Rethinking real numbers as infinite decimals

TL;DR

This paper constructs real numbers via infinite decimal expansions, analyzing their computability properties and showing that basic operations are weakly computable but not fully computable in this encoding.

Contribution

It provides a detailed construction of real numbers through decimal expansions and investigates the computability of arithmetic operations within this framework.

Findings

Addition and multiplication are not computable in the decimal encoding.

Both operations are weakly computable in the decimal expansion.

Certain shifts of addition and multiplication become computable after encoding permutations.

Abstract

We give a~detailed construction of the complete ordered field of real numbers by means of infinite decimal expansions. We prove that in the canonical encoding of decimals neither addition nor multiplication is {\em computable}, but that both operations are {\em weakly computable}; we introduce both kinds of computability in greater generality. We determine which additive and multiplicative shifts (restrictions of addition and multiplication to one variable) are computable, and prove that each of these shifts becomes computable after a~permutation of encoding. We ask if it is the case for the bivariate addition and multiplication.