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

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.