> dimensions are different from irrationals, vectors, complex numbers, etc, they are mostly a thing of their own. I harbor a terrible internal mental model of

> dimensions are different from irrationals, vectors, complex numbers, etc, they are mostly a thing of their own. I harbor a terrible internal mental model of dimensions which I have never really validated or explored fully, where I like to think they might be vector exponents, or something vaguely similar. If we assign each dimension to be a dimension of a vector - (length, mass, time, etc…) then a ‘distance’ might be e^((1,0,0,…)); a ‘duration’ e^((0,0,1,…)). These have the requisite properties that when we multiply and divide them, we end up adding and subtracting these vectors. So a distance times a distance is e^((2,0,0,…)) and a distance over a duration (a speed) is e^((1,0,-1,…)) They have the right basic algebraic behavior but who knows what terrible consequences they would have.