Dimensionless Units in the SI

TL;DR

This paper examines the coherence of the SI system concerning dimensionless units like radians and counts, proposing a consistent way to incorporate them and clarifying their impact on fundamental constants.

Contribution

It identifies incoherence issues with dimensionless units in SI and proposes a method to include them properly, emphasizing radians as the coherent angular unit.

Findings

Radian should be the coherent SI unit for angles.

Including counting units affects the values of fundamental constants.

Hertz is not a coherent SI unit.

Abstract

The International System of Units (SI) is supposed to be coherent. That is, when a combination of units is replaced by an equivalent unit, there is no additional numerical factor. Here we consider dimensionless units as defined in the SI, {\it e.g.} angular units like radians or steradians and counting units like radioactive decays or molecules. We show that an incoherence may arise when different units of this type are replaced by a single dimensionless unit, the unit "one", and suggest how to properly include such units into the SI in order to remove the incoherence. In particular, we argue that the radian is the appropriate coherent unit for angles and that hertz is not a coherent unit in the SI. We also discuss how including angular and counting units affects the fundamental constants.