Why are radians not more natural than any other angle unit ?

Blogger Zendmailer 2012 deserves huge compliments for also thinking about a circle with circumference 1, that I baptised the Angular Circle. See also the figure with both the Angular Circle and the Unit Circle (radius 1) on page 36 of Conquest of the Plane (COTP, 2011).

Zendmailer ponders the question “Why are radians more natural than any other angle unit?” In his words:

“I’m convinced that radians are, at the very least, the most convenient unit for angles in mathematics and physics. In addition to this I suspect that they are the most fundamentally natural unit for angles. What I want to know is why this is so (or why not). (…) Why not define 1 Angle as a full turn, then measure angles as a fraction of this full turn (in a similar way to measuring velocities as a fraction of the speed of light (c = 1). Sure, you would have messy factors of 2π in calculus but what’s wrong with this mathematically? I think part of what I’m looking for is an explanation why the radius is the most important part of a circle.” (Physics Stack Exchange, August 6 2012)

The main thing wrong with this is that “angle” already has been defined, so that it cannot be taken as a unit of measurement. It would have been better when he had chosen 1 Turn as the unit. It is not really very wrong because if he had focussed on this longer he might well have corrected it. It is a pity that he uses 2π instead of Θ = 2π, the unit that I call “Archi” (after Archimedes). (Others want to use tau (τ) for this, see the American Scientist, but this looks too much like r for the radius.)

By chance, if that exists, I applied the Angular Circle recently on Euclid’s fifth postulate. Check the idea in action. It is great to see that more people come up with the same kind of questions and solutions.

It is also great to see that there is room for debate. Zendmailer is convinced that radians are most convenient but there is no need for this conviction. My suggestion is to keep both circles and see which is handier on occasion. For teaching, I would start with the Angular Circle, since it would seem to be easier to calculate in 1 than in Θ. This, of course, needs testing for evidence based education.

Zendmailer rightly refers to sine and cosine functions. If we use radians, the derivative of the sine is the cosine function, so that the slope of the sine at 0 equals 1. When we use dynamic division (I refer to COTP again) then we can write Sin[φ] // φ = 1 at φ = 0, for φ measured in radians, using the Unit Circle. I already knew this, but Bob Palais alerted me to the phenomenon that many graphs do not show the proper slope 1 at 0.

These points arise:

  1. Radians are often called dimensionless, since they arise from dividing arc by radius, thus length / length, but the arc is two-dimensional with the aspect of a turn, whence the dimension is Turn. (Addition July 29 2014: It occurred to me that this shift in focus might also be regarded in terms of the procept-theory of Gray & Tall: as an object we have length / length but as a process we have (length in one direction) / (length around). This may explain the difficulty for some people to “get it”.)
  2. Zendmailer uses a limit expression for Sin[φ] // φ = 1 at φ = 0 but skip the need for limits here.
  3. Zendmailer writes Sin[x] but sine and cosine represent y and x values of an angle φ.
  4. For α on the Angular Circle we can find x and y values on the Unit Circle via Xur[α] = Cos[Θ α] = Cos[φ] for φ = Θ α, and similarly Yur[α] = Sin[Θ α] = Sin[φ], where the “ur” means that those x and y values are relevant for the Unit Circle. See COTP for pictures.
  5. The derivatives (slopes) of Xur and Yur have a proportionality factor since these angles are measured on the Angular Circle and not on the Unit Circle. E.g. Yur'[α] = Sin'[Θ α] = Θ Cos[Θ α] = Θ Xur[α].
  6. Such a proportionality factor also exists for the sine of angles measured on 360 degrees. Try to figure out whether its slope at 0 is higher or lower than 1. Hint: your unit of measurement will be 1 degree.
Angular and Unit Circles

Angular Circle (c = 1) and Unit Circle (r = 1), Conquest of the Plane p36

While trigonometry is less cluttered in using Turns and Xur and Yur, for derivatives it becomes less cluttered from using radians. Note that you can still define what the unit of measurement is, e.g. 1 cm or 1 inch, so there is no real limitation on that choice. The only limitation is the issue of consistency, that once you choose, say 1 m, then the used sine and cosine show such and such slopes.

With this established, the reading of Zendmailer’s questions and the reactions should be easier.

Perhaps these critical comments are still useful:

  1. There is an expression “1 rad = 1”. My impression is that you should not write expressions like this, since this creates confusion. When the rad measurement is transformed from the circle arc to a straight axis in another space (where you plot the sine) then this best be indicated by a functional relationship. Subsequently, keep track of the “turn” in the rad: 1 rad ⇔ 1 / Θ Turn. I also propose Turn = Unit (Measure / Meter) Around = UMA to link up to standard measures.
  2. Note Philip Oakley: “The difficulty in point 2 is that the two lengths are in independent dimensions (as in 3d space). One has just cancelled Lx/Ly and lost information for one’s dimensional analysis (this is a Physics question;-). If one did the same with Charge/Temperature it would be a gross error, but we tolerate it for length. Dimensional analysis is newer than the cubit, so the old inconsistency remains. –  Philip Oakley May 11 ’13 at 20:52” and “Anybody working in optics definitely cares. There are many measurements that have Angle(radians) as an integral part of their value, and it is a very common error, not spotted by dimension checking, for the angle part to be omitted, double counted, or wrongly applied. –  Philip Oakley May 14 ’13 at 7:30” I would like to agree but don’t know optics. Also, my impression is that Lx/Ly would cancel as straight lines though this might be different in optics; but then the better format is 2D / 1D = 1D.
  3. There is also mention of Euler’s equation, but this can also be created for Xur and Yur, and thus doesn’t carry weight for the choice between the Angular Circle and the Unit Circle.

Overall, I find that there is no “natural” choice of either Angular Circle or Unit Circle as the “natural” unit of reference. The Angular Circle seems to be best to understand how an angle is measured, the Unit Circle might reduce the clutter for who works a lot with derivatives. Dimensions however tend to arise from the field of application. Having more bodies circling a Sun at various radii destroys all simplicity anyway, especially when those appear to be no circles at all.


This Wikipedia article has a wrong statement on dimensions (today, July 2014): “Although the radian is a unit of measure, it is a dimensionless quantity. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is equal to the ratio of the length of the enclosed arc to the length of the circle’s radius. Since the units of measurement cancel, this ratio is dimensionless.” The arc is in 2D space while the radius is in 1D, and 2D / D still leaves a dimension. The proper dimension is Turn. Use 1 Turn ⇔ Θ radians, so that 1 radian ⇔ 1 / Θ Turn ≈ 16% Turn. Turns are measured on the Angular Circle and radians on the Unit Circle. See the earlier weblog entry. I suppose that mathematicians enjoy taking the ratio arc / radius, and then create a bit of mystery, while engineers directly use the Unit Circle, with r = 1 in the standard unit of measurement (meter, foot), with the magic of being practical without the mystery.

Dimensional analysis generally concerns the units such as meters and seconds and dollars, e.g. see here or on wikipedia again. I have been using it to good effect since early university, in particular since F.J. de Jong at Rijksuniversiteit Groningen had increased awareness there. In this case we apply dimensional analysis to our 2D space. I met a mathematician who thought that I thought that dimensional analysis applied only to 1D, 2D, 3D, … and who started lecturing me, and it continues to amaze me how easy it is that misunderstandings arise.

Another possible misunderstanding is this. If you take a circumference of an object in 2D, say an equilateral triangle with sides 1 meter giving a circumference of 3 meter, and divide that circumference by a side, then it is conventionally (3 meter) / (1 meter) = 3 dimensionless, but rather be aware of (3 meter around) / (1 meter straight) = 3 around / straight. Again it is 2D / 1D = 1D. Just like the circle, you can make a turn going around that triangle. As it stands, it is little use to make an issue of this for circumferences in general, and the conventional view has its advantages. But for the circle it is useful to bring it to the fore in the definition of angle and turn. Indeed, here we need it to get the polar co-ordinates {radius, angle}, which uses that Turn is a separate dimension indeed.


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