if and only if $\mu(\mathbf{app}) \geq 1$. 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.