On the dual nature of a plane angle

TL;DR

This paper argues that a plane angle can be both a dimensional and a dimensionless quantity depending on the context, advocating for its classification as a base quantity in SI units.

Contribution

It demonstrates that plane angles should be considered as base quantities in SI, with their own units, and clarifies their role in physical measurements and equations.

Findings

Plane angle is described as a dimensional quantity in measurements.

In theoretical physics, angles appear as dimensionless combinations in equations.

The paper advocates for including the radian as a base SI unit.

Abstract

For decades, metrologists have debated heatedly whether a plane angle is a dimensional or dimensionless quantity; whether it is a base quantity in the International System of Units (SI) or a derived quantity. Two main points of view have emerged in the international metrology community. Those who hold the first view believe that a plane angle is a dimensionless derived quantity equal to the ratio of two lengths, and its unit, the radian, is the dimensionless number one (1 rad = 1 m/m = 1). Those who hold the second view believe that a plane angle is a dimensional quantity with its own independent dimension, and its unit, the radian, is not the dimensionless number one, as is currently accepted in the SI. This article demonstrates that, depending on the physical situation, a plane angle is described by either a dimensional or a dimensionless quantity. When measuring, expressing, and communicating an angle's size, physicists use the dimensional quantity plane angle. Its dimension and unit are independent of the dimensions of other quantities and their units. This quantity, plane angle, should be classified as a base quantity, and its unit, radian, should be included in the class of base SI units. In theoretical studies of physical systems with angular quantities, the latter always enter into equations as a dimensionless combination of dimensional plane angles. This dimensionless combination, in turn, is also a physical quantity characterizing the plane angle in question. This new quantity is a dimensionless derived quantity, which physicists also call an angle.