| irreducible units of measure are checked in the sciences |
Inches are not meters, and by value cannot be added, subtracted, printed, interchangeably nor directly, till converted, inches/meter.
(~39.37 inches/meter or ~.0254 meters/inch)
Some units are convertible, some similar, some related, some unrelated;--
cf The distinction of foot-pounds for energy as well as for torque: Their pounds of force are very similar, but one persists in motion and the other static; while their feet of distance are perpendicular: one being progress traversed under force; the other, remoteness leveraged, at transverse, under force.
Something tighter than units-monitoring is needed in base arithmetic, or the method of checking becomes "forgetful".
Units act much like advanced computer digital-signatures under conversion by multiplication .... Addition, subtraction, display, of numbers must have the same units or be converted in some cases, while multiplication and division cycle to generate new units: Eg. inches plus inches only, but inches*meters become inch-meters or meter-inches, which then can add, subtract, print, the selfsame.
* (The units themselves may be stored and processed logarithmically additively.)
In the plethora of units - inches of rain, inches of string, inches of cloth - some units are near-similar, as inches of string and inches of bow; some units are fairly distinct, as inches of cloth are not used to measure inches of rain, unless the cloth may be a peculiar fiber-glass cloth used in building a boat for floating in rain-collection ponds; some units are similar by constant conversion coefficient, or by multiple constant conversions, squares, cubes. The computer, for process simplification by content-free statistical isolation, can let each curious unit take some 'hash-value', and thence match or generate new values by products and conversions: Each classification-category may contribute to the units digital-signature: inches/feet/meters, rain/string/cloth ... cloth might be the square of string - as also square-inches is the square of inches - conversion of square-inches to square-meters is squared multiplications.
Such a scheme is coarsely equivalent to single-factor error correction.
Overall, units are like circuitry, and we can prove them, piecewise and uniformly.
Units-order distinction can be kept by non-commutable multiplication, eg. matrix product, at slight increase in computational cost.
[under construction]
In calculus we find further unit challenges: The transformation of bases requires a change of units, but not always seen in the formulation - and, derivative skills, beyond mere multiplicative.