Notes on the Synthesis of Form
RRP: £23.75 Paperback: 216 pages; Publisher: Harvard University Press; New Ed edition (1 July 1974) ISBN: 9780674627512
A physical quantity can be expressed as the product of a number and a physical unit. A formula expresses a relationship between physical quantities. A necessary condition for a formula to be valid is that all terms have the same dimension, meaning every term in the formula could be potentially converted to contain the identical unit (or product of identical units).
There is vast educational training about retaining units in computations, and converting units to a desirable form, such as in units conversion by factor-label.
The vast majority of computations with measurements are done in computer programs with no facility for retaining a symbolic computation of the units. Only the numerical quantity is used in the computation. This requires that the universal formula be converted to a formula that is intended to be used only with prescribed units, meaning the numerical quantity is implicitly assumed to be multiplying a particular unit. The requirements about the prescribed units must be given to users of the input and the output of the formula.