arXiv:1604.06774v1 [.class-ph] 18 Feb 2016 n steui o oi nl xrse sm as expressed units solid base SI for of unit terms the in for as m/m unit and the as as expressed symbols,” and angle units names plane derived special “Coherent with 3, SI Table listed the in in are [1] Brochure they SI units, the in derived dimensionless as ednt eue nepesosfrohrS derived SI other for expressions in convenient.” but used as may, units, which be dimensionless of not, as symbols and need steradian, names the the units, SI, and derived the in the units supplementary namely the interpret “to decision osse fterda o ln nl n h steradian the and angle angle. plane solid units for for supplementary radian of the category of The consisted units, supplementary units. units, derived base and clas- classes, were three units into 1960, on sified in Conference (CGPM) General Measures the and of 12 Resolution by tablished hroe hspatc a e oerr npbihdre- frequen- published and in cies. involving errors quantities to physical for led sults has Fur- practice nonsense. this as statement thermore, italicized the view we phasis; quantities.” omitted dimensionless generally of is values but one the appropriate, unit specifying derived where in the used for practice are symbol sr In the and concerned. symbols quantity the the one about number information the for “ Table names that: to stated footnote is a In it respectively. 3, units, base SI of terms n o hs,smtmscle hs nl 2.This [2]. angle called It angle sometimes unit. plane phase, for base unit for a SI as coherent and de- the reclassified dimensionless considered be be a instead should considered therefore and be that unit longer and rived no angle) radian of the (the dimension ∗ † [email protected] [email protected] .Cretsau fpaeageadsldangle solid and angle plane of status Current 2. sarsl ftercasfiaino h ainand radian the of reclassification the of result a As n19,Rslto fte20 the of 8 Resolution 1995, In es- was (SI) Units of System International the When nteaoeqoain h tlc r de o em- for added are italics the quotation, above the In ti rpsdhr htage ecniee ohave to considered be angles that here proposed is It osqecso uhamdfiainaeexamined. are modification a such of consequences epooeta h Ib oie ota h aini aeun base a is radian the that so modified be SI the that propose We .INTRODUCTION I. .Background 1. rpslt lsiyterda sabs nti h SI the in unit base a as radian the classify to proposal A h ainadseainaespecial are steradian and radian The ainlIsiueo tnad n ehooy Gaithersb Technology, and Standards of Institute National .Proposal 3. htmyb sdt convey to used be may that th ee .Mohr J. Peter GMsae the stated CGPM 2 /m ∗ 2 n ila .Phillips D. William and in h au fapaeageo hs angle phase particular, or In angle plane a quantities. of value physical the measurable other to nteS as SI the in a mlctosfrteuiso rqec,a explained as , of units below. the for implications has where tagtlnsi h ai ftelnt ftearc the of length the intersecting of two ratio between the is angle lines the straight of value numerical ainad[ and radian we h ie facrl etrda h etxo the of vertex the at centered the a to angle of lines the tween uniis eas a ugse nteS rcue the Brochure) SI the in ratio suggested (as because quantities, 1. number the by replaced or omitted ntisl,rte hnadrvdui si h present the in base as a unit be derived should a than therefore SI. rather and derived not itself, units, is unit it base SI; SI; the the other in in from angle unit dimension dimensionless have a should it considered be not should angle is of radian. angle dimension of that the unit mean has the to it not with arc quantity; does subtended dimensionless for this the in a way Similarly, of angle length same the the radius. the of of the In ratio value the numerical is meter. the dimension of a angle, the the unit of two-radian has of the length a it length with the the quantity; length to that dimensionless of mean table is a not the meters is does of in table This table length stick. the the meter of of length ratio the the of value numerical dimensionless. itself for quantity brackets physical curly the in make number the fact, any In number. a just s[ is ln nlsadpaeage aepoete similar properties have angles phase and quantities angles physical Plane as phase and angle Plane 1. h trda ste eie ntwt h ntrad unit the with unit derived a then is steradian The ti oeie adta nlsaedimensionless are angles that said sometimes is It ti nesniltnto hsnt htteradian the that note this of tenet essential an is It The table. long two-meter a consider example, For θ a.I h urn I h nt[ unit the SI, current the In rad. = ] hsclqatt sjs ubr u htde not does that but number, a just is quantity physical s/r { θ I UPEETLMATERIAL SUPPLEMENTAL II. } salnt iie eghadi therefore is and length a divided length a is sthe is θ steui a.Frapaeage the angle, plane a For rad. unit the is ] r,M 09,USA 20899, MD urg, ueia value numerical t oeo h eal and details the of Some it. † r ftecircle the of θ = { θ } [ θ ] fteagei h unit the in angle the of { θ } = s/r θ θ a eeither be may ] a ewritten be can n h unit the and s be- (1) 2 . 2

2. Relations to non-SI units for angles SI is prevented. In particular, in the current SI, since the SI unit radian is considered a dimensionless unit, it As with any , angles may be expressed may be omitted. The same is true for the non-SI unit in units other than coherent SI units. However, units ”cycle,” which is omitted when the currently permitted −1 other than the radian cannot also be coherent SI units replacement Hz → s is made. As a result of these if the radian is. Some other units for angles are de- prescriptions, the units on both sides of Eq. (5) may be grees, , turns, cycles, and revolutions. Angles omitted leading to an inconsistent result. range anywhere from zero to a complete revolution; an- The other principal difference that results from re- gles greater than π rad are sometimes called reflex angles. garding angles as quantities with units that may not be A complete revolution is an angle of 2π rad. dropped is that Hz, i.e. cycles/, explicitly includes Some relations to the non-SI units are the non-SI unit cycle and is therefore not a coherent SI ◦ unit. Instead, radians/second is the coherent SI unit for 180 = π rad (2) frequency. As a consequence, Hz should be considered a π 100 grad = rad (3) unit “permitted for use with the SI.” 2 1 = 2π rad (4) 1 cycle = 2π rad (5)

1 revolution = 2π rad (6) 5. vs physics

3. Periodic phenomena In texts and monographs on mathematics, particularly and complex variable theory, there is little or A physical phenomenon that repeats in over a no discussion of units. Lengths and angles are simply regular time interval is periodic or cyclic. One repetition numbers with no units and with no mention of meters of the phenomenon that happens in a period is a cycle. or radians. As a result, mathematical functions such as The rate of repetitions is a physical quantity called the , the , spher- frequency. The numerical value of the frequency will de- ical Bessel functions, spherical harmonics, or any number pend on the units in which it is expressed. of other mathematical functions are defined as functions We recommend that the SI states that the value of any of a dimensionless real or . frequency of a periodic phenomenon include the complete On the other hand, in physics, it is necessary to include 1 unit such as rad/sec or Hz, but never just s− . units in order to make contact with measurements. The An example is a turning bicycle wheel. If the wheel ro- values of physical quantities are given by a number tates by 2 radians in one second, its rotational frequency a unit. Thus when, for example, the function is used in SI units is 2 rad/s. in physics, the argument often is written as a quantity Another example is electromagnetic radiation, where whose units are understood to be radians, as in Eq. (8). the electric field vector undergoes a periodic repetition However, since the sine function can be defined by its of one million cycles per second. This frequency is , and the terms must be homogeneous in their units, the argument is necessarily a number with ν = 1 MHz, (7) no unit. where MHz is one million cycles per second. Since the This leads to a problem with the use of the radian as electric field vector may be described as a unit in the argument of functions that are defined as having dimensionless arguments. An unambiguous res- E sin ωt, (8) olution of this conflict would be to write the argument of mathematical functions as the numerical value of the where the time dependence of the vector repeats when angle. That is, to write the sine function in Eq. (8) as ωt increases by 2π. (Here we take ωt to represent the nu- sin {ωt}, where ωt = {ωt} rad. merical value of the product.) If t is expressed in , then ω must be expressed in the coherent unit rad/s: However, in scientic publications, functions are written 6 with arguments without the curly brackets, and it would ω =2π × 10 rad/s. (9) be too disruptive to the common practice to insist that curly brackets should be present. So a compromise would be to recommend that when angles are present in math- 4. Difference with the current SI ematical functions and expressed in units of radians, the curly brackets be omitted. By the same token, it should There are two principal differences between the for- also be recognized that when quantities that are the ar- mulation above and the current SI. One consequence of guments of functions such as the sine function are used writing plane angles and phase angles as quantities with in equations outside of those functions, the radian unit explicit units is that an ambiguity present in the current must be explicitly restored. 3

6. Illustrative example this category. Another category of physical quantity, ex- amples of which are the proton-electron ratio and A simple example that illustrates the points made the fine-structure constant, are indeed dimensionless and above is the physics of a harmonic oscillator. The equa- unitless. They are simply numbers. Angles do not fall tion for a harmonic oscillator is into this category. Concerning units for frequency, some of the ambiguity d2 and confusion over frequency arises from the idea that m x(t)+ kx(t)=0, (10) dt2 frequency may be classified into types of frequency with different units being used for different types. The prime where m is the mass and x(t) is the coordinate of the example is Hz vs rad/s which might be considered dif- body and k is the spring constant of the spring. The ferent types of frequency and so are expressed in differ- solution for the motion of the body is ent units. However, frequency is a general concept and classifying types of frequency leads to confusion. For fre- k quency, there is one coherent unit, which is rad/s, and x(t)= x0 sin t + φ , (11) which applies to all types of frequency whether cycles (rm ) of electromagnetic radiation or rotations of a wheel or where x0 is the amplitude of the motion and φ is a heartbeats. In the latter case, the role of radians per phase that depends on the boundary condition. The second is easily seen by considering a Fourier decompo- curly brackets emphasize that the argument of the sine sition of the pattern of the heartbeats and realizing that function is the numerical value of the enclosed physical the fundamental frequency can be viewed as the rate of quantities. The frequency of the oscillations is change of the phase of a sine function with the argument being the numerical value of the phase angle expressed in rad. k k 1 frequency = = s− . (12) This is analogous to , for example. There are m m r (r ) different types of energy, mechanical energy, energy of electromagnetic radiation, gravitational potential energy, In the current SI, where both rad/s and Hz may be ex- but these are recognized to be various forms of energy, −1 pressed as s , this could be taken to mean both and so they all have the same unit, . Of course, this is based on physical principles and may be counter- k frequency = Hz (incorrect) (13) intuitive to people unfamiliar with these principles, but (rm) nevertheless is well understood to be true. While in our proposal the radian becomes a base unit and and angles have dimension, we recognize that historic and common practice often drop the unit radian where k −1 it would otherwise appear. With the exception of fre- frequency = rad s (correct). (14) quencies and only if no confusion can result, we would ( m) r continue to allow the unit radian to be dropped. The choice of Eq. (14) as the correct expression of the In the new SI, expected to be adopted in 2018, all units frequency is evident from the rule (see above) that when are defined by giving an exact value to a fundamental the argument of a trigonometric function (or the imag- physical constant, although not all the definitions strictly inary argument of an exponential function) is used out- adhere to that designation. In the case of angle, one could side of such functions, the unit radian must be explicitly specify the unit radian by fixing the numerical value of a restored. in radians by writing π θ = rad, (16) ⊥ 2 7. On the nature of units where θ⊥ a right angle. This note focuses on angles, radians and related quan- In this brief note, we have proposed that the radian be tities, because the ambiguity in the SI associated with viewed as a base unit in the SI. In this section, we clarify these concepts has often led to errors and confusion. some issues relating to units. Other similar quantities, such as counting units, may Values of most physical quantities q can be written as need to be addressed in a similar manner, but this is beyond the scope of this note. q = {q}[q], (15) Finally, we note that in the new SI, the distinction between base units and derived units may be unneces- where {q} is the numerical value of the quantity ex- sary. The same may be true of the distinction between pressed in the unit [q]. As seen in Eq. (1), angles fall into and units. 4

[1] BIPM. Le Syst`eme international d’unit´es (SI). Bureau [2] Here the term phase refers to a physical phase as in quan- International des Poids et Mesures, S`evres, France, 8th tum mechanics or electrodynamics. In mathematics, par- edition, 2006. ticularly complex variable theory, phase is regarded as a dimensionless number as is length.