## Units and Numerical niceties.

The discussion of what programming will look like in 2020 is getting excessively long, so I'm starting a new topic for a tangent instead of discussing it there.

This is a response to this post, which I fear clicking on the link will not bring up for you because due to the aforementioned excessive length of that topic it's already off the first page.

In brief, I'm responding to the Frink data file that defines units and dimensions for calculation with units in the language Frink, which provides dimensioned calculations with units.

As a meta-issue, I maintain that people defining such things for a language should use definitions they consider correct regardless of what a standards body might say; standards bodies are there to document correct usage, IMO, not to decide it. The recommendations of standards bodies ought not be followed when they are obviously incorrect. In particular, I am concerned with the following definitions and their interactions.

Radians are dimensionless. Alan editorializing goes on to say, "The angle subtended at the center of a circle by an arc equal in length to the radius of the circle" and then asserts that it is correct to regard it as dimensionless. As far as I can see, these statements are contradictory. If a radian is "The angle ...." then it is a unit of angle.

Hertz are considered by the SI committee to be the inverse of seconds. Alan agrees with me that this definition of Hertz is broken, but I maintain that the correct definition has a subtlety not so far captured by any proposed SI definition. Clearly, 1 Hertz is not equal to one Radian per second, which would be true given dimensionless radians and Hertz being the inverse of seconds. The issue is that all periodic functions of interest defined on (Radians as a unit of) angle are periodic with a period of 2pi radians, not with a period of 1 radian.

The notion of periodicity, or periods per second, is fundamental to what we mean by Hertz, but periodicity is a different fundamental notion than one of whatever measurement unit we happen to use for something per second. When an object is moving at 90 meters per second, we do not speak of it as having a frequency, because there are no periodic functions defined on velocity. And if we were measuring its rate of movement in feet instead, it would be unsatisfying to then claim it has a different frequency. On the other hand, if a sequence of identical objects two meters apart are all moving in the same direction and on the same path at 90 meters per second, we can speak of them as having a frequency of 45 Hertz, because the positions of objects relative to some fixed point is then a periodic function defined on velocity.

I maintain that Hertz is (ought to be) defined in periods-per-second of some periodic function of time. Thus if angle alpha is increasing by one radian per second, then the cosine of alpha is periodic with a frequency of 1/2pi Hertz. Conversely, the cosine of twice alpha would be periodic with a frequency of 1/pi Hertz. But the angle alpha, itself, would have no periodicity, because although it is a function of time it is not a periodic function. The point is that one cannot define Hertz (a unit of frequency, or an inverse unit of period) without reference to a specific periodic function of time.

Thus, expressing any measurement in Hertz requires a specific periodic function (or set of functions having the same period) of time. This makes Hertz a higher-order function, in that a function to calculate the Hertz of anything requires a function as one of its arguments. And I gather the SI committee really dislikes requiring people to define higher-order functions, and the community of mathematicians and engineers is very much accustomed to talking about Hertz (in angles, etc) correctly with respect to my definition but usually without saying what function the frequency describes.

As an aside, this points out that a useful mathematical library in any language that provides units of measure should have a higher-order function that gives the periodicity of another function in units of that function's input, and also a higher-order function that tells the units (or the dimension) in which another function is defined.

Ray

(edited 9:25 Pacific time, Sun 1 Jan 2013: simplified discussion of Hertz definition)

## Comment viewing options

### Dimensions are arbitrary

Radians are usually defined as arclength / radius and length / length is dimensionless. This definitely isn't wrong, but I would agree with you that it would also be correct and probably more useful to define radians as a unit of angle measure. One useful thing to notice about units is that the choice of what dimensions there should be is kind of arbitrary. For another example, you could count the animals at your farm and report the result as a Nat. Or you could introduce a new category of "animal counts" with values such as "5 animals". Dimensional analysis can then help you keep track of dollars/animal or legs/animal. These dimensions are just ad hoc tags that help us keep track of what the numbers mean.

It sounds like with your Hertz example that you want to stick with the familiar definition of Hertz: cycles per second. This leads to the question "what's a cycle?" and the answer is that it depends on the context (which is what I guess you're getting at by "higher order" units). That there shouldn't be an implicit conversion from radians to cycles is a good reason to have at least one of them not be dimensionless.

### Dimensions are arbitrary

" length / length is dimensionless"

That is one of the things I've very often disagreed with, they aren't dimensionless, they are unit invariant. You can use the number in any unit system you want. But it's still arc-length / radius. To me that indicates a slight brokenness in how we deal with units. A unit is a vector, you have a property like distance, and a standard, like meters. When you apply the unit to something, the value is applied to the property, but not the measurement standard.

So 100 m/s x-axis / 200 m/s y-axis != .5 dimensionless. It's 0.5 x-axis (distance) / y-axis (distance)

A classic is saying Gc is 32.2 ft/s^2; It's not.

### Those dimensions are also arbitrary

Let me elaborate a little more on what I mean by dimensions being arbitrary, because I think this example fits into that just fine. In a program that doesn't make use of dimensions we might have a variable vx that's the velocity of the ball along the x-axis in m/s. What happens when you introduce units into your language is that you reify part of that context. So we could now have vx is the velocity of the ball along the x-axis, in which case the 'm/s' bit is reified and belongs to vx. Or, we could go further and reify the 'along the x-axis' bit as well. That gets us the unit tags that you've suggested.

That might be useful or might be annoying. You'll have to explain why sqrt(vx^2 + vy^2) makes sense with your units. My point is that not having the extra reified context isn't wrong. Nor is the axis tag the only context you might want. If you're going to be multiplying by mass later to get momentum, you might want to reify "of the ball" so that you can check that you're multiplying by the mass "of the ball".

So there is no right or wrong here. It's a question of cost/benefit analysis. Reifying more context can avoid new classes of mistakes, but can also require annoying casts everywhere as you discard irrelevant context.

### Another example

I think that's getting close to the issue. But here's another datapoint. Consider the scale of a map; perhaps I'm currently showing the map at 5 cm/km. You might say that value is dimensionless and unitless. But suppose the map is plotting low-level ozone concentration, which might at some point be 42 nmol/mol. That's also dimensionless and unitless. Now, whatever the justification might be for maintaining separate types for measures of distance (centimeters, say) and amount of ozone (nanomoles) should apply mutatis mutandis to the scaling factors. It is just as meaningless to add a centimeter to a nanomole as it would be to multiply a mole by 5 cm/km.

In your terms, the scaling factors must have reified the dimension to which they apply, so in some sense they do have dimensions even though they are unitless. (And even the unitlessness is debatable, since it would logically show up as a default output format.)

### Another example

Isn't this just another case of wanting to reify more context? It sounds like you want the map dimensions to be (5 cm mcs / km wcs) where mcs is "map coordinate system" and wcs is "world coordinate system". Similarly, you want (42 nmol ozone / mol air). If all you reify is meters, then 5 cm / 1 km really does equal 0.00005.

### Maybe. The devil is in the details.

Sure, I think I said that above. It's just reifying more context, if you like. Or it's modelling the world more precisely.

The question which strikes me as interesting about the scaling issue is that the (unitless) scale still has some sort of phantom dimension. That is, it suggests that length/length is not exactly the same as, say, count, because length/length is certainly not the same as volume/volume.

All of the dimensional modeling systems I've seen (including one I wrote, years ago) assume that you can just do simple dimensional analysis and add units, done. But after trying to write such a system, and looking at frink, I started to wonder if that is really true, or if dimensional modeling is just searching where the light happens to be shining. Ray Dillinger's radian/hertz example is another great example.

I must be growing old. I just remembered that I raised this exact issue quite a few years ago, right here on LtU.

### Phantom dimensions

My point is that I don't think these phantom dimensions exist with the original units. They only exist within the context that you're modeling with the units.

It is just as meaningless to add a centimeter to a nanomole as it would be to multiply a mole by 5 cm/km.

A pipe of uniform thickness is a kilometer long and contains a mole of some gas. How much gas is in 5 cm of that pipe?

### Sleight of hand

You have an implicit conversion of length to volume in that question, so I don't regard it as a counter-example.

You could just as well ask about adding the length of pipe to a reservoir of 100 megalitres, but you'd still need to convert it to a volume if you were using dimensional types.

You could ask about applying the length scaling factor to the volume, but I'd argue that it shouldn't type-check either: it's ambiguous whether you're scaling just the length of the pipe (which is pretty well meaningless) or scaling each dimension of the reservoir (which is meaningful, but involves a cube).

### Implicit conversion is the point

You have an implicit conversion [...] so I don't regard it as a counter-example.

I think the reason you consider it incorrect to multiply a mole by 5 cm/km is because "5 cm/km" isn't a fully accurate description of the value you're considering. It isn't really the unitless scale you say it is. It's probably shorthand for 5 (cm on the map)/(km on the territory), and to take it at face value (0.00005) is an implicit conversion.

### Deletion vs. reification; explicit vs. implicit

I took care to not make a firm claim about 5 cm/km being unitless, but that's a side issue. Certainly you are right that if the scaling factor retains/reifies both co-ordinate systems, as Matt M suggested, then I get my desired type error.

On the other hand, you can clearly go too far: I don't want to associate the scaling factor with a *specific* map (probably, I'd want the map to have-a scaling factor) because that would make it impossible to ask whether two maps have the same scaling factor. I don't even want the scaling factor to reify units, because I might have a model with multiple units.

For example, this lovely walk above the city of Zurich. (Sorry for not having an English version; it's a scale model of the solar system using a scaling factor of 1mm/1000km; both the distance between the sun and planets and their diameters -- but not their volumes -- are scaled by the same 10^-9, and a comment from the astronomer who designed the walk points out that this scaling makes the scaled speed of light 30 cm/sec, allowing us to stroll along the model at about twice the speed of light.)

Anyway, I'm happy to accept the premise that the way to make scaling factors work the way I think they should would be to explicitly reify the "axes", but I don't know how far that goes towards clarifying the semantics of a reified dimensional expression.

That is, if I have a scaling factor of 1 mm model / 1000 km solarSystem, I would expect the dimensional infrastructure to reduce that to 10^-9 (model length)/(solarSystem length), thereby deleting the units but not the dimension, and to be prepared to generate the equivalent 10^-27 (model volume)/(solarSystem volume). What sort of beast, then, is "model volume"? And is it model^3 length^3, or just model length^3? Or do I have to explicitly spell out the axioms of my 'ad hoc' dimensions/labels/whatever when I declare them?

As one more datapoint, there is little difference between a scaling factor and a unit conversion factor. Indeed, all the scaling factors I've mentioned in this discussion have included unit conversions. Strictly speaking, unit conversions don't normally change the canonical value of a dimensioned quantity, but in some systems, applying a unit conversion to a value may result in its default output format changing.

### Good question

That is, if I have a scaling factor of 1 mm model / 1000 km solarSystem, I would expect the dimensional infrastructure to reduce that to 10^-9 (model length)/(solarSystem length), thereby deleting the units but not the dimension, and to be prepared to generate the equivalent 10^-27 (model volume)/(solarSystem volume). What sort of beast, then, is "model volume"? And is it model^3 length^3, or just model length^3? Or do I have to explicitly spell out the axioms of my 'ad hoc' dimensions/labels/whatever when I declare them?

For this particular example, I think the best answer would be that "Model Length" is an instantiation of the concept of Length. Volume being being cubic length would be a part of that concept and would come along for free. Note that you may or may not want to have the same units for those dimensions in this new system.

I don't want to associate the scaling factor with a *specific* map (probably, I'd want the map to have-a scaling factor) because that would make it impossible to ask whether two maps have the same scaling factor.

I didn't really understand this. Also, from your last paragraph, think we may be thinking of units a little differently. If you declare there to be three dimensions, World Length, Map1 Length, and Map2 Length, then values of those different dimensions are incomparable. The idea of a "scaling factor" doesn't make sense unless the map also exists at a specific size in the world, which it may not. For example, if the map is displayed on various size screens, then you don't have any fixed "scale factor". If you do have a conversion in mind (1 map inch / world inch), then you can just multiply by that conversion factor. There wouldn't have to be implicit conversion.

### Comparing scaling factors

If three dimensions -- World Length, Map1 Length, Map2 Length -- are incommensurate, then so are the fractions Map1 Length / World Length and Map2 Length / World Length. This incommensurability prevents me from asking the question: "Is Map1 higher resolution than Map2?", which I think is meaningful even if the units of Map1 Length and Map2 Length are both pixels -- i.e. neither exists at a "specific size in the world" -- although I admit, I was thinking of maps and models which have a physical and not just luminous presence. Your intuition is undoubtedly more actualized than mine.

### Convert first

To compare them, convert to a common dimension like pixels or world inches. If you're assuming one world inch per map inch then multiply by that.

### Scaling factor conversions

I took care to not make a firm claim about 5 cm/km being unitless, but that's a side issue.

Sorry, I didn't mean it was a firm claim. I was just targeting one half of the way you phrased "the (unitless) scale still has some sort of phantom dimension," trying to tip the ambivalence. :)

[...]impossible to ask whether two maps have the same scaling factor.

We could multiply or divide by a factor of 1 (map2 length)/(map1 length), which represents our ability to lay one map on another as a measuring tool.

Measuring a map with a map is a bit farfetched. Here's a slightly more practical example:

   20 (projected image length) in (hand lengths)
/ 0.5 (projector lens length) in (hand lengths)
-----
40 (projected image length) in (projector lens lengths)

We may not be able to lay a projector lens on its projected image to do a comparison--at least not without changing the image we're trying to measure--but a mundane object in between can help. Usually(?) it's a dedicated measuring device, and we don't bother to mention it.

Back to the map example, the 1 (map2 length)/(map1 length) factor might not represent the fact that we physically use one map to measure the other, but that we can measure them both using the same measuring stick. Unlike the projector example, in this case we're going to measure distances on the maps' surface rather than currently assessing the maps' own overall size, so we're doing zero "measurements," and the resulting product has a coefficient of 1. Or something like that!

[...]I would expect the dimensional infrastructure to reduce that to 10^-9 (model length)/(solarSystem length), thereby deleting the units but not the dimension[...]

I agree with this. It shouldn't matter if we specify it as 10^-9 (model miles)/(solarSystem miles).

[...]and be prepared to generate the equivalent 10^-27 (model volume)/(solarSystem volume).

We could "generate" that manually by taking the cube of the scaling factor, but would it really make sense for the language to infer exponents? If the point is to highlight unit errors, this means we're letting certain errors slip by.

But if we want to take advantage of unit notation to do unit-directed computation, then it makes sense that we could set up ambient conversion factors and then apply several of them all at once with a cast. A length factor like this could apply multiple times, letting us convert volumes.

And is it model^3 length^3, or just model length^3?

If it were model^3 length^3 as though they were separate units, then we could start with 10^-9 (model length)/(solarSystem length) and get 10^-9 (model mass)/(solarSystem mass). So it isn't that.

### That's what I meant by (unitless)

I agree with this. It shouldn't matter if we specify it as 10^-9 (model miles)/(solarSystem miles).

Precisely. (unitless) but still not a scalar.

We could "generate" that manually by taking the cube of the scaling factor

Yes, that's what I meant. And if we simply cube the scaling factor, it should turn into volume scaling, just like cubing a distance turns into volume. So we should now be able to apply the cubed scaling factor to a volume.

### What's the compiler error?

pipeL = 1 km
pipeGas = 1 mol
segL = 5 cm
segGas = pipeGas * segL / pipeL -- Error:  Illegal sleight of hand

Seriously, though, I don't think you want an error there. This calculation is dimensionally and semantically correct. What would you write for line 4 instead? Are you saying we should bake into the language some assumption about moles being proportional to volume?

### Challenge

Construct a meaningful example for 1 mmol / mol + 5 cm / km ;-) Note the + not *.

### OK...

You have two processes for producing a unit of a certain compound. The first process requires a mol of reagent and you have only 1 mmol. The second process requires a kilometer of cord and you have 5 cm. How many units of the compound can you produce?

### Hmm?

Sorry to jump in, but I don't see how this is a counterexample. It was my understanding of your proposal that you want to require explicit multiplication by the (1 unit-of-end-product / mol) and (1 unit-of-end-product / km) scale factors that are implicit in the proposed problematic example?

### Yes, but these scale factors

Yes, but these scale factors cancel out in Matt's example? i.e. it doesn't force them to appear.

### I see your point

OK, sometimes a scaling factor is just a scaling factor. But please don't use map scales to dilute your medicines, ok? Sometimes the difference is important, too.

Specifically, in your example you're really computing:

volume of pipe = length * cross-section
moles in the pipe = (moles / volume) * volume of pipe
moles in the subpipe = moles in the pipe *
((subpipe length * subpipe cross-section) /
(pipe length * pipe cross-section))

The second of the computations is pipegas, if I understand you correctly. So it already had an implicit volume/volume phantom component (in my admittedly idiosyncratic terminology). The third one explicitly states that the cross-section is consistent for the subpipe. If you didn't put that in the code, you'd have to put it in a comment, because it's important.

Admittedly, I'd probably be annoyed about having to insert the cross-section twice (although it will be hypothetically be optimized away by the hypothetical compiler) but I can think of equally annoying things I've provided to real compilers in order to demonstrate my typesafe credentials.

### Also, units/resource...

is a perfectly good dimension, so you can add (units/resourceA) * (quantity of resourceA) + (units/resourceB) * (quantity of resourceB). In this case, the resource/resource really should disappear.

I honestly don't know what the answer to the question of when units should really cancel out, and when they should leave behind some trace; if it were an easy question, it wouldn't be worth talking about.

### Buckingham Pi Theorem

I haven't seen yet in this discussion the mention of the Buckingham Pi theorem. This proves how in any system of measurements that a certain number of measurements can be considered dimensionless, while the others must have some dimensions to be orthogonal. I see much of the discussion leading up to this, but no mention of the fundamental theorem.

### Units and symmetries

Units are used to enforce measurement scale invariance: results of computations shouldn't depend on what length you call "one meter". This suggests that perhaps you can invent unit systems to capture other symmetries/invariances. For example results of computations should not depend on where you choose the origin of your coordinate system. Perhaps you can devise an analogous system to enforce this. Since addition has basically the same structure as multiplication, this doesn't seem too hard: just take logs everywhere. You can go further and enforce rotational invariance (it doesn't matter in which direction you choose your coordinate axes). Perhaps even discrete symmetries (it doesn't matter which electrical charge you call positive and which you call negative -- though this is not actually true in weak interactions, but a more complicated symmetry does hold). Does anybody have ideas on how to do this with units?

### Units, coordinate systems, and invariance

My paper Abstraction and Invariance for Algebraically Indexed Types with Bob Atkey and Patricia Johann that I presented last week at POPL'13 considers this idea.

For a readable introduction to previous work on units-of-measure, including scale invariance, see Types for Units-of-Measure: Theory and Practice. You can try this for real in the F# programming language.

### I haven't finished reading

I haven't finished reading the paper in detail yet, but I certainly will. That's exactly what I was looking for and this would make my life so much easier! Do you have any thoughts on typing angular functions in this framework like cos, sin, complex exp?

### Dimensional analysis has its limits

It's simply not reasonable to assume that if two numbers have the same dimensions, they are necessarily interchangeable. Energy and torque both have the same unit: newton meters, or kg m^2 s^2 when fully reduced. Yet they are not mathematically equivalent, because energy is a scalar and torque is a vector. You can also think of torque as joules (= newton meters), here a unit of work, per radian, in which case 1/radians can be seen as a dimensionless unit that converts a scalar to a vector (by the same token, radians themselves convert a vector to a scalar).

### Subtle radians vs angle

This comes down to a mathematically very subtle question about "are radians angles or dimensionless?" I can give you many many mathematically deep arguments that radians are dimensionless. They are a length (around a circle) divided by a length.

In the case you mention before about torque and energy, the two are very closely related. A torque applied through an angle (measured in in radians) (again, radians are dimensionless) is exactly an energy, dimensionally and measurably. That is (torque*angle = energy) So the units are indeed conformal in a system where radians are dimensionless.

If you can come up with a system where radians are not dimensionless, it would be interesting. I'll give the example of the Taylor series for sin[x]. Try to work out the dimensions of a radian in the Taylor series where sin[x] = x - x^3/3! + x^5/5! - x^7/7! + ... You have to invent a system where x - x^3 works dimensionally. The only obvious solution is to make the radian dimensionless.

Hint: You cant add/subtract x and x^3 unless x is dimensionless.

Edit: fixed signs in Taylor series and wording to match.

### What is a radian?

If you define radians as length/length, then they're dimensionless. If you postulate a new dimension "angle" that has radians as a unit of measure, then they aren't dimensionless.

Mathematicians usually take the former approach, but I suspect the latter approach is superior for most everyday programmers using angles. The definition of sin by Taylor series in this system is then given by sin[x rad] = x - x^3/3! + x^5/5! - x^7/7! + ... + x^odd/odd! + ... with the only difference from your formula being the pattern match on x.

### Let rad be a unit such that

Let rad be a unit such that rad^2 = 1. Now sin x works!

### I think we basically agree

I'm not saying that energy (or work) and torque aren't conformal: they are. What they aren't is mathematically interchangeable: a scalar is not a vector. IOW, conformality is not tantamount to interchangeability.

### Doesn't this complaint prove

Doesn't this complaint prove too much? What if I have a Taylor series approximation to some function in my problem domain, f, that takes a distance and returns a mass? Won't I have the same dimensional problem?

I think this problem arises because the Taylor series coefficients were derived from a chain of derivatives, and the derivatives carried the units that would have canceled out with the units from the increasing powers of x. But it seems to me that this is the case regardless of whether we are talking about sin :: Radian -> Dimensionless or f :: Distance -> Mass.

The Taylor series is actually sin[x] = (1 rad^-1) x + (-1/3! rad^-3) x^3 + (1/5! rad^-5) x^5 + ... (modulo whatever operator precedence mistakes I made). At least I think it is consistent to see it that way, and potentially useful because angles as a statically tracked dimension really do prevent a lot of silly forgot-a-180-over-pi-someplace errors.

### Taylor series

The Taylor series is actually sin[x] = (1 rad^-1) x + (-1/3! rad^-3) x^3 + (1/5! rad^-5) x^5 + ... (modulo whatever operator precedence mistakes I made).

In fact, I think that it is even more useful to view it as sin(x) = \sum_{i = 0}^\infty (-1)^i\prod_{j = 1}^{2i + 1} (x/(j rad)). I can't think of one off the top of my head, but I'm willing to bet that there is an interpretation of that formula (probably inclusion–exclusion flavoured; see http://www.cut-the-knot.org/arithmetic/combinatorics/CombinatorialPyth.shtml) that says something quite interesting about sines.

### The difference between angle and dimensionless quantities.

It is also not reasonable to assume that if two numbers have the same representation type, they are necessarily interchangeable. What type systems try to guarantee is always necessary, but never sufficient, to exclude the possibility of error.

That said, I think you've placed a finger on the pulse of what is capital-W Wrong with angles as being dimensionless.

Energy is scalar and Torque is vector, because Torque is constrained by a direction. And the constraint of direction is at the root of the difference between dimensionless and angular radians.

Radians are not just a ratio of lengths. They are a ratio of a length to a quantity that has a particular direction relative to that length. If the "length" along the arc was not curved in precise proportion to the length along the radius, then the ratio would not represent an angle. And while a distance can be characterized simply by its endpoints, a distance along an arc cannot be defined without reference to both its endpoints and its center of curvature. If you construct an arc around a different center point, you change the length of the arc just as surely as if you had constructed it with a different end point.

So one of these "lengths" has the usual meaning of absolute location-independent separation in space, and the other "length" has a different meaning of separation around some particular point of rotation. When you divide one by the other, the units ought not not cancel. When you multiply the quotient by a scalar, you get a vector not another scalar.

So ... what relationship does dimensional analysis indicate that the location of a point (the extra information needed to define the length of the arc) has to the definition of a rotational direction (the extra information needed to define the vector of torque)?

Ray

### Angles are not vectors

When you add angles that are defined in different planes, the convention is that means scalar addition.

This scalar/vector distinction is a red herring. Torques are vectors, but "torque magnitude", a scalar, is a useful concept in its own right, just as speed is useful as a scalar. In fact, when people say "torque" they often mean "torque magnitude". So, should torque magnitude be interchangeable with energy?

Radians are not just a ratio of lengths. They are a ratio of a length to a quantity that has a particular direction relative to that length. If the "length" along the arc was not curved in precise proportion to the length along the radius, then the ratio would not represent an angle. And while a distance can be characterized simply by its endpoints, a distance along an arc cannot be defined without reference to both its endpoints and its center of curvature. If you construct an arc around a different center point, you change the length of the arc just as surely as if you had constructed it with a different end point.

I think you'll have trouble formalizing anything coherent along this line of reasoning.

### Perhaps you want an "empty" or "counting" unit

That is, a trivial unit of measure such that "2 units" is distinct from a dimensionless "2", but that can still be treated as a multiplicative identity in dimensional analysis, so that "2 units * 3 meters" meaning 2 units of 3 meters each still becomes 6 meters, but "1 unit + 2" is an error.

A value dimensioned in "unit/s" would then mean occurrences per second of whatever is being measured, and while it would still be necessary to distinguish between e.g. "2 units" (of apples) and "2 units" (of oranges), any dimensional analysis involving one should make sense for the other, whereas a dimensionless 2 (e.g., a scaling factor) is rather different.

Starting from that perspective, when discussing frequency the occurrences being counted are very obviously "periodic repetitions", and so 1 Hz can be naturally defined as "1 unit/s" (of periodic repetitions) and everything else follows sensibly. Similarly, when measuring rotation the most natural baseline is a full rotation.

Allegedly dimensionless quantities can be seen either as conversion factors between things being counted, or as derived counting units defined as some multiple of a baseline, or as things being counted in their own right.

Then again, perhaps it would make more sense to consistently distinguish between what kind of thing is being counted. The idea that occurrences per second is simply 1/s independent of what's counted is patently inaccurate, as evidenced by the proliferation of distinct names for the "same" unit depending on whether one is measuring rotational speed, cycles per second, catalytic activity, modulation rate, radioactivity...

### Introduce wavelength-like things?

I don't think the higher-order Hz idea is necessary. I think the SI unit of frequency should be periods/second, and we just need to introduce a concept like wavelength into other frequency-generating equations. (OTOH Hz itself should remain 1/s since that's what Hz is - it just doesn't mean "frequency".)

Wave velocity is (say) 10m/s, but wavelength is no longer 2m, it's 2m/period. Then frequency = 5 period/s. 2m/period says we didn't just measure the length of one wave, but we made sure many waves had the same length, so it's not just meters, but meters over several periods = m/period.

In parallel, angular velocity is (say) 8*pi rad/s, but angular wavelength is the (fixed) quantity 2*pi radians/period, giving 4 period/s. That quantity again is not just "the angle of a period" but "the angle of every period", so radians/period not just radians. [We can make the constant 1 by using a different unit for angle (turns instead of radians) which is probably more SI-like than using radians for this.]

Now angular velocity has the same relationship to frequency as spatial velocity does, when there is periodic behaviour quantified by a periodic "wavelength".