# Talk:Dimensionless quantity

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## Atomic Mass

I think that "atomic mass" is not dimensionless and should be removed from the list. Atomic mass is measured in amu, atomic mass units. I am not knowledgable to know for sure. Can someone correct me on this if I am wrong, or correct the article if it is wrong? Thanks. --Atraxani

You're right - it's entirely dimensional, so i removed it. --Whosyourjudas (talk) 00:08, 14 Nov 2004 (UTC)
If it not express in units of mass (g, kg), but just in a relative way (amu or dalton (unit)), then it is surely dimensionless.--188.26.22.131 (talk) 11:21, 24 February 2014 (UTC)

Everyone I've said this to has disagreed entirely, but I'm not going to give up.

People like to elevate Avogadro's number as a "magical" number.

It is just the conversion factor between amu and g, nothing more. Both grams and amu are just arbitrary units of mass.

If you say the number is a dimensionless constant, than you have to say the convertion factor between all other units are too. In which case 2.45 cm/in is a dimensionless constant. And 4.45 lb/N is also a dimensionless constant.

There is a reason we don't include unit convertion factors here and instead on another page (see unit conversion); they are expressed in units over units, yet mass over mass should cancel out, but we aren't using the same units, so they're here to stay.

In other words, 2.45 isn't a dimensionless constant because it is expressed in cm/in. Similarly 4.45 isn't a dimensionless constant because it is expressed in lb/N. Finally, Avogadro's number isn't dimensionless either because it is expressed in amu/g. GWC Autumn 57 2004 13.20 EST

I'm convinced -- Tim Starling 05:01, Nov 18, 2004 (UTC)
I concur and do not need convincing. (If I were an extremist, I would do away with the candela and the mole.) – Kaihsu 14:30, 2004 Nov 18 (UTC)
No. You really misunderstand these things. They are expressed in units over units, yet mass over mass should cancel out, but we aren't using the same units, so they're here to stay. Yes, but what you're forgetting is that the entire thing is equal to ONE. Take 2.45 cm/in = 2.45 cm/1 in = 2.45 cm/2.45 cm = 1. It's equal to ONE. That's why you're allowed to multiply by it without changing the measurement! "2.45 cm/in" is literally equal to one. It is not only a dimensionless constant, it is a very specific dimensionless constant -- unity. Similarly, Avogadro's number is dimensionless.
Yes, and an 00 gauge in model railroading of 4 mm:1 ft or 4 mm/ft is also dimensionless, though not unitless and not equal to unity, but rather equal to 5/381 (more commonly expressed as 1:76.2). Gene Nygaard 06:27, 20 October 2005 (UTC)
I never claimed anything was "unitless" (or if I did, I was wrong to say that; I don't see it above). No quantity is "unitless", even dimensionless quantites have a unit, namely their unit is 1. And yes, 4 mm/ft is not equal to unity, I never claimed that all dimensionless quantities were equal to unity. Think about why it's okay in model railroading to multiply by something not equal to unity -- the model is much smaller than the real thing!! Whereas in science, you typically do change of units which are on the same scale.
Dimensionless numbers are often unitless; I was merely pointing out that that need not be the case. However, even if the units in the numerator and the denominator are the same (or differ only in metric prefixes), it is often a good idea to include them even if they are not necessary. An expansion of 3×10-6 m/m is a convenient and compact way to identify the expansion as linear expansion. A concentration of argon in air expressed as either 9.34 mL/L or as 12.9 g/kg tells us whether the comparison which ends up dimensionless is being made on the basis of volume or of weight. Gene Nygaard 06:56, 20 October 2005 (UTC)

### Amount of substance

Amount of substance is also dimensionless or pseudo-dimensional, although it appears dimensional. It can be compared to angle whose unit radian is a dimensionless unit. Mole has a similar dimensional status with radian. Amount is the ratio of two masses: a given value of mass of a substance divided by the molar mass which is a material/substance constant. Molar mass itself is the expression in units of mass of relative molecular mass which contains Avogadro number of particles.--188.26.22.131 (talk) 11:09, 24 February 2014 (UTC)

## Clarification needed

The article contains the following two statements:

1. A dimensionless number has the same value regardless of the measurement units used to calculate it. It has the same value wherether it was calculated using the metric measurement system or the imperial measurement system.
2. However, a number may be dimensionless in one system of units (e.g., in a nonrationalized cgs system of units with the electric constant ε0 = 1), and not dimensionless in another system of units (e.g., the rationalized SI system, with ε0 = 8.85419×10-12 F/m).

How can these two statements be reconciled? -- The Anome 23:57, Jan 5, 2005 (UTC)

I added the second one, and was trying to think of a good way of doing that also. Other comments would be appreciated.
It has to do with an entirely different system of units; with the choice of the number of base units to use, and how they are related to each other, and things like that. Switching from feet and pounds to meters and kilograms won't change the magnitude of a dimensionless number expressed in those units, as long as they are "compatible" (something I'm just using, not being able to think of a better term right now) systems of units. But a more fundamental change is how the system of units works can change the fact of whether or not a particular value is dimensionless. This doesn't really make all that much sense even to me; I have a general idea where I'm going, but am having difficulty explaining it concisely. Gene Nygaard 00:55, 6 Jan 2005 (UTC)
You're having difficulty explaining it because you're wrong. The reconciliation is that you are wrong to say the latter is not dimensionless. See the comments above.

### dimensionless ≠ pure

Our definition says: "a dimensionless number (or more precisely, a number with the dimensions of 1) is a pure number without any physical units". This is only superficially true. Beside the example given by The Anome there is an easier case, as Pdn pointed out on Category_talk:Dimensionless numbers:

The example given by Anome is not a counterexample, for reasons stated above. 216.167.142.251
'fraid "pH" is not dimensionless at all - as I remember (from Chem 2 at Harvard, 1953) it is the negative logarithm of the hydrogen ion concentration in moles per liter. I think it's based on the common logarithm, not the natural one. So it involves moles and liters, which are based on dimensional standards. OK, a logarithm is "dimensionless" perhaps to a mathematician, but not really. It a logarithm were always dimensionless we could take the distance to the Sun, about 1.5 *10^11 meters, take its logarithm (a little over 11), and claim the distance to the Sun is dimensionless. [...]

One might object that pH should be measured in bel, but we can't set standards for chemistry here. Fact is that our intruduction contains a wrong statement. If you're a quantity, having dimension 1 (or 0?) is not equivalent to being "pure" – you may still have physical units in your definition. — Sebastian (talk) 18:18, 2005 May 21 (UTC)

No, you're wrong. 100 cm/1 m has NO units, there is no such thing as a "unit" of "cm/m". This "unit" is equal to the dimensionless constant 1/100.
Dimensionless numbers can have units. A simple example: an angle is measured in radians or degrees. The angle is always a number with dimension 1, but radians and degrees are different units. Units tell you in which system of measurement one works. You may throw in that both are only "supplementary units" but this concept does not exist anymore since 1995. -- 86.32.108.65 (talk) 16:16, 31 January 2010 (UTC)

## Dumb questions from a newbie

Isn't any cardinal number dimensionless? For example, four, or twenty six? If "dimensionless numbers" refers to ratios which have been found to be useful in some discipline, wouldn't pi count? Perhaps someone can write an introductory paragraph to this article, which would explain this category of numbers a little better? Thanks! --24.190.141.119 14:37, 19 Apr 2005 (UTC)

That's not a dumb question at all. You're making three valid points:

1. We should speak of quantity in the introduction.
2. Currently, "this category of numbers" isn't explained very well. I'd circumscribe it as "interesting physical quantities that are "pure" (as defined above)". The word "interesting" is fuzzy; what i mean is: If you count peas for an Experiments on Plant Hybridization you might find very interesting numbers, but they are not interesting beyond the scope of your experiment.
3. There are a lot of numbers that this article isn't concerned about. Pi is interesting, and i don't see any reason not to include it here other than mere historical usage of the term and that it already has enough attention from mathematicians. — Sebastian (talk) 18:38, 2005 May 21 (UTC)

## number with the dimensions of 1 - shouldn't this be 0?

For all i know, dimensionless numbers are rather measured in m0 than in m1. But it's in such a prominent place in this page and this whole term is so fraught with inconsistent usage anyway that i rather ask before changing it. — Sebastian (talk) 18:55, 2005 May 21 (UTC)

Maybe this was meant as dimension(dimensionless numbers) = dimension(1) = 0 ? Still, the wording is misleading. — Sebastian (talk) 00:06, 2005 May 22 (UTC)

## A simple exercise

I don't know if this will help, but in my experiance a simple example often lends to better understanding of concepts. Consider for example Strain. Strain is considered to be a dimensionless quantity (as opposed to a dimensionless constant such as pi), and is defined by $delta L/L$, with L being a unit of length. So long as the units are kept the same in the numerator and denominator, then it doesn't matter whether you measure in metres, feet, or even snorkles, all that matters is that A), the units are identical in the equation, and B), that the unit used is well defined.

Another property which I feel has been neglected here, is what happens to other quantities and specifically their units when multiplied/divided by a dimensionless quantity. For example: If we take one unit (length) and divide it by another unit (time), the result will be in a unit (in this case speed) which is defined as $length/time$. However if we do the same thing, but replace one of the units with a dimensionless quantity, this no longer happens.

A simple example of this is in the equation for determining the Modulus of Elasticity of a material, which is defined as $stress/strain$. Stress here can be defined as load/area, which has dimensions to it. Going back earlier, I pointed out that strain is a dimensionless quantity, so therefore when we divide stress/strain, the resulting unit remains the very same stress unit that was used in the formula.

No, they are not the same unit. People are confusing units with dimensions. Feet and metre have the same dimension, but they are not the same unit. Similar for radian and degree, or second and minute.

Any mathamaticians out there would be able to clarify this in a way that follows a bit more convention (my math is pretty poor), but nonetheless I feel that these simple examples would help out the conceptual understanding. --Sjkebab 01:39, 3 Jun 2005 (UTC)

In this case, it would seem they have the same units, but obviously will differ by some constant multiple which for the scientists has some important physical meaning. This doesn't change the fact that the dimensions are identical.

## Rename to "Dimensionless quantity"

Encouraged by the comment of one contributor above, who suggests that "quantity" should be referred to in place of "number" in the introduction to this article, I propose that in fact the entire article (including its title) should be revised to refer to dimensionless quantities. All numbers are dimensionless. It is the physical quantity, expressed by a number and its unit, that may be dimensionless. Contradictions invited.

I concur

The article as it exists now is a conflation of the two somewhat separate ideas, that of a dimensionless numbers which are used to measure some property, and that of particular dimensionless numbers which are dimensionless physical constants rather than quantity (which I'd take to be something like "mass" for something measured in units of "pounds"). "Dimensionless quantity" is a less interesting concept, maybe applying to things like the abolishment of the class of "supplemental" units in SI and instead describing them as "derived units" of the quantity "one".
Just look at all the dimensionless numbers in the lists on this article's page which have the word number in their own article titles. That is a pretty good indication that the current title is a reasonable one. I say keep it here Gene Nygaard 07:27, 20 October 2005 (UTC)

I agree with the move

1. a number is always dimensionless, therefore "dimensionless number" is an unfortunate title. "Dimensionless quantity" or even "dimensionless physical quantity" is much better and also according to the international conventions (see IUPAC green book).
2. a physical quantity in general consists of a (dimensionless) number and a unit and thereby has a dimension
3. in special cases, a physical quantity has no dimension and also no unit and thereby becomes a pure number. Example: Reynolds number.
4. in other cases, a physical quantity has no dimension but still has a unit. Example: 1 mg/kg = 1 ppm

## Rockwell Hardness Scale

I would argue that Rockwell Hardness is not a true dimensonless quantity because without knowing the scale used it is useless. For that reason the scale itself becomes a dimension. For example the hardness number is referred to by the scale used, e.g. 60 HRB, which becomes a unit itself. Reading the way these are derived, the actual number is a constant minus a depth in mm, so the actual unit seems to be mm. If others concur we should remove this from the list.

You seem to be conflating unit with dimension. For example: length is a dimension which can be represented in different units such as meters. A scale in this context is just a method to assign meaning to otherwise arbitrary quantities. This is thus distinct from the concept of dimension.--217.84.29.188 (talk) 13:15, 20 July 2011 (UTC)

## E&M

Without being an expert on E&M, in regard to

However, a physical quantity may be dimensionless in one system of units and not dimensionless in another system of units. For example, in the nonrationalized cgs system of units, the unit of electric charge (the statcoulomb) is defined in such a way so that the permittivity of free space ε0 = 1/(4π) whereas in the rationalized SI system, it is ε0 = 8.85419×10-12 F/m.

I would say that you must be measuring different things here. In other words, these two quantities must be defined in different ways that give them different dimensions. There is no mathematical reason to explain how one physical quantity could have 2 different dimensions. The reason must be something to do with the physics.

## Notation: italics and brackets

Please, 217.84.175.39, stop messing around in the several "dimension" related articles. The square brackets mean "dimesion of" and in the articles where we have used italics, it is because they have been so used, at least, in the last 50 yrs. If there are "new" rules, show them . --Jclerman 11:03, 28 July 2006 (UTC)

did you even notice what I edited? The other ones were mistakes, which I accepted - no reason to pull out the lobe (is this the right expression in english?) -- 217.84.175.39 20:19, 28 July 2006 (UTC) PS: did I miss any other of your helpfull contributions?

## section Dimensionless Phyical Constants is a misunderstanding

In my understanding, this section is completely wrong. In the Planck system, constants like c are used as units, but this does not make them dimensionless. If you look up the Planck units article, the dimensions of all constants are even indicated. In case of c, the dimension is L/T. Therefore, those constants are by no means dimensionless and the section should be deleted.

What happens is the following: instead of the "normal" system of base quantities (which you find in the article about physical quantity) the Planck system uses another system of base quantities which is based on natural constants. That's all. It is also not true that the quantities have no units, as many people think. They have units but they unfortunately are not used. For example, when indicating a velocity v in Planck System the unit of v is c ([v] = c). Most people, however, do not say "v = 0.8 c", they only say "v = 0.8" which is convenient but lazy and inaccurate.

The notion that in the Planck system the constants are "eliminated" is also inaccurate. What in short is called a physical constant is more precisely a constant physical quantity. A physical quantity Q, however, consists of two things, a numerical factor {Q} and a unit [Q]. In the case where the numerical factor {Q} = 1 the quantity Q is by no means eliminated, it still has the unit [Q]. (Otherwise one could say that the original meter in Paris would be "eliminated" because its numerical factor is also 1, but in fact we need it because of its unit, the meter.)

--Kehrli 12:56, 31 July 2006 (UTC)

## Pure number redirects here.

I had never heard the term "pure number" until about a month ago (I had heard of dimensionless though), the mark scheme for a CCEA A level physics question "explain why the quantity x (or whatever it was actually called) does not have units" said that the correct answer was the obvious "it's in a log" and the weirder terminolgy "it is a pure number", so I assumed pure = dimensionless. So now pure number redirects here. Stuart Morrow 00:11, 30 December 2006 (UTC)

## Normalised Moment

Would it be correct to add normalized moments (see moment) to the list of dimensionless quantities? EverGreg 13:08, 20 September 2007 (UTC)

## Need a better example than radians

The example, given at the article's beginning, of radians as a dimensionless quantity is very confusing. While technically expressing a ratio, radians function a lot like units and are often seen combined with units (e.g. rad/s rather than Hz). It seems like the example is more complex and confusing than the concepts it's trying to clarify. jdg (talk) 19:40, 29 February 2008 (UTC)

radians are a unit of angle, and angle is a dimensionless quantity. This is what the article should say. I cannot think of a good way to write this 64.91.186.214 (talk) 16:29, 7 March 2008 (UTC)
I tried anyways, but it just seems to have made it more complicated... 64.91.186.214 (talk) 16:40, 7 March 2008 (UTC)
Using the illustration in the radians article might help EverGreg (talk) 22:32, 7 March 2008 (UTC)

Cannot radians be expressed with a small "r". I know it's optional, but couldn't one be put in parens or something. Asmeurer (talkcontribs) 15:15, 16 October 2008 (UTC)

Should not Chandrasekhar number be a part of list? ~Divij (talk) 11:17, 25 March 2009 (UTC)

Yes. You can just add it. -- Crowsnest (talk); 12:01, 25 March 2009 (UTC)
Finally did it after nearly 3 years ~Divij (talk) 11:24, 19 January 2012 (UTC)

## Berchak number

I removed Berchak number from the list, since both an article on the subject and reliable sources are missing. It can be re-instated once this entry is verifiable. -- Crowsnest (talk) 19:03, 15 April 2009 (UTC)

## Decibel

Regarding the table, entry "Decibel", "ratio of two intensities, usually sound". 1) There is a non-linear transformation involved (log, base 10), not just the ratio 2) dB is also commonly used for ratios in electronic systems, e.g. -3 dB frequency cutoff (ratio between input and output signal, possibly normalised) and signal-to-noise ratio. --Mortense (talk) 18:27, 15 November 2010 (UTC)

## First paragraph example - is it correct?

for instance, the number of people N in a room is a dimensionless quantity.

Wouldn't the dimension of the number people in a room be "people" or "persons"?

If I were calculating the ratio of apples to people in a room, and I had, say, 10 apples and 5 people, I would have 2 apples/person. Ktwombley (talk) 17:49, 4 February 2011 (UTC)

The key word here is "number" of people. Numbers themselves have no dimension. Your ratio is dimensionless. Sometimes they are considered as having a dimension for the purpose of dimensional analysis, for example to make sure conversion factors are applied correctly for dimensionless quantities. As an example of this: say I have a variable measured in degrees and another measured in radians; comparing them directly would be possibly meaningless even though they are dimensionless. To catch this mistake, I treat the units much like dimensions so I can make sure that the units are converted where appropriate.--217.84.29.188 (talk) 13:31, 20 July 2011 (UTC)

## unit versus dimension

People seem to be confused by the difference between these two related concepts. Any suggestions for how to make the article address this better?--217.84.29.188 (talk) 13:17, 20 July 2011 (UTC)