Editing Dimensional analysis (section)

=== 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.