Editing Dimensionless quantity (section)

== Ratios, proportions, and angles ==
Dimensionless quantities can be obtained as [[ratio]]s of quantities that are not dimensionless, but whose dimensions cancel out in the mathematical operation.<ref name="ISO 80000-1" /><ref>{{Cite web |title=7.3 Dimensionless groups |url=http://web.mit.edu/6.055/old/S2008/notes/apr02a.pdf |access-date=3 November 2023 |website=[[Massachusetts Institute of Technology]]}}</ref> Examples of quotients of dimension one include calculating [[slope]]s or some [[Conversion of units|unit conversion factors]]. Another set of examples is [[mass fraction (chemistry)|mass fraction]]s or [[mole fraction]]s, often written using [[parts-per notation]] such as ppm (=&nbsp;10<sup>−6</sup>), ppb (=&nbsp;10<sup>−9</sup>), and ppt (=&nbsp;10<sup>−12</sup>), or perhaps confusingly as ratios of two identical units ([[kilogram|kg]]/kg or [[mole (unit)|mol]]/mol). For example, [[alcohol by volume]], which characterizes the concentration of [[ethanol]] in an [[alcoholic beverage]], could be written as {{nowrap|mL / 100 mL}}.

Other common proportions are percentages [[%]]&nbsp;(=&nbsp;0.01), &nbsp;[[per mil|‰]]&nbsp;(=&nbsp;0.001). Some angle units such as [[turn (angle)|turn]], [[radian]], and [[steradian]] are defined as ratios of quantities of the same kind. In [[statistics]] the [[coefficient of variation]] is the ratio of the [[standard deviation]] to the [[average|mean]] and is used to measure the [[Statistical dispersion|dispersion]] in the [[statistical data|data]].

It has been argued that quantities defined as ratios {{nowrap|1=''Q'' = ''A''/''B''}} having equal dimensions in numerator and denominator are actually only ''unitless quantities'' and still have physical dimension defined as {{nowrap|1=dim ''Q'' = dim ''A'' × dim ''B''{{i sup|−1}}}}.<ref name="Johansson2010">{{cite journal |author-last=Johansson |author-first=Ingvar |title=Metrological thinking needs the notions of parametric quantities, units and dimensions |journal=[[Metrologia]] |volume=47 |issue=3 |date=2010 |pages=219–230 |issn=0026-1394 |doi=10.1088/0026-1394/47/3/012 |bibcode=2010Metro..47..219J |s2cid=122242959}}</ref>
For example, [[moisture content]] may be defined as a ratio of volumes (volumetric moisture, m<sup>3</sup>⋅m<sup>−3</sup>, dimension L{{sup|3}}⋅L{{sup|−3}}) or as a ratio of masses (gravimetric moisture, units kg⋅kg<sup>−1</sup>, dimension M⋅M{{sup|−1}}); both would be unitless quantities, but of different dimension.