Fundamental Units and Supplementary Units — What are the differences?

[Mr X]

Fundamental units are defined as - each of a set of unrelated units of measurement, which are arbitrarily defined and from which other units are derived.
while Supplementary units are - Supplementary units are the dimensionless units that are used along with the base units to form derived units in the International System.
But, both of the supplementary units can't be defined or expressed using any other units, which is the same as fundamental units.
And, a supplementary unit is not a necessity or a requirement to form derived units.

So, What exactly is the difference between fundamental and supplementary units?

 

[Baluncore]

Can you please list the supplementary units you are considering.

Supplementary units are usually the geometrical quantities of the circle and the sphere.

 

[DaveC426913]

What do you think of the various definitions that result from a quick Google search?

 

[Mr X]

Can you please list the supplementary units you are considering.

Supplementary units are usually the geometrical quantities of the circle and the sphere.

Steradian for solid angles and radian for angles.
And I did see the second sentence, but did not understand what exactly that meant. Does the fact that definition of these units are based on geometry change anything? since each of the fundamental forces are defined differently as per convenience.

 

[Mr X]

What do you think of the various definitions that result from a quick Google search?

the definitions I've put in there were all I could find. Most of the sites compare derived and fundamental units, But not fundamental and supplementary.

 

[Dale]

What exactly is the difference between fundamental and supplementary units?

Fundamental units are dimensionful and supplementary units are dimensionless in the SI. By the way, the “supplemental” designation is now discarded. The radian and steradian are now simply considered to be derived units

 

[Mr X]

Fundamental units are dimensionful and supplementary units are dimensionless in the SI. By the way, the “supplemental” designation is now discarded. The radian and steradian are now simply considered to be derived units

What exactly does dimension mean here ? What should I search to find more information? And since derived units are derived from fundamental units, How can steradian and radian be considered derived units?

sorry for the late reply

 

[Dale]

What exactly does dimension mean here ?

The term dimension refers (in this context) to the type of a physical quantity. Seconds, years, hours, and weeks are all different units that share the dimension time. They are commensurate quantities, meaning that any quantity you express in one can be expressed in another. 2 hours is exactly the same thing as 7200 seconds, but in SI units 2 hours is not the same as any number of kilograms because hours and kilograms have different dimensions (time and mass respectively).

And since derived units are derived from fundamental units, How can steradian and radian be considered derived units?

In the SI system the radian is a derived unit: $1 \ \mathrm{rad}=1 \ \mathrm{m/m}$

 

[Mr X]

The term dimension refers (in this context) to the type of a physical quantity. Seconds, years, hours, and weeks are all different units that share the dimension time. They are commensurate quantities, meaning that any quantity you express in one can be expressed in another. 2 hours is exactly the same thing as 7200 seconds, but in SI units 2 hours is not the same as any number of kilograms because hours and kilograms have different dimensions (time and mass respectively).

In the SI system the radian is a derived unit: $1 \ \mathrm{rad}=1 \ \mathrm{m/m}$

Got What dimensions are, thankyou

As for radians and steradians, they can be considered the name of a ratio rather than a unit then?

 

[Dale]

As for radians and steradians, they can be considered the name of a ratio rather than a unit then?

I don’t know what you mean by “rather than” here. Would you say that a katal is a ratio rather than a unit also? ($1\ \mathrm{kat}=1\ \mathrm{mol/s}$)

I would say that they are both a ratio and a unit, so I don’t understand the “rather than” which sounds like you are implying that ratios and units are mutually exclusive categories.

 

[Baluncore]

As for radians and steradians, they can be considered the name of a ratio rather than a unit then?

Angular units such as degree, radian and steradian are convenient geometric proportions of a whole. An angle is therefore a scalar ratio, without dimension. It would be nice if the world could remain simple and convenient, but sometimes it is necessary to accept more complexity.

The problem comes where an angle is specified by a number, then it must be accompanied by a unit. The angular unit implicitly identifies both the whole that was subdivided, and the number of subdivisions that were employed. An angle unit is therefore more than just a ratio.

It is sometimes convenient to treat angle, money, and information as fundamental dimensions since they have distinct units and conversion factors, but that elaboration is really only a data handling convenience.

 

[Mr X]

I don’t know what you mean by “rather than” here. Would you say that a katal is a ratio rather than a unit also? ($1\ \mathrm{kat}=1\ \mathrm{mol/s}$)

I would say that they are both a ratio and a unit, so I don’t understand the “rather than” which sounds like you are implying that ratios and units are mutually exclusive categories.

By a ratio, I mean that the units are canceled out. As for kat, this is the first time I've heard about it, but mol/s is a derived unit as per my understanding so they're different.
Sorry if it sounds stupid, but I still don't understand why they're considered derived units when there are no units as per the dimensional formula, but there is still a unit name.

 

[Mr X]

I think it might be more efficient and effective if anyone can provide any videos or simple articles about the topic, because I cannot comprehend the idea

 

[Mr X]

The problem comes where an angle is specified by a number, then it must be accompanied by a unit. The angular unit implicitly identifies both the whole that was subdivided, and the number of subdivisions that were employed.

I don't understand this, sadly.

 

[Dale]

By a ratio, I mean that the units are canceled out. As for kat, this is the first time I've heard about it, but mol/s is a derived unit as per my understanding so they're different.

Why would cancellation mean that something isn’t a unit? Either division is a legitimate operation on units or it isn’t. If unit division is not legitimate then the katal is not a legitimate unit, nor are Newtons, joules, hertz, or any other unit involving division. If division of units is a legitimate operation then m/m is legitimate.

I still don't understand why they're considered derived units when there are no units as per the dimensional formula, but there is still a unit name.

I think that you mean there are no dimensions. The radian is a dimensionless unit in the SI system.

The dimensions of a quantity is a matter of convention. A unit system can choose what it wants to use for the dimensions. The SI chose to make their unit of angle be dimensionless. There isn’t any deep reason that it has to be that way, it was simply a decision by a committee.

 

[Mr X]

Why would cancellation mean that something isn’t a unit? Either division is a legitimate operation on units or it isn’t. If unit division is not legitimate then the katal is not a legitimate unit, nor are Newtons, joules, hertz, or any other unit involving division. If division of units is a legitimate operation then m/m is legitimate.

Since the division of units are legitimate, we name derived units as per that, like kgm/s². But m/m is just one. Even if that is named as a unit, then shouldn't every unit that gives the result one have the same unit name? But that is apparently not the case since sterradian=m² / m²= 1, and radian = m/m = 1.

I think that you mean there are no dimensions. The radian is a dimensionless unit in the SI system.

The dimensions of a quantity is a matter of convention. A unit system can choose what it wants to use for the dimensions. The SI chose to make their unit of angle be dimensionless. There isn’t any deep reason that it has to be that way, it was simply a decision by a committee.

Alright, Got that part.[/SUP][/SUP]

 

[vanhees71]

If I remember right, there's some debate whether one should write $\alpha=\pi/4$ (as mathematically makes sense) or rather $\alpha=\pi/4 \text{rad}$ within the SI. The latter makes some sense, because then it's clear that the angle is expressed in radians, but on the other hand of course it's not a unit in the strict sense, because it's simply dimensionless, i.e., $\text{rad}=1$. Then also an equation like $45^{\circ}=\pi/4 \text{rad}$ is more clear.

 

[Dale]

shouldn't every unit that gives the result one have the same unit name?

No. For example, the SI units for torque and energy have the same dimensions, but we don’t use the joule as a unit of torque. Having the same dimensions is a necessary condition for the units to be the same, but it is not a sufficient condition. As another example, both a radioactive decay constant and a frequency have the same SI dimensions, but we don’t report decay constants in hertz. I am sure there are more.

 

[f95toli]

There is no simple answer to this because it still being debated within the metrology community
There are different views, but so far most seem to prefer to keep units for angles dimensionless

See section 9 in the report from the last CCU (consultative committee for units) meeting from September last year. Most participants seem to prefer to keep things as they are

https://www.bipm.org/documents/20126/63656783/25th+meeting/b485c4dd-19ec-61fb-b64a-780a17ad5666

However, the mere fact that this was brought up does illustrate that this is not a settled question,.

Note that this is yet another question that can be answered using purely technical arguments; to some extent this is always going to be a matter of opinion of what is the most convenient there is also always some political considerations.

 

[Lord Jestocost]

One should rely on the “Guide for the Use of the International System of Units (SI)” (SP 811) by the "National Institute of Standards and Technology" (NIST):
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwjgvd-nkcj2AhVRQ_EDHYGJB1IQFnoECAsQAQ&url=https://www.nist.gov/pml/special-publication-811&usg=AOvVaw0QhRbNBZMOxum-v6Dq0phg

 

[vanhees71]

This is the new one:

https://www.nist.gov/pml/special-publication-330

 

[Baluncore]

I prefer to keep angles dimensionless. That way I can evaluate transcendental functions using polynomials, without having to assign different dimensions to each coefficient, before evaluating and summing the terms.

 

[vanhees71]

Indeed, the SI is not always the most convenient choice, but indeed they explicitly say in the above quoted brochure that $\text{rad}=1$ ;-).

 

[Dale]

However, the mere fact that this was brought up does illustrate that this is not a settled question,.

Well, I would be a little more careful in this statement. It is a settled question for the current version of the SI, but it is something that may change for future versions of the SI. The discussion about possible changes in future versions of the SI does not make the current SI ambiguous or unsettled, but it does indicate that the committee recognizes that this is a topic where their current approach may change in the future.

I take a practical approach. When I am writing a paper or other document for public use then I use SI units and I treat the radian as dimensionless and equal to 1. But when I am privately doing a dimensional analysis using angular quantities (especially phase angles) then I often use "Dale units" which are just like SI units but where the radian has the dimension of angle and is often converted to cycles which also have the dimension of angle and the conversion factor of $2 \pi \ \mathrm{rad}=1 \ \mathrm{cycle}$. This is perfectly legitimate since even if a journal requires SI units in the publication I am free to use other units elsewhere, and since units are defined by humans and I am a human I can define my units as I like.

 

[Dale]

This is the new one:

https://www.nist.gov/pml/special-publication-330

Oh, this is good to see. I have been using the .pdf published by the BIPM directly, but the .html is much more convenient.

 

[berkeman]

This is the new one:

https://www.nist.gov/pml/special-publication-330

Sorry if I'm being dense, but what does the inner circle of symbols mean? I understand the outer circle of symbols as being fundamental units, but only the "e" in the current sector of the circle makes any sense to me...

 

[Dale]

what does the inner circle of symbols mean?

Those are the fundamental constants that define the unit. The second is defined by the frequency of the caesium hyperfine transition, the meter is defined by the speed of light, the kilogram is defined by Planck's constant, the ampere is defined by the charge on an electron, the kelvin is defined by Boltzmann's constant, the mole is defined by Avogadro's number, and I can never remember the candela.

 

[Motore]

Sorry if I'm being dense, but what does the inner circle of symbols mean? I understand the outer circle of symbols as being fundamental units, but only the "e" in the current sector of the circle makes any sense to me...

The International System of Units, the SI, is the system of units in which

• the unperturbed ground state hyperfine transition frequency of the cesium 133 atom ΔνCs is 9 192 631 770 Hz,
• the speed of light in vacuum c is 299 792 458 m/s,
• the Planck constant h is 6.626 070 15 × 10−34 J s,
• the elementary charge e is 1.602 176 634 × 10−19 C,
• the Boltzmann constant k is 1.380 649 × 10−23 J/K,
• the Avogadro constant NA is 6.022 140 76 × 1023 mol−1,
• the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz, Kcd, is 683 lm/W,

 

[f95toli]

Well, I would be a little more careful in this statement. It is a settled question for the current version of the SI, but it is something that may change for future versions of the SI. The discussion about possible changes in future versions of the SI does not make the current SI ambiguous or unsettled, but it does indicate that the committee recognizes that this is a topic where their current approach may change in the future.

Indeed, I just meant that it something that is still being debated internally among the NMIs. But the SI is indeed very clear about that fact that angles are dimensionless.