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I think that the reason is because one revolution or one turn is equal to $2 \pi$ rad or to $360$ degrees.

We can relate rads and degrees to two units of length that cancel each other.

rad $= \frac{arc\: length}{radius\: of\: the \:arc\: length}$

degree $=$ arc length$ * \frac {1}{360}$ of the total circunference.

In both cases the meters from the numerator cancel with the meters from the denominator. This implies that rads and degrees are dimensionless, but not unitless.

Is there another explanation why a revolution is dimensionless?

Is there an analogous explanation, that meters with meters cancel each other, for revolutions?

Or you can only explain it equating revolutions with degrees or radians?

Morover, Tipler's Physics for scientists and engineers explains what a dimension is in this way.

enter image description here

I can measure the number of revolutions (for instance with a photoelectric sensor) of a turning plate. So I have a number with units(revolutions or rads). Don't we have a dimension in this case, the angle?

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  • $\begingroup$ @G.Smith I can measure the number of turns (for instance with a photoelectric sensor) of a turning plate. So I have a number with units(revolutions or rads). Don't we have a dimension in this case?. I added to the O.P. the explanation of dimension given by Tipler's Physics for scientists and engineers $\endgroup$ – roy212 Jul 30 at 3:21
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    $\begingroup$ I've deleted a comment that should have been posted as an answer. $\endgroup$ – rob Jul 30 at 3:29
  • $\begingroup$ @Qmechanic The question you link to does not seem to be a duplicate (despite the title of the post). That question asks whether two different dimensionless units (radians and steradians) are compatible for comparison/addition. This question asks for why angles specifically are dimensionless. $\endgroup$ – Mark H Jul 30 at 4:58
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    $\begingroup$ I dispute the premise of the question. The October Revolution clearly had units of time, and the Arab Spring was in newtons per metre. $\endgroup$ – David Richerby Jul 30 at 16:31
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Is there another explanation why a revolution is dimensionless?

In the end the radian is dimensionless because the BIPM (the organization which defines the SI) decided that it is dimensionless.

Here is the official definition of the SI, updated earlier this year: https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf

On p 136 it defines each of the base units to have its own unique dimension, and specifies that all derived units have dimensions corresponding to the base units used to derive the derived unit. Then on p 137 it defines the radian to be a derived unit of m/m, implying that the radian is dimensionless.

The dimensionality of a unit is just as much a matter of convention as its size. For instance, in SI the ampere is a fundamental unit with dimension of current, $I$ meaning that charge has dimensions of $IT$. In contrast, in cgs units the statcoulomb has dimensions of $L^{3/2}M^{1/2}T^{-1}$.

So, although the radian is defined by the BIPM as dimensionless, there would be nothing logically wrong with a non-SI unit of angle that was considered to have dimensions. It is entirely a matter of convention. However, note that if you change your units then you may also need to change some of your physics formulas.

You have specifically asked about “turns” or “revolutions” rather than radians. As far as I am aware there is no governing body defining a system of units in which the unit of angle is a turn or a revolution. Therefore, the dimensionality is entirely up to you. If you like you may consider a turn to have dimension, and if you like you may consider it to be dimensionless. There is nothing which physically or mathematically prohibits either convention.

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    $\begingroup$ Note that the question is about revolutions ("turns") rather than radians. The radian is dimensionless because it's a ratio; a number of turns is dimensionless because it's a thing that you count. The defining logic is in the same section of the SI brochure, but the two angular measures (radians and revolutions) are dimensionless for slightly different reasons. $\endgroup$ – rob Jul 30 at 3:48
  • $\begingroup$ Good point. I have added a paragraph specific to “turns” and “revolutions”. Note that there is no requirement that they be dimensionless. $\endgroup$ – Dale Jul 30 at 3:55
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    $\begingroup$ I agree with everything you say here, but I feel like it misses the point. The OP is asking why we make revolutions unitless. They are not asking if it's valid to give revolutions units. $\endgroup$ – Aaron Stevens Jul 30 at 4:36
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    $\begingroup$ @AaronStevens The title of the OP says "dimensions" not "units". Units and dimensions are two different things. Miles and millimeters are different units, but they both have the same dimension - length. Similarly for radians and turns of revolutions $\endgroup$ – alephzero Jul 30 at 11:10
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    $\begingroup$ @Dale Right, and I feel like that is what the OP might be getting at. What is the reasoning behind this subjective convention. Your answer is still good though. No worries. $\endgroup$ – Aaron Stevens Jul 30 at 11:14

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