Metrologia PAPER • OPEN ACCESS Related content - On the units radian and cycle for the Dimensionless units in the SI quantity plane angle Ian Mills To cite this article: Peter J Mohr and William D Phillips 2015 Metrologia 52 40 - Comment on ‘Dimensionless units in the SI’ B P Leonard - Redefinition of the kilogram, ampere, kelvin and mole View the article online for updates and enhancements. Ian M Mills, Peter J Mohr, Terry J Quinn et al. Recent citations - D. Rossi et al - Angles are inherently neither length ratios nor dimensionless Paul Quincey et al - True versus apparent shapes of bow shocks Jorge A Tarango-Yong and William J Henney This content was downloaded from IP address 131.169.5.251 on 06/11/2019 at 19:19 IOP Metrologia Metrologia Metrologia Metrologia 52 (2015) 40–47 doi:10.1088/0026-1394/52/1/40 52 Dimensionless units in the SI 2015 Peter J Mohr and William D Phillips © 2015 BIPM & IOP Publishing Ltd National Institute of Standards and Technology, 100 Bureau Dr, Gaithersburg, MD 20899-8420, USA E-mail: [email protected] and [email protected] METRO Received 25 August 2014, revised 1 October 2014 Accepted for publication 8 October 2014 40 Published 18 December 2014 Abstract P J Mohr and W D Phillips 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 Dimensionless units in the SI consider dimensionless units as defined in the SI, 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 Printed in the UK ‘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 MET and that hertz is not a coherent unit in the SI. We also discuss how including angular and counting units affects the fundamental constants. 10.1088/0026-1394/52/1/40 Keywords: SI, international system of units, dimensionless units, radian, hertz Paper A parable of dimensionless units If Bert had given an uninformative response about the rota- Bert has a turntable operating at a rotation frequency of tion rate, Ernie might have asked ‘what is the concentration 0026-1394 331 revolutions per minute. His friend, Ernie, asks Bert, ‘At of oxygen in the air you’re breathing?' Bert could respond 3 25 −3 24 −3 what frequency is your turntable rotating?’ If Bert were to ‘about 10 atoms m ’ or ‘5 × 10 molecules m ’. These are answer ‘0.555 Hz’ Ernie would know the rotation frequency. clear answers. A factor-of-two ambiguity would arise if he had not specified the entity being counted; in fact, the current SI 1 Similarly, if Bert were to answer ‘3.49 radians per second’, Ernie would know the rotation frequency. And, if Bert were to says that ‘atoms’ or ‘molecules’ are dimensionless units that 24 −3 say ‘200 degrees per second’, Ernie would be well informed. should be set equal to ‘one’. If Bert had said ‘5 × 10 m ’, 40 On the other hand, if Bert were to respond ‘the rotation fre- Ernie might have interpreted that as being the atomic den- quency is 3.49’, we would all agree that Ernie would not know sity and wonder if oxygen deprivation had compromised his friend’s mental acuity. 47 the rotation frequency. Nor, would a response of ‘3.49 per second’ be useful to Ernie, any more than a response of Such situations allowed by ambiguous units are unten- ‘200 per second’. In order to convey useful information, Bert able, which is one of the main points of this paper. Fixing the must tell Ernie the units in which he is reporting the rota- problem is not going to be easy, as evidenced by the fact that tion frequency, including the so-called dimensionless units of it has persisted for so many years after the institution of the SI cycles or radians or degrees. The current formulation of the SI in 1960. This situation has led to such problematic pronounce- specifically allows the units ‘radian’ or ‘cycle’ to be replaced ments as ‘the radian and steradian are special names for the by the dimensionless unit ‘one’, and it allows both radians number one …’ [1]. One starting point is to recognize that per second and cycles per second (Hz) to be replaced with replacing radians or cycles or similar dimensionless units inverse seconds. Clearly, if Bert had followed this prescrip- by ‘one’ leads to trouble, and replacing molecules by ‘one’ tion, allowed by the current SI, he would have left Ernie in the in expressions for molecular concentration may also lead to dark about the rotation frequency. trouble. These and similar arguments are made explicit below, as are our suggestions for a revision to the SI that goes a long way toward solving the problem1. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further 1 The hypothetical conversation in this section is not meant to suggest that distribution of this work must maintain attribution to the author(s) and the title there actually was such a conversation between Albert Einstein and Ernest of the work, journal citation and DOI. Rutherford. 0026-1394/15/010040+8$33.00 40 © 2015 BIPM & IOP Publishing Ltd Printed in the UK Metrologia 52 (2015) 40 P J Mohr and W D Phillips 1. Introduction or The International System of Units (SI) defines units that are 1 22221 {Em}J=={}{}vmkg ms− {}{}v J. (4) used to express the values of physical quantities [1]. In the 2 2 foreseeable future, it is expected that there will be a redefini- tion of the SI based on specified values of certain fundamental In this way, calculations are separated into a purely numer- constants [2]. This constitutes a dramatic change with one of ical part and one involving units. For non-trivial equations, the consequences being that there will no longer be a clear working separately with only the numerical values provides distinction between base units and derived units [3, 4]. In view a practical way of carrying out the calculation. In particular, of this change, it is timely to reexamine units in the SI and when mathematical functions such as exponential, trigono- their definitions. One goal is to ensure that all such units are metric, or Bessel functions are involved, the arguments are coherent, i.e. they comprise a coherent system of units. necessarily numbers without units, and calculations are done In the current SI, various quantities are designated as being with only the numerical values. This is further simplified if dimensionless. That is, they are deemed to have no unit or a coherent system of units is used, so there is no additional have what has been called the coherent derived unit ‘one’. In numerical factor. some cases this designation leads to ambiguous results for Physical science is based on mathematical equations, these quantities. In this paper, we examine units in the SI that which follow the rules of analysis spelled out in numerous are considered dimensionless and other units not presently mathematical reference works. Generally, in mathematical included in the SI that might be added to bring it into closer reference texts, distances, areas, and angles, for example, are alignment with widespread scientific usage. all dimensionless. On the other hand, in physical science, one uses units. One consequence of this difference is that mathematics 2. Units and dimensional analysis provides no information on how to incorporate units into the analysis of physical phenomena. One role of the SI is to In general, units are used to convey information about the provide a systematic framework for including units in equa- results of measurements or theoretical calculations. To com- tions that describe physical phenomena. municate a measurement of a length, for example, the result is expressed as a number and a unit, which in the SI is the meter. The number tells the length in meters of the result of 3. Angles the measurement. For simple algebraic calculations involving units, one Angles play an essential role in mathematics, physics, and can write out the expression and separately collect the units, engineering. They fall into the category of quantities with which may be replaced by an equivalent unit for convenience. dimensionless units in the current SI, which leads to ambigui- For example, the kinetic energy E of a mass m = 2 kg moving ties in applications. This issue has been widely discussed in at a velocity v = 3 m s−1 is calculated as the literature, and arguments are given on both sides of the question of whether angles are quantities that should have 2 1 211 2 2·3 22 units [6]. E ==mv (2kg) (3 ms−−) ==kg ms 9J, (1) 2 2 2 In part because units are rarely considered in mathematics, where J is the symbol for joule, the SI unit of energy. This the unit of radian for angles is rarely mentioned in the math- calculation illustrates the important principle of the SI that the ematics reference literature, just as the meter is also rarely units are coherent. That is, when a combination of units is mentioned. Units are unnecessary in purely mathematical replaced by an equivalent unit, there is no additional numer- analysis.