A question in this month’s Mathematics Teacher asks about the range of \(\sin(\sin(x))\). My initial concern about this was over the units of the input and output of the sine function. I’ll summarize those briefly, but this post is about the resolution of those concerns by clarifying what a “degree” is in the first place.

By its historical definition, the sine of an angle is the ratio of two lengths: The length of the side opposite the angle in a right triangle divided by the length of the hypotenuse. Let’s say that a right triangle has legs of 3 meters, 4 meters, and 5 meters. Then the sine of the angle opposite the 3 is 3 m / 5 m, or 0.6. The angle with this sine is about 48.6º, or 0.85 rad. My complaint was that, if sine is a function that takes something in units (an angle) and returns something in a unitless value (a ratio), then \(\sin(\sin(x))\) ought to be malformed: The inner sine would output a unitless ratio, not an angle (which has a unit).

This led to a discussion on my G+ account. The details of that suggested that radians are unitless, being defined as the ratio of circumference to radius. However, that led me to a new concern: If radians are unitless, what are degrees? Degrees are not (traditionally) defined in terms of a ratio, but rather in terms of the amount of rotation from one ray to another. If radians are one sort of thing (unitless ratio) and degrees are another sort of thing (a unit of rotation), then how can we declare that \(2\pi rad = 360^\circ\)? Those are two types of things.

### My Resolution

I’ll preface this by saying what follows is my way of seeing things. I think it’s rather elegant, but others might well think it’s strange. Regardless, it satisfies my frustration above.

I’ll also quote Jordan Ellenberg here: “Mathematics is the study of things that come out a certain way because there is no other way they could possibly be” (*How Not to Be Wrong, *p. 12). For me, what follows is the only way it could possibly be.

Plane Euclidean Geometry consists of several different primitives which build on each other. Above the point, the next tier of objects are lines and portions thereof and angles. Above that are polygons and such and then solids. Lines, polygons, and solids have related measurements: Lines have length, polygons have area, and solids have volume.

This in itself is a cause of confusion for high school students. Let’s say we have a square with side length 1. Then its perimeter is 4 and its area is 2. Students naturally expect the area to be more than the perimeter, but the issue here is that perimeters and areas are measured in different units. If we make the units explicit, that becomes clear: A square with a side length of 1 meter has a perimeter of 4 meters and an area of 2 square meters. A square meter is much larger (in terms of infinities of points) than a meter. Since we often ignore units, though, this distinction is lost.

Furthermore, when we calculate area, we have to be consistent with the units of length we’re using (at least, best practice says so). The area of a rectangle that is 3 feet by 6 inches is not 18 foot-inches, even though we can say that. We don’t generally consider a “foot-inch” to be a valid unit, because it doesn’t make any sense.

So for the purposes of making the implied overt, let us assume that all measurements in geometry that do not have an explicit measure have an implicit one. Length is measured as \(L^1\); area, \(L^2\); and volume, \(L^3\). \(L\) can be any valid measurement (including inches, feet, yards, miles, meters, and so on, but also including half-inches any anything else we might devise).

Now, even though the units are parallel in name, they’re not completely parallel in what they represent. \(L^1\) is the distance between two points; \(L^2\) is the space taken up by an object; \(L^3\) is the volume of a subspace. Even so, we can define each in terms of their infinity of points… an area takes up more “points” than a line segment with the same numeric dimension, and a volume takes up more “points” than a plane shape.

A rectangle with sides \(3L\) and \(4L\) has an area of \(12L^2\). A prismic subspace with a volume of \(35L^3\) might result from a face with an area of \(7L^2\) and an extrusion spline of \(5L\). This approach allows us to reach into higher dimensions: 4D may not exist in the real world (unless we want to talk about time or something quantum), but mathematics is fine with it.

### What if…?

I’d made the assumption that \(L \div L = 1\), so ratios of lengths are unitless. This is based on the mathematical truth that \(x \div x = 1\) (except when \(x = 0\)). At the same time, though, \(x \div x = x^0\). What is \(L^0\)?

By the “point” definition of \(L\), \(L^0\) is the space taken up by a group of points represented by a single point. Of course, in “real space”, points don’t take up any space at all, whatever that means. I’ve already mentioned above, though that the practical definitions of \(L^1\), \(L^2\), and \(L^3\) differ from their theoretical “how many points is this?” definition. What if the same is true for \(L^0\)?

What if the unit category \(L^0\) represents the amount of displacement that occurs around a single point, in the same way that \(L^1\) represents the amount of displacement that occurs between points? Then \(L^0\) would represent angular rotation, because that’s the only sort of meaningful displacement that could possibly occur.

This is in fact related to how I teach angle measurements in the first place: If I’m standing at a specific spot and looking at a specific place, how much do I have to turn in order to look at a second specific place? I’m not allowed to relocate, I’m only allowed turn my body on a single point.

So that’s my current perspective: Radians and degrees are units of displacement on a point. Because there is no length involved in that displacement, their unit category is \(L^0\). That’s not the same thing as being unitless.

### Sidebar on Multiplication

Let’s look at the arithmetic senses of unitlessness. There are two major ways that multiplication is used prior to Calculus or so:

- To find the aggregate count of
*n*groups of*m*things. - To find the area of a shape with dimensions related to
*m*\(L^1\) and*n*\(L^1\).

The second one I have discussed above: The multiplication results in an object with a different unit than the input.

In the first case, the multiplication results in an object with the same unit as the input. Five groups of four students yields twenty students. Eight groups of six books yields forty-eight books. For the students and the books, we can generalize to an implied unit \(T\) which is “things”. But what of the groups?

Two choices immediately come to mind: Create an implied unit \(G\) which has the rules \(TG = T\) and \(T/G = T\), or say that groups are unitless. Positing a unit that does nothing seems less elegant than simply arguing that groups have no units. This latter approach also allows for statements like: \(3 \times 5 = 3 + 3 + 3 + 3 + 3\), although it does apparently lead to Internet arguments about \(3 \times 5 = 3 + 3 + 3 + 3 + 3 = 5 + 5 + 5\) (since “times” in the realm of positive integers and elementary mathematics is a shortcut for addition… but, in turn, these arguments are poorly founded, since \(5 \times 3 \times T = 3 \times 5 \times T\) regardless).

Anyway, “groups” is, in my mind, an example of a “unitless” quantity. We don’t arbitrarily break any random aggregate into some constant number of groups. But we *do* break circular rotations up into some constant number of wedges; indeed, there are two constants we use, depending on the level of the students and the audacity of the teacher.

### Conclusion

Practically speaking, \(L^0 = 1\). Once we’ve taken this long first step, it’s not a huge leap to declare that degree and radian markers are only needed to disambiguate which of the two common wedgifications of the circle we’re taking. We don’t need to mark it at all. And of course things like Euler’s Formula equate trigonometric function outputs with non-trigonometric function outputs, although perhaps that’s coming from the further abstraction of \(\{T, L^0, \emptyset\}\) to a single “doesn’t matter” unit.

Regardless, I think acknowledging \(L^0\) as being philosophically different from \(T\) is a useful thing to do. The former is for amounts of rotation, the latter for counts of objects, even if they can be used effectively interchangeably in many contexts.

Credit: Significant aspects of these thoughts were inspired on G+ by Carlos Castillo-Garsow and Christopher Rasmussen.