Editing Dimensional analysis (section)

=== Combining units and numerical values ===
{{main|Physical quantity#Components}}

The value of a dimensional physical quantity {{math|''Z''}} is written as the product of a [[Unit of measurement|unit]] [{{math|''Z''}}] within the dimension and a dimensionless numerical value or numerical factor, {{math|''n''}}.<ref name=Pisanty13>For a review of the different conventions in use see: {{cite web |url=http://physics.stackexchange.com/q/77690 |title=Square bracket notation for dimensions and units: usage and conventions |last1=Pisanty |first1= E|date=17 September 2013 |website=Physics Stack Exchange  |access-date=15 July 2014}}</ref>
: <math>Z = n \times [Z] = n [Z]</math>

When like-dimensioned quantities are added or subtracted or compared, it is convenient to express them in the same unit so that the numerical values of these quantities may be directly added or subtracted. But, in concept, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 metre added to 1 foot is a length, but one cannot derive that length by simply adding 1 and 1. A [[conversion factor]], which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed:
: <math> \mathrm{1\,ft} = \mathrm{0.3048\,m}</math> is identical to <math> 1 = \frac{\mathrm{0.3048\,m}}{\mathrm{1\,ft}}.</math>

The factor 0.3048&nbsp;m/ft is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to the same unit so that their numerical values can be added or subtracted.

Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units.