[adrian_b]

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

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.

[eigenket]

Ok, you've replied to quite a lot of my comments, with some fairly confusingly worded responses. Some of your claims appear to be incompatible as far as I read them.

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.

[adrian_b]

No, even if you have reproduced a definition of the plane angle that is encountered in many textbooks "the angle subtended by an arc in a circle is the ratio of the arc length and radius", this definition is very incorrect.

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.