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.
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 $ounce \cdot inch / ms^2$.
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.
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 $\hbar$.
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.
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.
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.
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".
"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?
(There is such a thing as a maximum angle against which you can compare, but not a maximum length.)
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.
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.
By speaking of angles and lengths, the question is asking about the dimensions of parameters we use in doing transformations such as rotations, boosts, and translations.
The sin(x) answer recognized that angle x must be expressed in the very special mathematical quantity of radians in order for the expansion of sin(x) or cos(x), found in trigonometry or rotation matrices, to make sense. The underlying reason for this is that rotations form a non-abelian Lie Group. If you make a small rotation of something (eg: a stick) about the x axis by $\theta_x$ and then about the y axis by $\theta_y$, the final stick will not point in the same direction as one first rotated about y and then x. The two final sticks are related by a rotation about the z axis by $\theta_z = \theta_x \theta_y$ where the $\theta$ must be expressed in radians for this product to be mathematically correct. Rotations do not commute. Non-commutativity makes rotation angles, measured in radians special and dimensionless. If we insist on using a dimensioned quantity such as d [degrees] to measure rotation angles, then we need a new fundamental constant $\kappa = \frac{180}{\pi}$ [degrees] to convert d angles to radians.
$$ \theta = \left(\frac{d}{\kappa}\right) \ \text{[radians]} $$
Similarly, non-commutativity of velocity boosts (Special Relativity and the Lorentz Group) causes boosts to be done in dimensionless radians. The Lorentz Boost parameter $\lambda$ [radians] replaces velocities $v$ [m/sec].
$$\lambda = \tanh^{-1} \left(\frac{v}{c}\right) \ \text{[radians]} $$
Like $\kappa$ for dimensioned rotation angles, the purpose of the fundamental velocity $c$ is to turn a dimensioned boost velocity into dimensionless radians. Because $c$ is the same everywhere, the meter or second could be removed from the Bureau of Standards, since the other can be obtained using the fundamental constant $c$. Let’s say the standard meter is still left in the Bureau.
This still leaves length being a dimensioned quantity, which means it is measured as multiples of the standard meter (an arbitrary length stick) stored in the Bureau of Standards (or more modernly as so many wavelengths of light from a particular atomic transition). If translations are one day found not to commute, then they too will have to be done by radians, and there will be some new fundamental constant length L to convert meters to radians. There will then be no dimensioned quantities left, therefore no dimensional analysis, and therefore no need to ask the very perceptive question that started this long winded answer.
Angles carry no information about location in space-time.
