Editing Dimensional analysis (section)

=== Affine quantities ===
{{further|Affine space}}
Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. In mathematics scalars are considered a special case of vectors;{{citation needed|date=September 2013}} vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to define a position, this assumes an implicit point of reference: an [[origin (mathematics)|origin]]. While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change).

Consider points on a line, each with a position with respect to a given origin, and distances among them. Positions and displacements all have units of length, but their meaning is not interchangeable:
* adding two displacements should yield a new displacement (walking ten paces then twenty paces gets you thirty paces forward),
* adding a displacement to a position should yield a new position (walking one block down the street from an intersection gets you to the next intersection),
* subtracting two positions should yield a displacement,
* but one may ''not'' add two positions.
This illustrates the subtle distinction between ''affine'' quantities (ones modeled by an [[affine space]], such as position) and ''vector'' quantities (ones modeled by a [[vector space]], such as displacement).
* Vector quantities may be added to each other, yielding a new vector quantity, and a vector quantity may be added to a suitable affine quantity (a vector space ''[[Group action (mathematics)|acts on]]'' an affine space), yielding a new affine quantity.
* Affine quantities cannot be added, but may be subtracted, yielding ''relative'' quantities which are vectors, and these ''relative differences'' may then be added to each other or to an affine quantity.

Properly then, positions have dimension of ''affine'' length, while displacements have dimension of ''vector'' length. To assign a number to an ''affine'' unit, one must not only choose a unit of measurement, but also a [[Origin (mathematics)|point of reference]], while to assign a number to a ''vector'' unit only requires a unit of measurement.

Thus some physical quantities are better modeled by vectorial quantities while others tend to require affine representation, and the distinction is reflected in their dimensional analysis.

This distinction is particularly important in the case of temperature, for which the numeric value of [[absolute zero]] is not the origin 0 in some scales. For absolute zero,
: −273.15&nbsp;°C ≘ 0&nbsp;K = 0&nbsp;°R ≘ −459.67&nbsp;°F,
where the symbol ≘ means ''corresponds to'', since although these values on the respective temperature scales correspond, they represent distinct quantities in the same way that the distances from distinct starting points to the same end point are distinct quantities, and cannot in general be equated.

For temperature differences,
: 1&nbsp;K = 1&nbsp;°C ≠ 1&nbsp;°F = 1&nbsp;°R.
(Here °R refers to the [[Rankine scale]], not the [[Réaumur scale]]).
Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1&nbsp;°F / 1&nbsp;K (although the ratio is not a constant value). But because some of these scales have origins that do not correspond to absolute zero, conversion from one temperature scale to another requires accounting for that. As a result, simple dimensional analysis can lead to errors if it is ambiguous whether 1&nbsp;K means the absolute temperature equal to −272.15&nbsp;°C, or the temperature difference equal to 1&nbsp;°C.