#### Briefly about the problem of dimensionless physical quantities

The term "dimensionless quantity" according to the Note 1. Sec. 1.8 of International vocabulary of metrology JCGM 200:2012 "*kept for historical reasons*". In the vocabulary the term "quantity of dimension one" literally means "quantity of dimension 1". In Note 1 Sec 1.8 of JCGM 200:2008 said: "*The term “quantity of dimension one” reflects the convention in which the symbolic representation of the dimension for such quantities is the symbol* 1".

Analysis of this concept is directly related to the problem of generalization and systematization of physical quantities. Actually, the introduction of the dimensions and units for some "dimensionless quantities" in metrology is already a fait accompli, has been a lively debate as to how it all order. Since the site's author does not agree with the phrase itself "dimensionless quantities," it will continue to be quoted.

Follows from this definition that dimensionless quantities are relations of physical quantities with the same dimensions. As will be shown below, this definition does not include all of the "dimensionless quantities".

The article of L.Bryansky et al (1999) indicated that some metrologists to "*not operate with dimension equal to 1, put a dash in corresponding columns*". Clearly, this is not the solution of problem. In the same article states that there are "*examples in which the dimensionless quantities in one system of units are dimensional in another system*".

#### Classification of "dimensionless physical quantities"

As pointed M.Foster (2010), **SI** identifies four different groups of quantities with dimension 1:

1. various kinds of similarity criteria;

2. rotation angles and angular displacements;

3. numbers representing the numbers of entities;

4. logarithmic relationship.

Each of these groups should have its own decision regarding the used dimensions and units. At Note 2 Sec. 1.8 of dictionary JCGM 200:2012 said: "*The measurement units and values of quantities of dimension one are numbers, but such quantities convey more information than a number*". For example, in article of J.Kogan (1998) was invited to decipher the symbol of dimension 1. For example, the dimension of the Mach number are not written as 1 but as (LT^{-1})^{0}. However, the article of L.Bryansky and others (1999) the proposal was ridiculed.

1. **Similarity criteria** very different in nature, as are relations of different physical quantities (the ratio of forces, powers, intensities, velocities, pressures, area, temperature, etc.). And this difference should be reflected somehow. The International vocabulary of metrology JCGM 200:2012 on this occasion to have a cautious Note. 3 to sec. 1.8: "*Some quantities of dimension one are defined as the ratios of two quantities of the same kind*". Analysis of the similarity criteria used is given in a separate section of site.

On the criteria of similarity with by the term "dimensionless" there is a serious drawback. If the content of the physical dimensional quantities can form some idea of their dimensions, the physical content of "dimensionless similarity criteria" can be judged only on their defining equation (if provided) and indirectly by name criterion. But the name is it's just a word at best verbal formulation. She, too, to a certain extent arbitrary.

2. Solution for **angular quantities** has been a long time, but it is still seen, but not be accepted by for execution. This solution is that the rotation angle is the base physical quantity and it should have its own dimension. Unit for this quantity in physics is available, although it should be changed. Need only use this unit for all quantities of the rotational movement forms, not only for the angular velocity and angular acceleration. This decision is ground still in the article of J.Kogan (1998) and argued in detail in an article devoted rotation angle. Such a decision also suggested by M. Yudin (1998). It is supported in the article of M.Foster (2010), but indicated that such a decision "*would require a revision of the defining equations for coherent derived units of other angular quantities*". The debate has not yet led to the result.

3. Solutions for **numbers of entities** has several. One of them considered by the authors article devoted number of entities, proposes to make a number of entities the base physical quantity. A similar proposal is contained in the article of R.Dybkaer (2004). Discussion on this issue boils down rather to how to name unit of such quantity (I.Mills, 1995, T.Quinn and I.Mills, 1998, A.Mitrokhin, 2005).
In the International vocabulary of Metrology JCGM 200:2012 already appeared note 3 to sec. 1.4: "*Number of entities can be regarded as a base quantity in any system of quantities*". So that the permission is already there, but a practical solution yet not.

4. For **logarithmic relationship** rational decision yet not (M.Foster, 2010).

With respect to the angular quantities and the numbers of entities when they become, finally, the base quantities cannot be talking about the absence of dimension or of dimension equal 1. I.Yohanson (2010) suggests that such quantities should be called unitless, but the units they have. As adopted (radians), such (piece), about which there is a discussion.

Replacing the term "dimensionless quantity" to adequate terms for each group of such quantities would eliminate considerable terminological confusion. Can offer, such as a version of the definition: "*quantity with the dimension 1 - it's such a physical quantity, which, being a factor of any of the defining equation, has no effect on dimension formula of the quantity, which is determined by this equation*".

#### Conclusion

In an article on analysis units of dimensionless quantities, it is shown that *dimensionless quantities does not exist*. There are similarity criteria having both size and dimension. Similarity criterion dimension is the dimension of its basis (of the denominator of the equation defining criterion).

As shown in the above-mentioned article, **the basis of a physical quantity** are system and off-system units, as available in the adopted system of units, and are not available to her. Basis of the physical quantity can be fundamental physical constants, the characteristic parameters of a phenomenon or that type of technical devices, as well as the physical quantities characterizing certain phenomenon.

#### References

1. Брянский Л.Н., Дойников А.С., Крупин Б.Н., 1999, О “размерностях” безразмерных единиц. – Законодательная и прикладная метрология, **4**, с.с. 48-50. (Bryansky L.N., Doynikov A.S., Krupin B.N., 1999, The "dimensions" of dimensionless units. - Legislative and Applied Metrology, **4**, 48-50.)

2. Коган И.Ш., 1998, К вопросу о размерности и единицах измерений безразмерных физических величин. – Законодательная и прикладная метрология, **4**, с.с. 55-57. (Kogan J.Sh., 1998, On the dimensions and units of dimensionless physical quantities. - Legislative and Applied Metrology, **4**, 55-57.)

3. Митрохин А.Н., 2005, Качественная единица как элемент размерностного анализа или к вопросу о размерности ”безразмерных” величин. (Mitrokhin A.N., 2005, Qualitative unit as an element of dimensional analysis or to the question for dimensions of "dimensionless" quantities.) – http://www.metrob.ru/HTML/stati/kachestv-edinica.html

4. Чертов А.Г., 1990, Физические величины. – М.: Высшая школа, 336 с. (Chertov A.G., 1990, Physical quantities - M.: High School, 336.)

5. Dybkaer R., 2004, Units for quantities of dimension one Metrologia **41**, р.р.69–73

6. Foster M.P., 2010, The next 50 years of the SI: a review of the opportunities for the e-Science age. Review article. – Metrologia, **47**, p.р. R41-51.

7. Johansson I., 2010, Metrological thinking needs the notions of parametric quantities, units, and dimensions. Metrologia, **47**, р.р.219–230

8. Yudin M.F., 1998, The problem of the choice of the basic SI units. Meas. Tech., **4**, 873–875

9. JCGM 200:2012 International vocabulary of metrology – Basic and general concepts and associated terms (VIM). 3rd ed. 2008 version with minor corrections. URL: http://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf,

© J. Kogan Date of the first publication 01.06.2006

Date of last updating 12.02.2014