On the dimension of angles and their units

Abstract

We show the implications of angles having their own dimension, which facilitates a consistent use of units as is done for lengths, masses, and other physical quantities. We do this by examining the properties of complete trigonometric and exponential functions that are generalizations of the corresponding functions that have dimensionless numbers for arguments. These generalizations provide functions that are independent of units in which the angles are expressed. This property also provides a consistent framework for including quantities involving angles in computer algebra programs without ambiguity that may otherwise occur. This is in contrast to the conventional practice in scientific applications involving trigonometric or exponential functions of angles where it is assumed that the argument is the numerical part of the angle when expressed in units of radians. That practice also assumes that the functions are the corresponding radian-based versions. These assumptions allow angles to be treated as if they had no dimension and no units, an approach that can lead to important difficulties such as incorrect factors of $2\pi$, which can be avoided by assigning an independent dimension to angles.

Unit one is intrusive

“…In 2016, Mills argued that the radian and the cycle should be treated as units of dimension angle, thereby clarifying that certain pairs of definitions refer to the same quantity expressed with different units [24]. Discussion of proposals to add dimension angle continued as a separate thread from the treatment of dimensionless quantities and has remained active [25][26][27][28][29][30][31].…”

Comment on ‘On the dimension of angles and their units’

“…Metrology cannot be a Humpty Dumpty world where 'when I use a word, it means just what I choose it to mean.' The authors of the paper [4] state in their Introduction that 'frequency is the time rate of change of the phase angle associated with a periodic series of events.' This may be a working definition used by some spectroscopists.…”

“…It is revealing that neither of the symbols f or ν appear within the paper [4], while the symbol ω is used for quantities described as 'frequency' or (in section 5.1) 'rotational frequency'. The standard definition of 'rotational frequency' is the number of rotations per unit time, with SI unit s −1 , whereas in the paper it is used for the quantity most people would call 'angular velocity'.…”