Yeah, you're right. I think I just got caught up trying to remove special cases and have a more general abstraction without fully thinking it through. I see that this was a mistake, now. The result is no longer radians, it's just an encapsulated ratio (encapsulated because all expressions are encapsulated, since they are represented as expression templates, and so that way operator usage is still performed through the provided policies). Conversion to the raw value is implicit, so you can use the result of such expressions intuitively. However, I would like some input on my talk of decibels, only now removing my foolish talk of radians with "untransformed ratios" (encapsulated raw numbers which when multiplied together result in another "untransformed ratio" just like "regular" numbers operations result in more "regular" numbers). I'm unsure if you were confused by my explanation of the relationship of decibels to regular numbers and how point decibels are convertible to regular number points yet not vectors and scalars, so I hope this clarifies, and I would appreciate some feedback. To explain this better I'll try to give some more examples relating bels to untransformed ratios. Firstly, think of bel points and points without no units (untransformed ratio units). Bels are just ratios represented with a base 10 logarithmic scale, so concerning point space, these would be the relationships (bels are approximated): Ratio: 1 = Bel: 0 Ratio: 2 = Bel: 0.301 Ratio: 3 = Bel: 0.477 Ratio: 4 = Bel: 0.602 Ratio: 5 = Bel: 0.699 ... Ratio: 10 = Bel: 1 ... etc. I'm sure you understand the above, but it is provided for clarity and a point of reference. Note that while the space between a ratio (encapsulated raw number) increases by the number 1 in each grouping, the value in bels increases more slowly the further away from 0 you get. You can convert a point ratio to a point bel no problem, since the mapping is done by just applying base 10 logarithm to the ratio. However, this relationship cannot be use with vectors and scalars! In other words, a bel vector can not be converted to a "ratio" vector (raw vector) and a bel scalar can not be converted to a regular scalar. This is because how a ratio vector or magnitude relates to a bel vector or magnitude depends on "where" in point space those quantities lie. Since vectors and scalars have no spacial location, the conversion is impossible. Firstly, say you have two bel vectors with a value of 1 and a bel vector with a value of 2. If you were to assume that bel vectors were convertible to ratios, you would think that you could convert those bel vectors with value 1 to ratios, add them together, then convert them back to bels to have a value of 2. As I said, the conversion wouldn't exist, but I'll follow through with the operation to show the results: 1 bel = 10 10 + 10 = 20 20 = 1.301 bels 1.301 bels != 2 bels Again, this is the case because what a translation of "1 bel" or magnitude of "1 bel" is as a raw number changes depending on spacial location, which vectors and scalars don't have. In actuality, converting a bel vector to a raw number vector isn't possible, even though you can convert a bel point to a raw point! This also means that given two vectors, one of bels and one as an untransformed ratio, you can't add the two together, since conversion from one to the other is impossible. Now, what about adding a raw point with a bel vector? The operation can be done and makes logical sense. Rather than converting the bel vector to a raw vector, which is impossible, you have to convert the raw point to a bel point. Now the operation can be performed and results in a bel point. This bel point can also be converted back to a raw point if needed. This relationship is the same for nepers and the richter scale, etc. making it perfectly fine to have such unit types in an environment as other units, given proper use of points, vectors, and scalars. -- -Matt Calabrese