Dimensionless quantity

Dimensionless quantities, also known as quantities of dimension one^[1] are implicitly defined in a manner that prevents their aggregation into units of measurement.^[2]^[3] Typically expressed as ratios that align with another system, these quantities do not necessitate explicitly defined units. For instance, alcohol by volume (ABV) represents a volumetric ratio. Its derivation remains independent of the specific units of volume used; any common unit may be applied. Notably, ABV is never expressed as milliliters per milliliter, underscoring its dimensionless nature.

The number one is recognized as a dimensionless base quantity.^[4] Radians serve as dimensionless units for angular measurements, derived from the universal ratio of 2π times the radius of a circle being equal to its circumference.^[5]

Dimensionless quantities play a crucial role serving as parameters in differential equations in various technical disciplines. In calculus, concepts like the unitless ratios in limits or derivatives often involve dimensionless quantities. In differential geometry, the use of dimensionless parameters is evident in geometric relationships and transformations. Physics relies on dimensionless numbers like the Reynolds number in fluid dynamics,^[6] the fine-structure constant in quantum mechanics,^[7] and the Lorentz factor in relativity.^[8] In chemistry, state properties and ratios such as mole fractions concentration ratios are dimensionless.^[9]

^ "1.8 (1.6) quantity of dimension one dimensionless quantity". International vocabulary of metrology — Basic and general concepts and associated terms (VIM). ISO. 2008. Retrieved 2011-03-22.
^ "SI Brochure: The International System of Units, 9th Edition". BIPM. ISBN 978-92-822-2272-0.
^ Mohr, Peter J.; Phillips, William Daniel (2015-06-01). "Dimensionless units in the SI". Metrologia. 52.
^ Mills, I. M. (May 1995). "Unity as a Unit". Metrologia. 31 (6): 537. doi:10.1088/0026-1394/31/6/013. ISSN 0026-1394.
^ Zebrowski, Ernest (1999). A History of the Circle: Mathematical Reasoning and the Physical Universe. Rutgers University Press. ISBN 978-0-8135-2898-4.
^ Cengel, Yunus; Cimbala, John (2013-10-16). EBOOK: Fluid Mechanics Fundamentals and Applications (SI units). McGraw Hill. ISBN 978-0-07-717359-3.
^ Webb, J. K.; King, J. A.; Murphy, M. T.; Flambaum, V. V.; Carswell, R. F.; Bainbridge, M. B. (2011-10-31). "Indications of a Spatial Variation of the Fine Structure Constant". Physical Review Letters. 107 (19): 191101. arXiv:1008.3907. doi:10.1103/PhysRevLett.107.191101.
^ Einstein, A. (2005-02-23). "Zur Elektrodynamik bewegter Körper [AdP 17, 891 (1905)]". Annalen der Physik. 14 (S1): 194–224. doi:10.1002/andp.200590006.
^ Ghosh, Soumyadeep; Johns, Russell T. (2016-09-06). "Dimensionless Equation of State to Predict Microemulsion Phase Behavior". Langmuir. 32 (35): 8969–8979. doi:10.1021/acs.langmuir.6b02666. ISSN 0743-7463.

[1] "1.8 (1.6) quantity of dimension one dimensionless quantity". International vocabulary of metrology — Basic and general concepts and associated terms (VIM). ISO. 2008. Retrieved 2011-03-22.

[SI_Brochure-2] "SI Brochure: The International System of Units, 9th Edition". BIPM. ISBN 978-92-822-2272-0.

[3] Mohr, Peter J.; Phillips, William Daniel (2015-06-01). "Dimensionless units in the SI". Metrologia. 52.

[4] Mills, I. M. (May 1995). "Unity as a Unit". Metrologia. 31 (6): 537. doi:10.1088/0026-1394/31/6/013. ISSN 0026-1394.

[5] Zebrowski, Ernest (1999). A History of the Circle: Mathematical Reasoning and the Physical Universe. Rutgers University Press. ISBN 978-0-8135-2898-4.

[6] Cengel, Yunus; Cimbala, John (2013-10-16). EBOOK: Fluid Mechanics Fundamentals and Applications (SI units). McGraw Hill. ISBN 978-0-07-717359-3.

[7] Webb, J. K.; King, J. A.; Murphy, M. T.; Flambaum, V. V.; Carswell, R. F.; Bainbridge, M. B. (2011-10-31). "Indications of a Spatial Variation of the Fine Structure Constant". Physical Review Letters. 107 (19): 191101. arXiv:1008.3907. doi:10.1103/PhysRevLett.107.191101.

[8] Einstein, A. (2005-02-23). "Zur Elektrodynamik bewegter Körper [AdP 17, 891 (1905)]". Annalen der Physik. 14 (S1): 194–224. doi:10.1002/andp.200590006.

[9] Ghosh, Soumyadeep; Johns, Russell T. (2016-09-06). "Dimensionless Equation of State to Predict Microemulsion Phase Behavior". Langmuir. 32 (35): 8969–8979. doi:10.1021/acs.langmuir.6b02666. ISSN 0743-7463.

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