I feel like this is mostly an artifact of notation. The thing that is not allowed with addition or subtraction is simplifying to a single term; otherwise it is fine. Consider:
10x + 5y −5x −10y = 10x − 5x + 5y −10y = 5x − 5y
So, everyone reasons to themselves, what we have here is two numbers. But hark, with just a little more information, we can see more clearly we are looking at a two-dimensional number:
5x − 5y = 5
5x = 5y +5
5x − 5 = 5y
x − 1 = y
y = x − 1
Such as a line.
This is what is happening with vectors, and complex numbers, quarternions, etc.
Brainteaser for anyone who doesn’t regularly think about units.
Why is it that I can multiply or divide two quantities with different units, but addition or subtraction is generally not allowed?
According to AnthonyC, units, for some reason, behave like variables in algebra, e.g.
2x×4y simplifies to (2×4)xy, with arithmetic this computes to 8xy.
2x4y simplifies to 24xy, with arithmetic this computes to 0.5xy.
2x+4y or 2x−4y don’t simplify so that we could perform addition or subtraction.
So this algebraic behavior does indeed look like what we would expect from units.
For those who would like a hint.
In English, “And” generally indicates addition, “Per” division.
Now consider which of the following makes sense:
Ferrets and seconds
Ferrets per second
I feel like this is mostly an artifact of notation. The thing that is not allowed with addition or subtraction is simplifying to a single term; otherwise it is fine. Consider:
10x + 5y −5x −10y = 10x − 5x + 5y −10y = 5x − 5y
So, everyone reasons to themselves, what we have here is two numbers. But hark, with just a little more information, we can see more clearly we are looking at a two-dimensional number:
5x − 5y = 5
5x = 5y +5
5x − 5 = 5y
x − 1 = y
y = x − 1
Such as a line.
This is what is happening with vectors, and complex numbers, quarternions, etc.