Is "nondimensionalization" a misnomer?

[Grefus]

TL;DR Summary

Why mass expressed in atomic units is considered dimensionless, but expressed in grams it is dimensional? (don't we have ratios in both cases?)

While doing some unit conversions for a task at hand, I understood that, even though I can do the conversions without any problems, there is something I find hugely confusing about terminology.

By and large, the confusion is related to the interpretation of "nondimensionalization".

As an example, consider relative atomic mass. To quote the wiki:

This comparison is the quotient of the two weights, which makes the value dimensionless (having no unit).

For me, this sounds self-contradictory. Why having no unit, when, by very construction, 1/12 of C¹² mass is chosen as the unit?

And why dimensionless? Mass expressed (say) in kilograms has the dimension of M to the first power, but switching to another unit turns it into M to the zeroth power? The argument goes that here we have the ratio of two quantities with the same dimension, so the result is dimensionless. But when we express masses in kilograms, doesn't that mean that we take the ratio of that mass over the mass of some standard object which is set to have the mass of 1 kg? Why then in this case we still say that the mass is dimensional?

Probably, I'm missing something obvious, but anyway the topic seems to be full of confusion. E.g., while googling for the answers, I found this related question. To quote the professor in the answer:

Since it’s a ratio of two quantities with the same units, those units cancel and you get a unit-less quantity

My impression was that having a dimension and having a unit are somewhat different things, aren't they?

The same goes with the procedure of nondimensionalization. I perfecly understand what it does, and I employed it many times. But, again, is it correct to say that it actually "eliminates" the dimension of a physical quantity? In my understanding, this procedure is just a rescaling, where you choose a unit for measuring that quantity (e.g., length) which is not related to any metric system of units, but is inherenet to the problem at hand (e.g., some characteristic length-scale of a model). This doesn't mean that we eliminate the dimension of length, we just switch to units which "make best sense" for the current problem.

So, I'll greatly appreciate a clarification from someone with experience in this topic.

 

[topsquark]

Summary: Why mass expressed in atomic units is considered dimensionless, but expressed in grams it is dimensional? (don't we have ratios in both cases?)

While doing some unit conversions for a task at hand, I understood that, even though I can do the conversions without any problems, there is something I find hugely confusing about terminology.

By and large, the confusion is related to the interpretation of "nondimensionalization".

As an example, consider relative atomic mass. To quote the wiki:

For me, this sounds self-contradictory. Why having no unit, when, by very construction, 1/12 of C¹² mass is chosen as the unit?

And why dimensionless? Mass expressed (say) in kilograms has the dimension of M to the first power, but switching to another unit turns it into M to the zeroth power? The argument goes that here we have the ratio of two quantities with the same dimension, so the result is dimensionless. But when we express masses in kilograms, doesn't that mean that we take the ratio of that mass over the mass of some standard object which is set to have the mass of 1 kg? Why then in this case we still say that the mass is dimensional?

Probably, I'm missing something obvious, but anyway the topic seems to be full of confusion. E.g., while googling for the answers, I found this related question. To quote the professor in the answer:

My impression was that having a dimension and having a unit are somewhat different things, aren't they?

The same goes with the procedure of nondimensionalization. I perfecly understand what it does, and I employed it many times. But, again, is it correct to say that it actually "eliminates" the dimension of a physical quantity? In my understanding, this procedure is just a rescaling, where you choose a unit for measuring that quantity (e.g., length) which is not related to any metric system of units, but is inherenet to the problem at hand (e.g., some characteristic length-scale of a model). This doesn't mean that we eliminate the dimension of length, we just switch to units which "make best sense" for the current problem.

So, I'll greatly appreciate a clarification from someone with experience in this topic.

I don't think I've ever seen 1 amu to be considered no unit. Can you show me a source that does that?

There are "unitless" units out there. Angular measure in radians, for example. 1 rad = (1 m)/(1 m). The coefficients of friction are unitless, etc.

There is something of a movement in unit systems for some advanced work that redefines some constants to be 1... for example Particle Physics and Relativity often set c =1, which "equates" the units 1 m = 1 s, and makes the unit for c (1 m)/(1 m) = 1. But aside from the logic of wanting to measure space and time the same way it's also a way to get rid of some of the clutter of constants that tend to appear in the equations. It can take some getting used to.

-Dan

 

[Dale]

Mass expressed (say) in kilograms has the dimension of M to the first power, but switching to another unit turns it into M to the zeroth power?

Yes. The dimensionality of an expression depends on the system of units that we use. However, it is important to understand that the equations of physics also depend on the system of units used, and changing to a system of units with different dimensionality usually entails a change to the equations of physics also.

For example, in such units Newton’s 2nd law would be $$\Sigma \vec F=k m \vec a$$ where $k$ is a “universal” constant with dimensions of $[k]=F^1\ L^{-1}\ T^{2}$.

In my understanding, this procedure is just a rescaling, where you choose a unit for measuring that quantity (e.g., length) which is not related to any metric system of units, but is inherenet to the problem at hand (e.g., some characteristic length-scale of a model).

This sort of non-dimensionalization tends to be more useful. In this case you are deliberately using the fact that changing the dimensionality of a problem changes the form of the laws of physics. You are essentially using that fact to simplify the laws of physics for a specific problem.

 

[sophiecentaur]

Summary: Why mass expressed in atomic units is considered dimensionless, but expressed in grams it is dimensional? (don't we have ratios in both cases?)

But, again, is it correct to say that it actually "eliminates" the dimension of a physical quantity?

I can't see that it "eliminates" anything.

Instead of xkg of a substance you can describe it in terms of 'the mass of Y Carbon atoms'. The kg is just hidden inside the name of Carbon. You can't do any calculations without including the kg in there somewhere.

Actually, nearly every unit of measurement used to be expressed in terms of One Ear of Corn or The King's Thumb etc. Even Time was expressed in terms of Days and it's only recently that the second (defined in terms of an atomic transition) became the primary unit of time.

 

[Grefus]

Unfortunately, none of the answers thus far clarified things to me...

I don't think I've ever seen 1 amu to be considered no unit. Can you show me a source that does that?

I had provided two quotes with links in the OP ("having no unit" and "unit-less").

Yes. The dimensionality of an expression depends on the system of units that we use

So let me restate the question: In what way am I wrong, saying that there is no difference in measuring mass in kilograms and in atomic units, and that in both cases we measure mass with respect to some standard, and so it makes no sense to say that mass is dimensional in one case and dimensionless in the other?

I can't see that it "eliminates" anything.

This is also the way I see it. But why then is it called nondimensionalization?

 

[Dale]

In what way am I wrong, saying that there is no difference in measuring mass in kilograms and in atomic units, and that in both cases we measure mass with respect to some standard, and so it makes no sense to say that mass is dimensional in one case and dimensionless in the other?

You are wrong because you are comparing the dimensionality of two different units in two different unit systems and incorrectly asserting that they must be the same. The dimensionality of a quantity depends on the system of units. It is a description of the units, not a description of anything physical.

The dimensionality of a unit is just as conventional as the size of the unit. The respective conventions are authoritatively set by the respective governing bodies. So $\mathrm{kg}$ has dimensions of mass because the BIPM says so, and $\mathrm{A_r}$ is dimensionless because the IUPAC says so. There is no deeper reasoning here.

 

[Grefus]

Ok, that seems getting closer to the issue I'm having.

So it all boils down to convention. It could be perfectly valid for IUPAC to call A_r dimensional (rather than dimensionless). But here is the problem: in most, if not all, cases the explanation goes that this mass is dimensionless not because IUPAC decided so, but because it represents the ratio of two dimensional quantities with the same dimension. And my argument is that it's no different in any other system (e.g., SI), where mass is taken to be dimensional. So, where am I wrong, again?

 

[PeroK]

I was thinking of currencies as an analogy. In general, currencies like US dollars, Euros or GB pounds are abstract. But, if we had a currency based on the price of an egg, say, and goods were valued in eggs, then that would no longer be an abstract measure but tied to something physical.

In that case $5$ would have units of dollars and a dimension of "value". Whereas, something worth $5$ eggs would be dimensionless, as it would be the ratio of the abstract value of the object to the abstract value of an egg.

The standard kilo is not directly the physical thing that all masses are multiples of. Instead, it represents the abstract quantity that is the kilogram.

I must confess it seems quite a subtle distinction to me.

 

[Dale]

the explanation goes that this mass is dimensionless not because IUPAC decided so, but because it represents the ratio of two dimensional quantities with the same dimension. And my argument is that it's no different in any other system (e.g., SI), where mass is taken to be dimensional. So, where am I wrong, again?

You are wrong for the same reason that I already said. Repeating the argument doesn't change its validity, regardless of what politicians try to convince everyone.

The explanation you mention is, roughly speaking, the IUPAC's justification for their decision. But the fact that you don't like their justification doesn't change their decision one iota. It is the decision that matters in the end.

Now, nothing is forcing you to use the IUPAC unit system in your own work, so if you don't like it then use a different system. You can even use Grefus units where you are the governing body and you can have your units have the same magnitude, but give it the dimensions of mass. It is their prerogative to make it dimensionless in their units and it is your prerogative to make it dimensional in yours.

 

[sophiecentaur]

But why then is it called nondimensionalization?

The dreaded Classification Monster rears its ugly head again.If the classification of something seems to disagree with Physics then the Physics is correct.

 

[Grefus]

The explanation you mention is, roughly speaking, the IUPAC's justification for their decision. But the fact that you don't like their justification doesn't change their decision one iota. It is the decision that matters in the end.

I have no trouble accepting that calling a quantity dimensional or dimensionless is a matter of convention. My question was about how this relates to the fact that this quantity represents a ratio. You never explicitely mentioned the word "ratio", but if I understand correctly, you say that this fact (being a ratio) is used (by IUPAC) as a justification for the quantity to be called dimensionless. Sorry, but I do not quite get it. If calling a quantity dimensional or dimensionless is a convention, why should it need justification? The more so that in another system of units it might be used to justify the opposite choice...

The atomic unit was taken just as an example. Let's switch to another example: N-body units used in astrophysical modelling. Here, the unit of mass is chosen such that the total mass of the stellar system is 1, and it is said that mass expressed in this system is dimesnionless, because it represents (again) the ratio of the mass of an object (e.g., a single star) over the total mass of all stars in the system. As far as I know, no institution like IUPAC decided this mass to be called dimensionless; everyone just accepts this as an obvious fact. No one comes in and says "Hey! I will call this mass dimensional and measured in the units of zelches, where one zelche equals the total mass of all stars in question". This is what bugs me most.

 

[Frabjous]

I would suggest that the Buckingham π Theorem and dimensional analysis can be used to provide insight between unit and dimension. I would say that relative atomic mass and the N-body mass are unitless but have the dimension of mass. I recognize that I am being a little bit sloppy.

 

[Grefus]

After reading the comments, at least I am a bit comforted by seeing that I am by far not the only one lost in confusion

 

[PeroK]

After reading the comments, at least I am a bit comforted by seeing that I am by far not the only one lost in confusion

It was an interesting question, IMO.

 

[Dale]

My question was about how this relates to the fact that this quantity represents a ratio.

Ratios of like-dimensioned quantities are dimensionless. There is no avoiding that. It is how unit dimensions work.

If calling a quantity dimensional or dimensionless is a convention, why should it need justification?

It doesn’t need a justification, but often giving a justification anyway helps motivate others to adopt the same convention.

Besides, most people are unaware of the fact that the dimensionality of a unit is a matter of convention. They tend to think that the SI dimensional conventions are physical and they struggle with cgs units, geometrized units, or other units with different conventions.

As far as I know, no institution like IUPAC decided this mass to be called dimensionless

That was Michel Henon’s decision. Conventions can be decided by individuals, not just groups of individuals.

No one comes in and says "Hey! I will call this mass dimensional and measured in the units of zelches, where one zelche equals the total mass of all stars in question". This is what bugs me most.

If it bugs you then you are free to do so yourself. You don’t need to wait for anyone else to do it. You have as much right as Henon to choose your conventions to your liking.

 

[Grefus]

I appreciate your attempts to clarify things for me, Dale, but the point still hasn't been hit. Sorry.

In particular, I find some of your statements self-contradictory.

On the one hand, you say "Ratios of like-dimensioned quantities are dimensionless. There is no avoiding that". - i.e. no conventions. Period.

On the other hand, you say "That was Michel Heron’s decision" - i.e., whether to call a quantity dimensional or dimensionless is a matter of convention, despite the fact that it is also defined as a ratio (normalization by the total mass).

If these are not self-contradictory, then there is something you fail to explain so that I could comprehend it :)

 

[Dale]

I find some of your statements self-contradictory.

On the one hand, you say "Ratios of like-dimensioned quantities are dimensionless. There is no avoiding that". - i.e. no conventions. Period.

On the other hand, you say "That was Michel Heron’s decision" - i.e., whether to call a quantity dimensional or dimensionless is a matter of convention, despite the fact that it is also defined as a ratio (normalization by the total mass).

I am not sure why you think those statements are self contradictory.

When someone, whether it is the BIPM, IUPAC, Heron, or you, decides to make a system of units they choose both the size and the dimensions of the units. Having done so, any time that you take the ratio of two like dimensioned quantities, the result is dimensionless. Dimensions can be chosen, and once chosen they follow the usual rules of dimensional quantities.

If these are not self-contradictory, then there is something you fail to explain so that I could comprehend it :)

In what way is any of that self-contradictory? Please be as explicit as you can. I am glad to help explain, but I need you to be more clear on what you are not comprehending. It would also help if you can focus on the concepts rather than your emotions. Whether something "bugs you" isn't something that we can help with, but where you are hitting a conceptual block is something that we can work on.

 

[DaveC426913]

A bit late to the game but:

For me, this sounds self-contradictory. Why having no unit, when, by very construction, 1/12 of C¹² mass is chosen as the unit?

Mass is not a unit; it's a property.

kg would be a unit, but C¹² does not make use of it. It is a valid ratio regardless of whether you're using kg, pounds or stone.

 

[sophiecentaur]

A bit late to the game but:

Mass is not a unit; it's a property.

kg would be a unit, but C¹² does not make use of it. It is a valid ratio regardless of whether you're using kg, pounds or stone.

That may be a way out of the problem.
I have already made the point that, whatever system you choose to operate by, somewhere in there the value of the mass in kg (or grains or pounds) must be hidden. If you want to calculate how many Joules you will need to perform something then there has to be some implicit or explicit value of the quantity Mass.

I challenge the OP to find a system of measurement in which actually does away with kg (or equivalent).

 

[Dale]

By the way, I looked at the official IUPAC document regarding their unit conventions, etc.: Quantities, Units and Symbols in Physical Chemistry.

On p. 4 they say "By convention physical quantities are organized in a dimensional system" and "The number and choice of base quantities is pure convention". So they are very clear about the conventional nature of dimensions. However, they have explicitly adopted the SI convention on units and dimensions.

The IAPUC does not consider relative atomic mass $A_r$ (defined on p. 47) to be a mass. It is just a specific quantity which is a ratio of masses $A_r=m_a/m_u$ where $m_u=m_a({}^{12}C)/12$. So given their conventions indeed $A_r$ is dimensionless even though the quantity that $A_r$ represents is measured using the same experiments as you would use to measure mass.

The reason this convention was chosen is that the SI $\mathrm{kg}$ has historically been rather unstable. The ipk, which historically defined the $\mathrm{kg}$, has varied significantly (several parts per million) in mass over the past century. If you can only determine the mass of the $\mathrm{kg}$ to within a few parts per million then any atomic mass expressed in $\mathrm{kg}$ can likewise be known at best to within a few parts per million.

However, a quantity like $A_r$ can be known to much higher precision. Exactly because it is a dimensionless ratio it is not subject to the instability in the mass of the ipk. Any variation in the size of the $\mathrm{kg}$ due to a variation in the mass of the ipk cancels out in the numerator and the denominator of $A_r$.