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So my friend asked me why angles are dimensionless, to which I replied that it's because they can be expressed as the ratio of two quantities -- lengths.

Ok so far, so good.

Then came the question: "In that sense even length is a ratio. Of length of given thing by length of 1 metre. So are lengths dimensionless?".

This confused me a bit, I didn't really have a good answer to give to that. His argument certainly seems to be valid, although I'm pretty sure I'm missing something crucial here.

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    $\begingroup$ Well, yuo can define angles on circles of any radius, so what is this "reference radius" to be used as a unit? In the case of length, I can show you a sample and it will work for any lenght. Can you do the same with the radius used to compute an angle of an arc? Any arc? $\endgroup$ – Peltio Jul 12 '15 at 16:53
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    $\begingroup$ The comment above was to show the asymmetry between the two situations. The compelling reason angles have to be dimensionless lies in the power series argument (see comment below). $\endgroup$ – Peltio Jul 12 '15 at 16:58
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    $\begingroup$ @ignis 90deg is also dimensionless. $\endgroup$ – Rishav Jul 12 '15 at 20:08
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    $\begingroup$ If you can see it, hear it, taste it, feel it, or in any way measure it, it has dimension and you can assign to it some name of units under its class of dimension. Angle has dimension and units. $\endgroup$ – docscience Jul 13 '15 at 3:55
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    $\begingroup$ Your friend's question about length being a ratio suggests that he's thinking not about length but about its expression in some chosen units; that expression is indeed dimensionless. For example, my height is 1.98 meters; it has a dimension, namely length. But the number of meters in my height is the pure number 1.98, the ratio of my height to the length of the meter stick. And the number of inches in my height is another pure number, 78, obtained as another ratio of lengths. In summary, after you take a ratio, as your friend suggests, you get a pure number, but that number is not a length. $\endgroup$ – Andreas Blass Jul 13 '15 at 4:31

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Your friend's question is perceptive but not at odds with your earlier answer.

When you compare the length of something with a unit (1 meter), the ratio is indeed a unitless number.

But then all numbers (1.5, $\pi$, 42) are unitless. When you want to determine speed you divide displacement by time - each of which has units. But what you enter into you calculator are just the numbers - you handle the units separately.

"The runner covered 100 meter in 10 seconds. What was his average speed?" Is solved by calculating the numerical ratio 100/10 and adding the dimensional ratio m/s to preserve the units. Most calculators don't have (or need) a means to enter units (some sophisticated computer programs do - to help you avoid mistakes by mixing units).

For some physical calculations you need to take the logarithm - when you do, you ALWAYS have to divide the quantity by some scale factor with the same units as it is not possible to take the $\log$ of a unit.

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    $\begingroup$ There is no mathematical definition of the logarithm of a second. You can write it but it is meaningless. $\endgroup$ – Floris Jul 12 '15 at 16:28
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    $\begingroup$ In general functions require dimensionless arguments because if you substitute them with their power series expansion, you'll end up mixing apples with oranges (if oranges are apples to some power). $\endgroup$ – Peltio Jul 12 '15 at 16:34
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    $\begingroup$ @RishavKundu that question may be interesting to you but to me it is meaningless. Units are not numbers. Mathematics deals with numbers. Your question cannot be answered AFAIK. $\endgroup$ – Floris Jul 12 '15 at 16:48
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    $\begingroup$ For more on taking the log of a "unit"-full value, see this Physics.SE post as well as this one. $\endgroup$ – Kyle Kanos Jul 12 '15 at 17:42
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    $\begingroup$ @BlueRaja-DannyPflughoeft - note that the dB is defined relative to a standard sound pressure (or energy depending on which definition you use) and therefore mathematically the log itself is once again taken on a dimensionless number. As Kyle's link points out, sometimes physicists are "lazy" but convenient and meaningful are not the same thing. $\endgroup$ – Floris Jul 12 '15 at 18:41
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$\newcommand{\t}{[\text{time}]}\newcommand{\e}{[\text{energy}]}\newcommand{\a}{[\text{angle}]}\newcommand{\l}{[\text{length}]}\newcommand{\d}[1]{\;\mathrm{d} #1}$Dimensions vs Units:

I want to take an educational guess as to why angles are considered to be dimensionless whilst doing an dimensional analysis. Before doing that you should note that the angles have units. They are just dimensionless. The definition of the unit of measurement is as follows:

A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same physical quantity.

There are in fact a lot of units for measuring angles such as radians, angles, minute of arc, second of arc etc. You can take a look at this wikipedia page for more information about units of angles.

The dimension of an object is an abstract quantity and it is independent of how you measure this quantity. For example the units of force is Newton, which is simply $kg \cdot m/s^2$. However the dimensions of force is

$$[F] = [\text{mass}] \frac{ [\text{length}]} {\t^2}$$

sometimes denoted as

$$[F] = [M] \frac{[X]}{[T]^2}$$

but I'll stick to the first convention. The difference between units and dimensions is basically that the dimensions of a quantity is unique and define what that quantity is. However the units of the same quantity may be different eg. the units of force may perfectly be $ounce \cdot inch / ms^2$.


Angles as Dimensionless Quantities

As to why we like to consider angles as dimensionless quantities, I'd give to examples and consider the consequences of angles having dimensions:

As you know the angular frequency is given by

$$\omega = \frac{2 \pi} T \;,$$

where $T$ is the period of the oscillation. Let's make a dimensional analysis, as if angles had dimensions. I'll denote the dimension of a quantity with square brackets $[\cdot]$ as I did above.

$$[\omega] \overset{\text{by definition}}{=} \frac{[\text{angle}]}{[\text {time}]}$$

However using the formula above we have

$$[\omega] = \frac{[2\pi]}{[T]} = \frac{1}{[\text{time}]} \; , \tag 1$$

since a constant is considered to be dimensionless I discarded the $2\pi$ factor.

This is a somewhat inconvenience in the notion of dimensional analysis. On the one hand we have $[\text{angle}]/\t$, on the other hand we have only $1/\t$. You can say that the $2\pi$ represents the dimensions of angle so what I did in the equation (1) i.e. discarding the constant $2\pi$ as a dimensionless number is simply wrong. However the story doesn't end here. There are some factors of $2\pi$ that show up too much in equations, that we define a new constant e.g. the reduced Plank's constant, defined by

$$\hbar \equiv \frac{h}{2\pi} \; ,$$

where $h$ is the Plank's constant. The Plank's constant has dimensions $\text{energy} \cdot \t$. Now if you says that $2\pi$ has dimensions of angles, then this would also indicate that the reduced Plank's constant has units of $\e \cdot \t / \a$, which is close to nonsense since it is only a matter of convenience that we write $\hbar$ instead of $h/2\pi$, not because it has something to do with angles as it was the case with angular frequency.

To sum up:

  • Dimensions and units are not the same. Dimensions are unique and tell you what that quantity is, whereas units tell you how you have measured that particular quantity.

  • If the angle had dimensions, then we would have to assign a number, which has neither a unit nor a dimension, a dimension, which is not what we would like to do because it can lead to misunderstandings as it was in the case of $\hbar$.


Edit after Comments/Discussion in Chat with Rex

If you didn't buy the above approach or find it a little bit circular, here is a better approach: Angles are nasty quantities and they don't play as nice as we want. We always plug in an angle into a trigonometric function such as sine or cosine. Let's see what happens if the angles had dimensions. Take the sine function as an example and approximate it by the Taylor series:

$$\sin(x) \approx x + \frac {x^3} 6$$

Now we have said that $x$ has dimensions of angles, so that leaves us with

$$[\sin(x)] \approx \a + \frac{\a^3} 6$$

Note that we have to add $\a$ with $\a^3$, which doesn't make any physical sense. It would be like adding $\t$ with $\e$. Since there is no way around this problem, we like to declare $\sin(x)$ as being dimensionless, which forces us to make an angle dimensionless.

Another example to a similar problem comes from polar coordinates. As you may know the line element in polar coordinates is given by:

$$\d s^2 = \d r^2+ r^2 \d \theta^2$$

A mathematician has no problem with this equation because s/he doesn't care about dimensions, however a physicist, who cares deeply about dimensions can't sleep at night if s/he wants angles to have dimensions because as you can easily verify the dimensional analysis breaks down.

$$[\d s^2]= \l^2 = [\d r^2] + [r^2] [\d\theta^2] = \l^2 + \l^2 \cdot \a^2$$

You have to add $\l^2$ with $\l^2 \cdot \a^2$ and set it equal to $\l^2$, which you don't do in physics. It is like adding tomatoes and potatoes. More on why you shouldn't add too different units do read this question and answers given to it.

Upshot: We choose to say that angles have no dimensions because otherwise they cause us too much headache, whilst making a dimensional analysis.

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  • $\begingroup$ In equation 1, you didn't discard the $2\pi$ so much as divided by $2\pi$, which gives you a dimensional value. For Plank's constant, it is defined using the frequency of light; so yes, it can be considered as [energy]⋅[time]/[angle]; where the angle is considered the portion of the wavelength. $\endgroup$ – LDC3 Jul 12 '15 at 16:16
  • $\begingroup$ @LDC3: That would however assign a number a dimension, which is not a very good thing to do. Then you would also have to say that 360 is a dimenional value since it also describes angle. $\endgroup$ – Gonenc Jul 12 '15 at 16:33
  • $\begingroup$ That you did not bother to write "radians" after $2 \pi$ is not relevant to whether the number is dimensionless or not. But you could also write $\omega = {{360 \,\textrm{degrees}} \over T}$, couldn't you? $\endgroup$ – Rex Kerr Jul 12 '15 at 20:36
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    $\begingroup$ @gonenc - You're assuming the conclusion by writing $\omega = {{2 \pi} \over T}$ without any units on $2 \pi$. $\endgroup$ – Rex Kerr Jul 12 '15 at 20:51
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    $\begingroup$ Because $2\pi$ is a number and doesn't have any units! Following your logic every number has units of angle, because I can divide the circle to $n\in \mathbb R$ equal parts, which is again nonsense that all numbers have units of angles. $\endgroup$ – Gonenc Jul 12 '15 at 20:54
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It's possible to express anything as a dimensionless number. Vitruvius, an ancient author who wrote a surviving book on Roman architecture, reveals that the ancient Romans made their hydrostatic and architectural calculations based on rational fractions, which are ratios of one quantity with another.

By convention, and because working with all quantities as ratios would prove cumbersome and difficult, physical quantities such as length, time, velocity, momentum, electrical current, pressure, etc. are expressed in agreed-upon units.

Another reason for expressing physical quantities as agreed-upon units is that dimensional analysis might not be possible if all physical quantities were expressed as ratios. So, working with units instead of ratios provides another tool to check and validate physical equations, which must have the same dimensions on left and right sides.

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  • $\begingroup$ This is quite interesting. $\endgroup$ – Rishav Jul 12 '15 at 14:39
  • $\begingroup$ In grad school, all of my fluid dynamics courses non-dimensionalized the problem before any calculations were done. $\endgroup$ – Rick Oct 20 '15 at 16:06
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An article published in Control Systems Magazine by Bernstein, et. al., Dec 2007, and one which focuses on the algebraic structure of dimensional quantities argues that angle should not necessarily be considered a dimensionless quantity, but rather a dimensionless unit - accounted for throughout a calculation. The article extends unit analysis to matrices, linear and state space systems.

Furthermore, and contrary to what Floris says, there is no reason why you can't have nonlinear functions of units (e.g. log of radians) in getting from point A to point B. (But you do have to watch out for singularities!) You just need to make sure when you get to point C, where C is an observable quantity, that the units have transformed into ones that are physically meaningful. Algebraic consistency is what ultimately matters.

And in the article's conclusion "Physical dimensions are the link between mathematical models and the real world". I've found that many (at least among engineers) don't give enough attention to that fact.

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  • $\begingroup$ Yes, but at point B, the nonlinear function doesn't make much sense, does it? $\endgroup$ – Rishav Jul 12 '15 at 17:40
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    $\begingroup$ @RishavKundu It doesn't need to make sense at point B. Just as the units of states in a state space system don't need to make sense. It's the output coupling matrix that makes sense out of the states as physical outputs. Physical models expressed in state space can have many different internal realizations. $\endgroup$ – docscience Jul 12 '15 at 21:56
  • $\begingroup$ "but rather a dimensionless unit - accounted for throughout a calculation." I.e. keep writing $\mathrm{rad}$ whenever you write an angle. This is what I tell my students in intro courses to do just to help them keep things straight, and because it delivers a consistent message on units throughout that first course. $\endgroup$ – dmckee Jul 13 '15 at 23:13
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You must check whether a quantity is scale invariant or, more generally, invariant with respect to a change of units. Angles have this property: they are defined as a ratio of lengths that both scale in proportion to the geometric figure. A uniform dilation of a circle, sphere or any other geometric figure (equivalent to multiplying our length units by a conversion factor from meters to feet of feet to Standard Snozfurgles) leaves the ratio of any two distances between any two pairs of points unchanged.

The same is not true of the ratio of a dimensioned length and the unit length. Represent the length as a line segment on a co-ordinate chart. Dilate the co-ordinate chart as above and watch it shrink / grow. Now, the unit length does not scale in the same way: it is defined in terms of a physical length: a unit measuring rod, a number of wavelengths and so forth. These natural things do not change with arbitrary dilations we choose to make on our co-ordinate systems.

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It's true that length can be expressed as a ratio with respect to 1 metre (or any other unit). But that unit itself, i.e. the idea of "1 metre" itself is not dimensionless. What I mean to say is that the unit "1 metre" itself can't be expressed as a ratio of similar quantities of same dimensions; whereas, "1 radian" can be expressed as the ratio of arc length of "1 metre" to radius of "1 metre".

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  • $\begingroup$ Please do see my answer on dimensions and units. They are two different things. $\endgroup$ – Gonenc Jul 12 '15 at 16:13
  • $\begingroup$ Sorry, I meant dimension. Made the change. Thanks. :) $\endgroup$ – Arkya Chatterjee Jul 12 '15 at 16:19
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"In that sense even length is a ratio. Of length of given thing by length of 1 metre. So are lengths dimensionless?"

No, if they where then where did the 1 meter come from, why wasn't it 1 feet, or 1 mile?

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Because length is relative, but angle is absolute.

(There is such a thing as a maximum angle against which you can compare, but not a maximum length.)

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  • $\begingroup$ Wrong you there is no maximum angle! Note that $4\pi$ is also an angle although it is equal to $0,2\pi,4\pi$ under some consequences. For example for the function $f(x)=\sin(x)/x$ $f(0) \neq f(2\pi)$. What I'm trying to say is that angles aren't limited to $[0,2\pi]$ $\endgroup$ – Gonenc Jul 13 '15 at 9:23
  • $\begingroup$ @gonenc: I know exactly what you're trying to say but I think you also know what I'm trying to say. $\endgroup$ – Mehrdad Jul 13 '15 at 9:25
  • $\begingroup$ No I don't. There is e.g. a minimum temperature that I can compare but still temperature isn't a dimensionless quantity. $\endgroup$ – Gonenc Jul 13 '15 at 9:26
  • $\begingroup$ @gonenc: Minimum ≠ Maximum $\endgroup$ – Mehrdad Jul 13 '15 at 9:30
  • $\begingroup$ Obviously but why does the maximum matter, when you only want to compare? $\endgroup$ – Gonenc Jul 13 '15 at 9:33
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When performing dimensional analysis, some terms will have physical quantities associated with them (e.g. time, charge, and length), some have quantities and directions (e.g. force, torque, and distance), and some have neither. A few such as slope and rotation have encapsulate directions but no physical quantities [slope is a ratio of movement in one direction to movement in another, and is only meaningful in the context of those directions]. In general, when applying to the real world terms where only the direction is meaningful, it will be necessary to combine a vector associated with the term with another vector in the real world, in a process which normalizes their lengths in relation to a unit vector.

For example, suppose a lake bed has a 75% downward slope in the north/south direction, and someone wishing to travel 4 meters north along the surface wishes to know how much deeper the water will be. Start by dividing the distance "4 meters north" by a unit vector in the north direction, yielding a length of 4 meters. Then multiply that length by the slope (0.75) and then by a unit vector in the downward direction, to yield a distance of 3 meters downward.

Note that it's possible to convert a distance into the sum of two or more other distances in different directions, but distances may only be added if they're going in the same direction. If someone travels 14.14 meters northeast and then 14.14 meters northwest, those translate into "10 meters north plus 10 meters east" and "10 meters north plus -10 meters east", yielding a sum of 20 meters north.

Rotational angles are a little tricky because applying rotation to a system will change all of the vectors therein, including those used to define rotation. Still, the same principles apply: the length of rotational vectors is significant in relation to the length of a unit vector. Since all unit vectors have the same length (it's defined as being precisely one), it's only necessary to define unit vectors in contexts where direction is important.

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Meter refers to something quite physical. Two people should be able to measure something called a "meter" and agree they are the same. NIST says:

The meter is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.

Angles come in units e.g. degree or radian. $1^\circ$ is $\tfrac{1}{360}$ of a full rotation of a circle. Hopefully we can all agree on what that means. Possibly not.

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In the SI system of units, angles are dimensionless. There is a system of units where the radian is a distinct base unit and the steradian is an actual derived unit. See https://www.ncbi.nlm.nih.gov/pmc/articles/PMC61354/.

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Angles carry no information about location in space-time.

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    $\begingroup$ This answer doesn't carry much information either. $\endgroup$ – Rishav Jul 12 '15 at 14:45
  • $\begingroup$ angles are indifferent to space or time metric thus dimensionless. $\endgroup$ – user12811 Jul 12 '15 at 16:11

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