Why I think radians should *not* be dimensionless

[Gmen]

As most know and rest can find out, radians are a dimensionless unit. Actually, it's even worse: angle in general is a dimensionless quantity. That stems from it being defined as a ratio of the arc versus the radius. I now present my opinion of disagreement to this concept.

In the strict scientific core of things, I believe no opinion is right or wrong. Units of measurement are there to help us. What does really make them dimensionless or not? Common examples like the aforementioned angle definition, or another classic I like: the efficiency of something, may at times seem "right" to define as dimensionless, their definition including ratios of same units and all. But is it all really not just our choice of convenience?

When doing nuclear physics homework at university, very frequently we assumed c=1 and dimensionless. The results weren't wrong. Obviously some units needed appropriate adjustments afterwards...to return to SI. But it's not like we hadn't already fully understood (after a bit getting used to) what the results said.

How about the second (s). Imagine a place full of strange sentient robots with processors that run at 1Hz. Their perception of life may be so linked with this unit, the second, that they might just as well not define any unit for time...it's just 1. Everyone understands this and the full laws of physics having it dimensionless.If until this point we have an agreement that dimensions are actually subjective, I can proceed to my humble personal view: it sits better with me that angle is not dimensionless. Rads, degrees...makes perfect sense. Notice how, even while saying angle is dimensionless, we have units for it? Isn't it a little strange? Or even suspicious? I can hardly believe, when talking about angle, that one TRULY thinks of it as a ratio. I think of it as..a wideness thing, an openess thing, a cheese pie piece thing. Contrast this to efficiency: J/J? W/W? Sounds messy. 100%? Yeah baby! 5%? Aw come on! Interestingly enough, although efficiency is a "quantity", there are no "units" of it around.

So what's everybody's view on the subject?

 

[anorlunda]

https://www.physicsforums.com/threads/can-angles-be-assigned-a-dimension-comments.893108/

Big recent discussion on that in that thread.

 

[Dale]

If until this point we have an agreement that dimensions are actually subjective, I can proceed to my humble personal view: it sits better with me that angle is not dimensionless

I wouldn't use the word "subjective". I would say "conventional". The dimensionality of a quantity is a matter of convention for a given system of units.

The BIPM (the organization in charge of defining the SI conventions) had decided that angles are dimensionless. You can certainly choose a different convention, but it isn't SI.

Personally, when I am doing dimensional analysis I often consider angles to have a dimension. But not when I am writing a paper or communicating results, then I use SI units.

 

[A.T.]

I can hardly believe, when talking about angle, that one TRULY thinks of it as a ratio.

This might be partly because the use of pi in radian angles. If we would express angles as the fraction of a full circle (rather than half a circle), the bare numeric values would be more intuitive.

 

[Jamison Lahman]

If I am not mistaken, angles do have dimension, particularly in non-Cartesian coordinate systems.

Units of measurement are there to help us. What does really make them dimensionless or not?

What makes them dimensionless is the the fact they can't be described by base units; angles are ratios of the base units of length. That is to say, any fundamental quantity can be expressed without them. For example, luminosity is the amount of energy released per time and it does not concern itself with where the energy is being released. However, we may observe the radiant intensity which includes units of solid angle^-1. We use this observation for convenience which can then be used to express the luminosity of the object. Treating them as a dimension can make observations or calculations possible (that is why we give them units), but they are not fundamental.

When doing nuclear physics homework at university, very frequently we assumed c=1 and dimensionless. The results weren't wrong. Obviously some units needed appropriate adjustments afterwards...to return to SI.

c is a constant and can be treated dimensionless however it is not unit-less. It will still have units of m/s in SI convention. Per wikipedia on Gaussian units: "As another example, quantities that are dimensionless (loosely "unitless") in one system may have dimension in another."

How about the second (s). Imagine a place full of strange sentient robots with processors that run at 1Hz. Their perception of life may be so linked with this unit, the second, that they might just as well not define any unit for time...it's just 1. Everyone understands this and the full laws of physics having it dimensionless.

I would imagine these robots would have a hard time grasping special relativity. Instead of seconds, I imagine they would invent some unit such as "per computation" so that they could measure velocities accurately.

Isn't it a little strange?

After my introductory quantum class, everything seems a little strange.

 

[zinq]

There is a very good reason for the units assigned to an angle. An angle is most simply defined as the number of revolutions. That is measured by the ratio of a) the arclength along a circle (according to the number of revolutions, which is often not an integer), and b) the arclength, along the same circle, of one revolution.

By dividing two quantities having length as their dimension, one gets a so-called "dimensionless" quantity of angle. But that is definitely not because dimensions have been ignored when taking account of an angle, but because they have been taken into consideration.

Perhaps for maximum clarity, we ought to be saying that the units of an angle are length⁰.

 

[Stephen Tashi]

it sits better with me that angle is not dimensionless.

Are you using the word "angle" to refer to a particular type of physical phenomena? - or are you using it to refer to a property of a particular type of physical phenomena ?

A big semantic confusion in such discussions is whether "angle" is used to denote a physical phenomena or whether it is used to denote a property.

Consider the similar example of the term "mass". It's common to read problems in textbooks that say things like "A mass of 2 kg is resting on an inclined plane...". But the property "2 kg mass" can't rest on an inclined plane. A teapot or a circular saw or some other physical object can rest on an inclined plane and have the property that its mass is 2 kg.

The semantics is further complicated by the fact that different properties of a physical situation can be measured in the same units. (For example, 3 Newton meters , could quantify the property of "work" or the property of "torque". Likewise, a property measured by meters/sec isn't necessarily the velocity of something, etc. ) As @zinq indicates, the fact that a unit of measure is dimensionless doesn't imply that all dimensionless units refer to the same physical property.

If we wish to consider the question of whether an "angle" is dimensionless, we should clarify whether "angle" refers to a physical situation ( e.g. a ladder leaning against a wall, a rotating wheel , etc.) or whether "angle" refers to a particular property of a physical situation (i.e. a "dimension" in the sense of dimensional analysis).

For example, in the case of a rotating wheel, we might be interested in the property of "the angle a radius sweeps out in in 2 seconds" and that angle might be, say, 780 deg, which we would consider a different physical result than 60 degrees. But in the case of a ladder leaning against the wall, in measuring the property we call "the angle the ladder makes with the wall", we would consider 780 deg the same result as 60 deg.

 

[zinq]

I agree with Stephen Tashi that there are basically two types of angles to consider: a) the angle between two rays in a plane having a common origin, and b) the total angle swept out by a ray in a plane rotating about its origin, perhaps over a specified period of time. (Or a situation similar to b), like how many times a wire has been wrapped about a cylinder.)

Cases a) and b) are certainly distinct geometrical concepts. But in my opinion they are not philosophically distinct kinds of things. (So I would not call one "referring to a physical situation" and the other, by contrast, a "property".) Possibly good terminology would be a) angle and b) total angle. But it's hard to change long-established ambiguity!

Either one of these cases a) and b) would nevertheless be measured in terms that describe the number of revolutions that is relevant. (Whether the units be radians, degrees, revolutions, or something else.) In the case a), this number will always be at least 0 but less than 1. In the case b), this number could be any real number (assuming the plane in question has been given an orientation).

 

[ImNotAnAlien]

I know this is an older thread, but I have a thought regarding this.
At every point, the circumference of a circle is perpendicular to the radius vector between the point on the circumference and the center of the circle. When you say an angle is the ratio between a given arc segment of the circumference and the radius, you're dividing two perpendicular things. While the units of length do cancel out, they're not in the same direction, and you could interpret that as them not being the same units of length. There's still that information about the perpendicular nature of the two lengths that you need to account for. One very useful application of this is dimensional analysis of torque or moment about a point.

Torque (According to my physics 1 textbook, "Sears&Zemansky's University physics with modern physics" ch 10 section 1) "-is not work or energy, and should be measured in Newton-meters, not Joules". It's confusing because normally N m is the definition of a joule. The difference in this case is that your force, starting at a given point, goes through a displacement along the torque arm which is perpendicular to your force. According to that same textbook, work is given by a force only when the displacement is either parallel or anti-parallel to the direction of force. This is why centripetal force does no work. When you multiply torque by an angle of rotation, the unit of length of the torque arm is then multiplied by a radian which gives you a length of arc which is parallel to the force at every point in that rotation (The torque arm is the radius of that arc). Now that you are multiplying the force by a length of displacement which is parallel to it at every point along the motion, you have done work. This suggests that the units for torque should be J/rad.

J/rad is a suggested unit of torque in SI literature

"For example, the quantity torque may be thought of as the cross product of force and distance, suggesting the unit
Newton metre, or it may be thought of as energy per angle, suggesting the unit joule
per radian" (Si brochure 8th edition 2014)

From the two sources ( 1. My textbook, and 2. the SI literature) we have

1. Units_of( torque )=/= J
2. Units_of( torque )=J/rad

From equation 2, we can say that

J/rad=Units_of( torque ) =/= J
J/rad =/= J
1/rad =/=1
rad =/= 1

Radians cannot be dimensionless, or at the very least can't be unity.

Personally I like to think of radians as being the complex number "i" just with a different symbol, but that's really up to interpretation. It does make eulers identity look cool, e^(pi rad)=-1
That's different from e^pi because of the rad unit.

 

[PeroK]

I know this is an older thread, but I have a thought regarding this.
At every point, the circumference of a circle is perpendicular to the radius vector between the point on the circumference and the center of the circle. When you say an angle is the ratio between a given arc segment of the circumference and the radius, you're dividing two perpendicular things. While the units of length do cancel out, they're not in the same direction, and you could interpret that as them not being the same units of length. There's still that information about the perpendicular nature of the two lengths that you need to account for. One very useful application of this is dimensional analysis of torque or moment about a point.

Torque (According to my physics 1 textbook, "Sears&Zemansky's University physics with modern physics" ch 10 section 1) "-is not work or energy, and should be measured in Newton-meters, not Joules". It's confusing because normally N m is the definition of a joule. The difference in this case is that your force, starting at a given point, goes through a displacement along the torque arm which is perpendicular to your force. According to that same textbook, work is given by a force only when the displacement is either parallel or anti-parallel to the direction of force. This is why centripetal force does no work. When you multiply torque by an angle of rotation, the unit of length of the torque arm is then multiplied by a radian which gives you a length of arc which is parallel to the force at every point in that rotation (The torque arm is the radius of that arc). Now that you are multiplying the force by a length of displacement which is parallel to it at every point along the motion, you have done work. This suggests that the units for torque should be J/rad.

J/rad is a suggested unit of torque in SI literature

"For example, the quantity torque may be thought of as the cross product of force and distance, suggesting the unit
Newton metre, or it may be thought of as energy per angle, suggesting the unit joule
per radian" (Si brochure 8th edition 2014)

From the two sources ( 1. My textbook, and 2. the SI literature) we have

1. Units_of( torque )=/= J
2. Units_of( torque )=J/rad

From equation 2, we can say that

J/rad=Units_of( torque ) =/= J
J/rad =/= J
1/rad =/=1
rad =/= 1

Radians cannot be dimensionless, or at the very least can't be unity.

Personally I like to think of radians as being the complex number "i" just with a different symbol, but that's really up to interpretation. It does make eulers identity look cool, e^(pi rad)=-1
That's different from e^pi because of the rad unit.

So, what you are saying is that if $\theta$ is an angle expressed in radians, then $\cos \theta$ should really be $\cos \theta i$?

 

[ImNotAnAlien]

So, what you are saying is that if $\theta$ is an angle expressed in radians, then $\cos \theta$ should really be $\cos \theta i$?

I take it you're asking this because of 3 main things:

1. Peoples sometimes include radians when they write angles inside of cos e.g. $\cos( \pi/3 \ rad )$ but sometimes it's omitted because rad is conventionally considered to be unitless
2. I've just redefined rad to be i instead of unitless
3. $\cos(\theta \ i)$ is considered to be the hyperbolic trig function and eulers equation uses $cos( \theta )$ but it depends on $\theta$ being just a unitless real number

And you're wondering how 2 and 3 don't contradict each other when you start including the rad unit instead of omitting it from the angle inside of cos()?

To avoid confusion, I'll use $cos(\theta )$ to mean the traditional function that everyone is familiar with. $\theta$ in this case would just be some real number, and when you plot it you get the familiar oscillating cosine wave.

Imagine some other function $cos$~$(\theta )$ Where "$\theta$" is of the form "$x \ rad$" (x is a real amount though)

An example of this function is
$cos$~$(\pi \ rad )=-1$
where in this case I've just plugged in $\pi \ rad$ for theta, and as I've suggested before rad is a complex unit. I can actually define $cos$~$(\theta )$ in terms of the traditional cos()

$cos$~$(\theta )$=$cos(\theta / rad )$

This works because when you plug in a value for theta containing rad as a unit e.g. $\theta = x \ rad$, that rad cancels out with the denominator inside of the cos() function which leaves you taking the cosine of the real number x.

You can also define $cos$~$(\theta )$ using exponentials in a manner similar to the traditional cos except with radians being included

$\cos$~$( x \ rad ) = (e^{ x \ i} + e^{- x \ i}) / 2$
And x is a real number

Funny thing is, since I've got radians, and I've defined them as "i", you can substitute

i=rad
$\cos$~$( x i ) = (e^{ x \ i} + e^{- x \ i}) / 2$

substitute $x \ i=\phi$
$\cos$~$( \phi ) = (e^{ \phi} + e^{- \phi }) / 2$

And in case this doesn't stand out to you
$(e^{ \phi} + e^{- \phi }) / 2=\cosh ( \phi )$
where $\cosh ( \phi )$ is the hyperbolic cosine

which means
$\cos$~$( \phi )= \cosh ( \phi )$Don't misunderstand though. I'm not saying the trig and hyperbolic functions are somehow flipped and therefore radians have dimensions.
That's basically as pointless as getting mad at Benjamin Franklin for getting charges and current direction flipped.

I am saying from my previous post that I can't think of a way to make dimensional analysis consistent for torque and energy unless you make radians be dimensional. Since angles are the ratio of two perpendicular things it seems natural to say that rad=i. From that, it seems the definition of $\cos$~$( \phi )$ is just swapping trig and hyperbolic functions and the signs* of the input so that you don't have to omit the radians unit.

* for $\sin$~$( \phi )=sin(\phi / rad )$ you have to recognize that $1/rad = 1/i = -i$ and really
$\sin$~$( \phi )=i \sinh(- \phi )=- i \sinh(\phi)$
but you don't notice that for the cos~(x) example because cos(x)=cos(-x)