Editing Dimensional analysis (section)

== Concrete numbers and base units ==

Many parameters and measurements in the physical sciences and engineering are expressed as a [[concrete number]]—a numerical quantity and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities; for example, speed is a combination of length and time, e.g. 60&nbsp;kilometres per hour or 1.4&nbsp;kilometres per second. Compound relations with "per" are expressed with [[Division (mathematics)|division]], e.g. 60&nbsp;km/h.  Other relations can involve [[multiplication]] (often shown with a [[centered dot]] or [[Juxtaposition#Mathematics|juxtaposition]]), powers (like m<sup>2</sup> for square metres), or combinations thereof.

A set of [[Base unit (measurement)|base unit]]s for a [[system of measurement]] is a conventionally chosen set of units, none of which can be expressed as a combination of the others and in terms of which all the remaining units of the system can be expressed.<ref>{{citation |author=JCGM |author-link=Joint Committee for Guides in Metrology |url=https://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf |title=JCGM 200:2012 – International vocabulary of metrology – Basic and general concepts and associated terms (VIM) |year=2012 |edition=3rd |access-date=2 June 2015 |archive-url=https://web.archive.org/web/20150923224356/http://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf |archive-date=23 September 2015 |url-status=dead}}</ref> For example, units for [[length]] and time are normally chosen as base units. Units for [[volume]], however, can be factored into the base units of length (m<sup>3</sup>), thus they are considered derived or compound units.

Sometimes the names of units obscure the fact that they are derived units. For example, a [[newton (unit)|newton]] (N) is a unit of [[force]], which may be expressed as the product of mass (with unit kg) and acceleration (with unit m⋅s<sup>−2</sup>). The newton is defined as {{nowrap|1=1 N = 1 kg⋅m⋅s<sup>−2</sup>}}.

=== Percentages, derivatives and integrals ===
Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since {{nowrap|1=1% = 1/100}}.

Taking a derivative with respect to a quantity divides the dimension by the dimension of the variable that is differentiated with respect to. Thus:
* position ({{math|''x''}}) has the dimension L (length);
* derivative of position with respect to time ({{math|''dx''/''dt''}}, [[velocity]]) has dimension T<sup>−1</sup>L—length from position, time due to the gradient;
* the second derivative ({{math|1=''d''{{i sup|2}}''x''/''dt''{{i sup|2}} = ''d''(''dx''/''dt'') / ''dt''}}, [[acceleration]]) has dimension {{dimanalysis|length=1|time=−2}}.
Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator.
* [[force]] has the dimension {{dimanalysis|mass=1|length=1|time=−2}} (mass multiplied by acceleration);
* the integral of force with respect to the distance ({{math|''s''}}) the object has travelled ({{tmath|\textstyle\int F\ ds}}, [[Work (physics)#Mathematical calculation|work]]) has dimension {{dimanalysis|mass=1|length=2|time=−2}}.

In economics, one distinguishes between [[stocks and flows]]: a stock has a unit (say, widgets or dollars), while a flow is a derivative of a stock, and has a unit of the form of this unit divided by one of time (say, dollars/year).

In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example, [[debt-to-GDP ratio]]s are generally expressed as percentages: total debt outstanding (dimension of currency) divided by annual GDP (dimension of currency)—but one may argue that, in comparing a stock to a flow, annual GDP should have dimensions of currency/time (dollars/year, for instance) and thus debt-to-GDP should have the unit year, which indicates that debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged.