What is the algebraic difference between arithmetic operations, that prevents entities with different units from being summed or subtracted, but allows them to be multiplied or divided?
This looks more like a question for Physics, but lengths and areas, for example, are in the domain of pure mathematic.
Now, I cannot sum or subtract an area and a length, but I can multiply and divide an area with a length!
Reading Wikipedia, it looks like this is a property of the dimensions set. Does it just depend on the definition of the dimensions, or is it something intrinsic in the operations of add, subtract, multiply and divide?
Please explain with simple words, if possible.
