1 radian has different units than 1 steradian and if they didn't there wouldn't be a need for two different words to denote them.

The quantity is a ratio of two lengths, and the length measure does "drop out". But it's not just any ratio, it's a very particular ratio, and the unit defines the particularness of that ratio.

The appropriate canonical representation of a rotation is a unit-magnitude complex number z = exp iθ = cos θ + i sin θ, which has a planar orientation (whatever plane i is taken to represent; if you want to represent a 3D rotation you can replace i with an arbitrary unit bivector) but is unitless.

This is similar to the way the "ten" in "scale by ten" is unitless, but if you take the logarithm you get "scale by 10 decibels" or "go up by 3 octaves and 3.9 semitones", which have the base of the logarithm as a kind of unit.

But you seem to be drawing a distinction between meters and angles in your analogy where I assert none exists. The base of a number system only affects representations.

This is not true for divisions of lengths. 1 meter divided by 2 meters is 0.5 as a number. But it is only 0.5 radians under (1ish) specific arrangements of those lengths in a particular metric space

This is analogous to the way a scalar logarithm can have a base of "octaves" (doublings), "decibels", or "powers of the golden ratio" (as found in the Zometool construction toy). Or pick your favorite other logarithmic system.

Both are "units" in a certain sense, but neither one is quite the same kind of "unit" as light years or foot–pounds or amperes.

(For example, you can measure an octave by marking out a particular fret on your guitar; it will make an octave change whichever particular note you start with.)

You can feel free to "reject" whatever you want. You'll just be wrong/confused. ;-) (But you'll be in good company. Most working engineers, scientists, and mathematicians don't have or need a particularly clear philosophical understanding of angles.)

A) multiply two degrees together

B) multiply two dBm values together

The output "units" in A change but do not in B. dB and angles are very different.

Edit: the units in B do change, but the dB part doesn't. Tired.

But you can compose rotations.

Multiplying decibels together is also not meaningful, but you can compose scaling.

Curious what you think of the d_theta * d_phi term in a spherical coordinates integral...

While rotation is naturally oriented like a bivector (plane), solid angle is naturally oriented like a trivector (3-space).

The natural representation is as a kind of (unitless) ratio formed from 3 vectors, not the product of two vector–vector ratios.

Essentially, I think that whatever angles are, they are not like other dimensionful physical quantities. I have two arguments.

The first: Someone mentioned symmetries in a reply. I wanted to mention them too but didn't have time to structure my thoughts into a coherent argument. But the gist of it is that dimensionality is just a kind of scale invariance, and the scale invariance of angles is fundamentally different from that of linear quantities due to their periodicity — to apply a unit transformation, you have to scale the quantity _and the period_.

The second: Consider units from a "type theory" perspective instead. If you are considering exclusively linear trigonometry (no arcs), it's trivial to assign a dimensional type structure to expressions (e.g. cos takes angle type and maps it to dimensionless type). But as soon as you allow arc lengths, it becomes cumbersome to type common expressions.

I think these distinctions form the crux of the disagreement. Ultimately, it depends on your intuitive notion of what "dimensionality" actually means, and how it ought generalise to other kinds of quantities.

Here is an example to highlight my point. Let there be a circle C of centre O and radius r. Let A be a point on the circle. Let there be a point M outside the circle such that (AM) is tangent to C. Let B be the intersection of C and [OM]. Let s be the arc length along C from A to B. Then we want to write AM = r tan(s/r).

How does one get s/r to resolve to an angular dimension? Ought we instead ascribe s dimensions of length-angle? Imagine, then, that the circle is in fact a pulley, and we wish to measure a change x in length of rope as the pulley rotates through the angle of the arc from A to B. We would want to write x = s. But this is now dimensionally inconsistent.

It's certainly possible to make all these expressions correctly typed by introducting appropriate conversion constants. But this seems to me to be cumbersome. Since in physics, arc and linear lengths can convert freely into one another, it seems more economical to just let angles be dimensionless.

Edit: in other words, you've encoded tons of information in the problem statement about the relationship between r and s and you aren't properly encoding that in your type system allowing s/r to output an angle

    360 - (360^3)/6 = -7M degrees

    2*pi - (2 * pi)^3 / 6 = -35 radians = -2k degrees

    1 - (1^3)/6 = 0.8 turns = 300 degrees

As far as you three examples go, which is "correct" depends on what you are trying to calculate - if you want this to approximate the power series for sin close to 0 you should use radians. Otherwise you use something else.

As another example for the second assertion, you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed, etc, which comes up all the time. But that doesn't mean `dimensionless + seconds + seconds^2` implies seconds are dimensionless any more than sin's series with `angle + angle^3` implies that angles are dimensionless.

> Angles aren't dimensionless any more than lengths are dimensionless

They are dimensionless, but they still have units. The concepts are orthogonal.

As for the question about adding an angle to its cube, I would say the enormous usefulness of computing trig functions by power series suggests strongly that this is meaningful.

That argument holds just as true for dimensional quantities frequently computed by power series though, which means it can't be valid.

A better argument that angles are dimensionless is that dimensionless quantities are formed by the ratio of two quantities with the same dimension, and the angle subtended by an arc in a circle is given by the ratio of the arc length and the radius.

The concepts are not orthogonal, they are incompatible.

A dimensionless quantity (which angles are not) is by definition the ratio of two quantities that are measured with the same unit.

When you compute the ratio by division, the two identical units disappear from the result, therefore the result is indeed dimensionless.

There is no way to choose a unit for a dimensionless quantity in the usual sense.

At most you could define a new different dimensionless quantity, as the ratio of two dimensionless quantities, i.e. as a ratio of ratios, but because it needs a different definition this should better be viewed as a different quantity, not as the same quantity with a different unit.

For example you claim that angles are not dimensionless, but that dimensionless quantities are formed by the ratio of two quantities with the same unit. Since the angle subtended by an arc in a circle is the ratio of the arc length and radius, it would seem that these two claims contradict each other.

I do agree that without some aditional work the power series argument I wrote above does seem to be wrong.

This very wrong definition forced upon many students is the root of all misconceptions about plane angles.

The reason why this definition is wrong is because some words are missing from it and after they are added it becomes obvious that its meaning is different from what many teachers claim.

First there is no relationship whatsoever between the magnitude of a radius and the magnitude of an angle. What that definition intended to say was:

"the angle subtended by an arc in a circle is the ratio of the arc length and of the length of an arc whose length is equal to the radius".

By definition, a radian is defined as the angle subtended by an arc whose length is equal to the radius.

To explain how plane angles are really defined would take more space, but the only thing that matters is that the characteristic property of plane angles is that the ratio between two plane angles subtended by two arcs of a circle is equal to the ratio of the lengths of the two arcs.

Introducing this characteristic property of the angles in the so-called definition from above reduces it into the sentence "the angle is measured in radians". This is either a trivially true sentence when the angle is indeed measured in radians, or it is a trivially false sentence when the angle is measured e.g. in degrees. It certainly is not a definition.

If that had been the definition of plane angle, that would have meant that the plane angle was discovered only in the second half of the 19th century, together with the radian, while in reality plane angles have been used and measured with various units for millennia.

To measure plane angles, it is necessary to first choose an arbitrary angle as the unit angle, for instance an angle of one degree.

Then you can measure any other angle by measuring both the length of the corresponding arc and the length of the arc corresponding to the chosen unit angle. Then the two lengths are divided, giving the numeric value of the measure of the angle.

The argument of exp does have to be dimensionless, exactly because adding seconds to seconds squared doesn't work. If t has units of time, there has to be another factor with units of inverse time, for example continuous compound interest is exp(rate*time).

A dimensional formula of a quantity just writes its unit as a function of the fundamental units.

In any equality of physical quantities, in the two sides not only the dimensionless numeric values must be equal, but also the units must be equal, which is usually expressed by saying that the dimensions must be the same, and it is verified by writing in both sides the dimensional formulae, i.e. the units of both sides as functions of the fundamental units.

A dimensionless quantity is a ratio of two quantities that are measured by the same unit, so that the units simplify during the division.

There may be different but related dimensionless quantities, which are differentiated by different definitions of those quantities, but a dimensionless quantity cannot have different units.

This is just meaningless mumbo-jumbo that has been sadly introduced in the documents of the International System of Units, in 1995, after a shameful vote of the delegates, who have voted automatically, without thinking or discussing, a vote equivalent with establishing by vote that 2 + 2 = 5.

Any physical quantity, for instance length, can appear as an argument of a nonlinear function that can be developed in a Taylor series. So your example would be identical for any other quantity not only for angle. I can make an analog computing element where a voltage is equal to the sinus of another voltage, so after your theory, voltage is dimensionless.

The reason why this is possible is that the arguments of such nonlinear functions are either explicitly or implicitly not the physical quantities, but their numeric values, i.e. the ratios between those quantities and their units, which are dimensionless.

In the case of the nonlinear sinus function, what is usually written as sin(x) is just one member of a family of functions where the arguments are angles implicitly divided by units of plane angle:

sin(x) is the sinus function with the angle implicitly divided by 1 radian

sin(x * Pi/2) is the sinus function with the angle implicitly divided by 1 right angle

sin(x * Pi*2) is the sinus function with the angle implicitly divided by 1 cycle a.k.a. turn

sin(x * Pi/180) is the sinus function with the angle implicitly divided by 1 sexagesimal degree

It is very sad that the logical thinking about angles of most people has been perverted by what they have been taught in school, which is just a bunch of nonsense copied again and again from one textbook to another.

This to me sounds like the most natural explanation. For example, in a sibling comment someone mentioned that "you can calculate e^(-t)", but I disagree: in physics it's always e^(-t / T), where T is some time constant, so that the argument of the exponential is dimensionless. Same applies to sin(x): usually we write something like sin(2pi f t), where the units of f and t cancel out, and the 2pi is there to cancel out the invisible implicit 1 radian. sin(ft) would be wrong, at t = 1 / f you wouldn't have advanced by a full cycle.

There is a natural reason for pi occurring in physics that makes little sense to ignore.

Treating it as something to be dealt with misses the forest for the trees.

Radians are the naturally occurring Euclidean unit of angular measurement.

I will discuss only the plane angle, because it is the most important, but the situation is the same for solid angle and logarithms.

The justification commonly given is that the plane angle is dimensionless because it is the ratio of two lengths, the length of the corresponding arc and the length of the radius. This justification is stupid, because that is not the definition of the plane angle, but it already includes the choice of a particular unit.

As formulated. this justification only states the trivial truth that the numeric value of any physical quantity is the ratio between that quantity and its unit. By the same wrong justification, length is dimensionless, because it is the ratio between the measured length and the length of a ruler that is one meter long.

Correct is to say that the plane angle is a physical quantity that has the property that the ratio between two plane angles is equal to the ratio between the lengths of the corresponding arcs.

This is a property of the same nature like the property of voltage that the ratio of two voltages across a linear resistor is equal to the ratio of the electric currents passing through the resistor. This kind of properties are frequently used in the measurement of physical quantities, because few of them are measured directly but in most cases ratios of the quantities of interest are converted in ratios of quantities that are easier to measure.

This property of the plane angle allows the measurement of plane angles, but only after an arbitrary unit is chosen for the plane angle. Because the choice of the unit is completely free, i.e. completely independent of the units chosen for the other physical quantities, the unit of plane angle is by definition a fundamental unit, not a derived unit.

The freedom of choice for the unit of plane angle is amply demonstrated by the large number of units that have been used or are still used for plane angle, e.g. right angle (the unit used by Euclid), sexagesimal degree, centesimal degree, cycle a.k.a. turn, radian.

The fundamental units of plane angle, solid angle and logarithms must never be omitted from the dimensional formulae of the quantities, otherwise serious mistakes are frequent & such mistakes have delayed the progress of physics with many years (e.g. due to confusions between angular momentum & action; the Planck constant is an angular momentum, not an action, as frequently but wrongly claimed). This is a problem especially for the unit of plane angle, which enters in the correct dimensional formulae of a great number of quantities, including some where this is not at all obvious (e.g. magnetic flux).

The synathroesmic writing style and inapt analogies obfuscates the claims and deters others from refuting them, but does not make them true.

Angles are defined in the abstract realm of mathematics not physics, which use them to describe physical phenomenon. In Cartesian space, the basis dimensions are defined using fixed perpendicular oriented lines, called coordinate lines on coordinate axes. Angles are not a basis dimension and as defined in Cartesian space are dimensionless. Angles are a dependent measure. Arc length is measured based on the units of the system. x radians or x degrees cannot be physically measured in and of themselves. In coordinate systems using an angle as a basis like spherical coordinates, angles exist as a dimension and the unit for them is free to chosen. Once the units are chosen, then the angles can be physically measured.