some numbers^* cannot be meaningfully multiplied or divided. For example, 3^rd place, 30° F, or 37.388N,-122.067S.
Some numbers can only be added to by numbers of a different type. For example x°C + y°K, 3rd grader + year, absolute position + offset. Similarly, subtracting two of these numbers results in a different unit. e.g. position - position = offset
it would be great if someone could comment the exact terminology for what I'm trying to describe.
* Types/Units
- nominal scales (e.g. man/woman),
- ordinal scales (elementary school/middle school/high school),
- interval scales (degrees Celsius) and
- ratio scales (human height, degrees Kelvin).
Only ratio scales may be multiplied and divided. Interval scales may be subtracted and added. Ordinal scales may only be compared (less/greater/equal). Nominal scales may only be compared in terms of (in)equality.
You may also be looking for fields: https://en.m.wikipedia.org/wiki/Field_(mathematics). You could probably also describe the relation between different number types as linear operators over a vector space
There are libraries that implement this concept in other programming languages (C++, Java, C#, etc.)