Why are angles dimensionless and quantities such as length not?

$\newcommand{\t}{[\text{time}]}\newcommand{\e}{[\text{energy}]}\newcommand{\a}{[\text{angle}]}\newcommand{\l}{[\text{length}]}\newcommand{\d}[1]{\;\mathrm{d} #1}$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.

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.

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.