Quantity with no physical dimension
Dimensionless quantities , or quantities of dimension one,[ 1] are quantities 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 value remains independent of the specific units of volume used, such as in milliliters per milliliter (mL/mL).
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): 537–541. Bibcode :1995Metro..31..537M . 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 . Bibcode :2011PhRvL.107s1101W . doi :10.1103/PhysRevLett.107.191101 . PMID 22181590 .
^ 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 . PMID 27504666 .