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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 physical quantity, 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, gradians, 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/second, 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 turn= 2π rad (4) 1 cycle = 2π rad (5)

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

3. Periodic phenomena In texts and monographs on mathematics, particularly calculus and complex variable theory, there is little or A physical phenomenon that repeats in time 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 trigonometric functions, the exponential function, 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 complex number. 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 times tates by 2 radians in one second, its rotational frequency a unit. Thus when, for example, the sine 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 Taylor series, 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 seconds, 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 mass 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 energy, 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, joule. 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. right angle 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 dimensions 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.