One of the things I like about science is that it’s not frozen. It’s a stew of ideas, and when a better one comes along, it gets embraced over older ideas with worse flaws. Of course, that can be a relatively slow process compared to the speed of news in this day and age, and sometimes its more complicated, but we tend to get there more often than not.

So, I’ve been rather frustrated to discover that the basic theory of one of the core ideas in applied mathematics remains a mess. I speak of the subject of variables, types, units, and dimensions.

Variables are a necessary ingredient in mathematical and computational models. We cann’t build a model without them, and in some sense, we’re all experts – we understand estimation, precision, evaluation, representation, types, units, and dimensions. We understand these things practically at an intuitive level – we just know how things fit together without having to invoke an explicit theory. Yet, when we pay attention to the details, we find some things are rather incoherent. Terminology is poorly defined. Implementation is often over-looked. Variety seems to be swept under the rug, without serious concern.

Other authors have said similar things – in fact, it seems every serious effort on the topic of dimensional analysis spanning decades has said the same thing. And each has made valuable contributions. Yet, a basic and satisfactory classroom theory still seems to elude us.

Perhaps we shouldn’t care too much – certainly I didn’t until recently. Experts know what they’re doing without a need for such a theory. But that argument strikes me now as dark parallel to that of the “Practical men” of electrical engineering back before Maxwell’s laws were accepted.

Angles are one spot where our theory seems confused. Here are some quotes from several authors on the topic.

A question may arise in connection with the dimensions of \(\theta\). We have said that it is dimensionless, and that its numerical magnitude does not change when the size of the fundamental units of mass, length, or time are changed. This of course is true, but it does not follow that therefore the numerical magnitude of is uniquely determined, as we see at once from the possibility of measuring in degrees or in radians. Are we therefore justified in treating as a constant and saying that it may enter the functional relation in any way whatever? – (Bridgman, 1922, page 3)

An angle, as can be seen from its expression in radians as the ratio of two lengths, is of zero dimensions. Hence the dimensions of angular velocity are \([T^{-1}]\); of angular acceleration, \([T^{-2}]\); and of angular momentum (moment of inertia x angular velocity), \([ML^2T^{-1}]\). (Pankhust, 1964, page 23)

(Langhaar, 1967) gives no discussion but requires angles to be dimensionless.

All angles are dimensionless (ratio of arc length to radius) and should be taken in radians for this reason. – (White, 1986, 249) - still in print in 2015.

The words radian and steradian merely serve as reminders of how the corresponding numbers were obtained. Nevertheless, there is no harm in including ‘radian’ or ‘steradian’ in any mathematical expression provided that we remember that we may put unity in their place whenever we wish. – (Massey 1986, page 37)

In 1995, the radian and steradian were abolished from the SI system of units, being deemed dimensionless and hence not needing official designation.

A dimension is a measure of a physical quantity (without numerical values), while a unit is a way to assign a number to that dimension. For example, length is a dimension that is measured in units such as microns (μm), feet (ft), centimeters (cm), meters (m), kilometers (km), etc. (Fig. 7-1). There are seven primary dimensions (also called fundamental or basic dimensions) … Dimensional variables are defined as dimensional quantities that change or vary in the problem. For the simple differential equation given in Eq. 7-4, there are two dimensional variables: z (dimension of length) and t (dimension of time). Nondimensional (or dimensionless) variables are defined as quantities that change or vary in the problem, but have no dimensions; an example is angle of rotation, measured in degrees or radians which are dimensionless units – (Cengal, 2006, pages 270 and 299)

On the other hand, there are quantities whose dimensions are 1 (one). These always occur when all the dimensions associated with the quantity cancel out. For example, the magnitude of a plane angle \(\varphi\) may be expressed in radians. But the radian (abbreviated “rad”) itself is the ratio of the length of the subtended arc \(\lambda\) of a circle and that circle’s radius R. Thus, we have \(\varphi = \lambda/R\), and hence we can write for the dimension of the angle \[[\varphi] = \text{rad} = \left[ \frac{\lambda}{R} \right] = \frac{[\lambda]}{[R]} = \frac{\text{meter}}{\text{meter}} = 1.\] Therefore, the dimension of the radian is 1. In this respect, a dimension of 1 is analogous to a coefficient of 1, or an exponent of 1. – (Szirtes, 2007, page 34)

Also included sometimes among the base quantities are two dimensionless quantities, plane angle and solid angle, which are measured in radians and steradians, respectively. We consider them derived quantities because, though dimensionless, they are defined in terms of operations involving length, much like area is defined in terms of length operations. – (Sonin 2001, page 30)

(Conejo, 2021) makes no mention of angles at all.

Even though frequency, angular velocity, angular frequency and radioactivity all have the dimension 1/T, of these only frequency is expressed in hertz. Thus a disc rotating at 60 revolutions per minute (rpm) is said to have an angular velocity of 2π rad/s and a frequency of rotation of 1 Hz. The correspondence between a frequency f with the unit hertz and an angular velocity ω with the unit radians per second is ω = 2πf ... – Wikipedia entry on Hertz in 2022

What are we to make of this?

It was, in some sense, an abolishment of the identity of angles to a nether-world of non-dimensionlessness – existing in every practical manner, but the legal one. Angles have units (degree, radian, turn, grade), we often have to convert between units, they are directly measurable using a protractor, and they have a unique topology (\(-180^{\circ} = 180^{\circ}\)) not shared by other dimensionless ratios like aspect ratio.

It is commonly asserted in engineering and applied mathematics that angles are dimensionless quantities as Bridgman did a century ago. The argument is typically something likes this (I have changed symbol names for convenience) …

… On the other hand, there are quantities whose dimensions are 1 (one). These always occur when all the dimensions associated with the quantity cancel out. For example, the magnitude of a plane angle may be expressed in radians. But the radian (abbreviated “rad”) itself is the ratio of the length of the subtended arc \(p\) of a circle and that circle’s radius \(r\). Thus, we have \(\theta = p/r\), and hence we can write for the dimension of the angle

\[[\theta] = \text{rad} = \left[\frac{p}{r}\right] = \frac{[p]}{[r]} = \frac{\text{meter}}{\text{meter}} = 1\]

Therefore, the dimension of the radian is 1. …

Now, that’s … reasonable. There’s nothing formally wrong with it. But it’s not a proof. This argument only demonstrates consistency – if we measure arc \(p\) in meters and radius \(r\) in meters and define the angle \(\theta\) as the ratio of arc to radius, then angle must be dimensionless. However, it also viloates one of the guiding principles of dimensional analysis – to create formulas that are true independent of the units selected. The equation \(\theta = p/r\) is **only true** when we measure angles in radians! It is not true if we use degrees, or any other convention.

Under the dimensionless assertion, angles are no different than **aspect ratios**. This has inspired authors to remind readers that “… just because quantities have the same dimensions doesn’t mean the are the same things …” This usually arises when we discuss torques and frequencies, for which the SI dimensions do not identify the rotational component. Because, after all, angles are periodic, and one turn in either direction brings things back to their starting point. And that is something clearly not present in aspect ratios.

I contend that this type of hedging/waffling/dancing flaunts the very purpose of augmenting numbers with units and type systems. Units and dimensions were created to enrich numbers – to make it easier to communicate information and context and to avoid ambiguity. Is it really worth the ambiguity to have torques and energies measured in the same dimensions when they are naturally different things?

To insist that angles are “dimensionless” when they clearly are their own kind of thing, and can be measured directly with their own units in complete independent parallel with lengths? That seems like a classic misguided attempt to extend a framework beyond its natural scope.

There is another way to think of the relationship between angle and arc – as a conversion. Suppose we measure an angle between two bearings. That angle corresponds to an arc that is part of a circle, and that arc has a length. Formally, this should be calculated as an arc-length integral. Calling the arc a curve given by \(\gamma(t)\) for \(t \in [0,\theta]\), the arc-lenth formula from calculus is \[p = \int_{\theta_1}^{\theta_2} \left|\frac{d\gamma}{dt} \right| dt.\] Observe that this formula is independent of the choice of units of \(t\) because the local arclength is a derivative with units of distance per \(dt\). So, let’s take length in units of meters and angle (our integration parameter) in units of radians (rad). Applying with \(\gamma(t) = r e^{it}\), \(d\gamma/dt = r i e^{it}\) and \(| r i e^{it}| = r\) with units of meters per radian. Then \[\begin{align*}
p = \int_{0}^{\theta} | r i e^{it}| dt
= \int_{0}^{\theta} r dt
= r \int_{0}^{\theta} dt
= r \theta.
\end{align*}\] where \(r\) *still has units of meters per radian*! \[[ p = r \theta] \rightarrow
[ p ] = [ r ] [ \theta ] \rightarrow
(\text{meter}) = (\text{meter}/\text{rad}) (\text{rad})\] If radians are dimensionless \(( \text{rad} = \text{1})\), then this reduces to the standard formula. However, radians need not be *DIMENSIONLESS*. The relationship holds even without this assumption. Thus, it is not necessary to treat radians as dimensionless as long as one treats the necessary conversions as lengths per unit of angle.

The length of the outer perimeter \(p\) of a sector of a circle with angle \(\theta\) \[\begin{align*} p \; \text{meters} &= (r \; \text{meters/radian}) &&(\theta \;\text{radians}) \\ &= (2 \pi r \; \text{meters/turn}) &&(\theta \;\text{turns}) \\ &= (\pi r /180 \; \text{meters/degree}) &&(\theta \;\text{degrees}) \\ &= (\pi r /200 \; \text{meters/grad}) &&(\theta \;\text{grads}) \end{align*}\]

or when we explicitly introduce the necessary conversion factors,\[\begin{align*} p \; \text{meters} &= (r \; \text{meters}) (1 \; \text{per radian}) &&(\theta \;\text{radians}) \\ &= (r \; \text{meters}) (2 \pi \; \text{per turn}) &&(\theta \;\text{turns}) \\ &= (r \; \text{meters}) (\pi /180 \; \text{per degree}) &&(\theta \;\text{degrees}) \\ &= (r \; \text{meters}) (\pi /200 \; \text{per grad}) &&(\theta \;\text{grads}) \end{align*}\] where \[ (2 \pi \;\text{radians}) =(1 \;\text{turn}) =(360 \;\text{degrees}) =(400 \;\text{grads}). \]

\[ p = r f \theta \] where \(f\) is the angular frequency = the circumference of a unit circle per unit radius divided by the number of angle units in a unit circle, with net dimensions of reciprocal angle.

When I look at scalars – continuous one-dimensional variables –, I see 6 natural situations that your system of variable dimensions should acknowledge. (I’ve never seen this picture anyplace before, but perhaps somebody knows a reference.)

Dimensional analysis only applies to these first two.

- things on a half-line with a zero and addition (mass,charge)
- things on a line with a zero and addition (displacements/translations)

These next four do not have the additivity necessary for dimensional analysis.

- things between two extremes (probability, black-to-white)
- things that define a position on a line and have ordering, but no natural addition operation (dates,coordinates)
- things that define a position on a circle and have betweenness but no addition (compass bearings)
- things turns about a circle, with a zero and addtion (angles)

I’m joining a certain conversation late, here, I now see. But I seem to be in general agreement with Angles in the SI: a detailed proposal for solving the problem by Quincey, Paul. Metrologia, 2021, v. 58 (5), pp. 053002. DOI:10.1088/1681-7575/ac023f