Is it possible to prove that units can be manipulated algebraically?

Question

With expressions such as

$$4\ \mathrm{\frac{m}{s}} \times 2\ \mathrm{kg} = 8\ \mathrm{\frac{m}{s}} \times 1\ \mathrm{kg}$$

We can justify that a $2\ \mathrm{kg}$ mass moving at $4\ \mathrm{m/s}$ has the same momentum as a $1\ \mathrm{kg}$ mass moving at $8\ \mathrm{m/s}$. This might make sense at an intuitive level, but is there a fundamental argument that says units can be manipulated algebraically such that in this case all we've employed is the commutative property?

Or take for instance when we have to do unit conversion and all we do is just cancel stuff according to some proportionalities. We're manipulating units algebraically.

Is this way of doing things blindingly obvious? Or did we have to go out there and find out that it works?

Also, if I have a differential equation such as

$$Lq''+Rq'+\frac{q}{c}=E(t)$$

I usually solve it as if it were an empty math problem and had no units. That is, I only care about the numbers involved. But how do I prove to myself that this way of doing things is right and won't produce contradictions in terms of its units?

Answer 1

I suppose that the best way to argue for this is to consider the units as indicative of the vector space from which the quantities in question originate. The algebra that we have defined in physics is one such that these quantities behave under the natural rules of commutative and associate multiplication, and so when we multiply quantities $m$ and $v$ (to stay consistent with your example above), we have a quantity with units $[m] \times [v]$, which is indicative of that quantity belonging to a different vector space than either of the two original spaces (specified by units $[m]$ and $[v]$). This is my interpretation, however.

It’s somewhat (crudely) analogous to the case of matrix multiplication; that is, you cannot add matrices of unlike dimensions. But, the product of two matrices (not necessarily commutative) can belong to a different vector space than the matrices originated from, in analogy to the above scenario.

Too long, didn’t read: units specify the vector/Hilbert (to be precise) space that to what a quantity belongs and it so happens that we use the unit algebra to specify exactly to what space the quantity in question belongs. We happen to treat them as nifty tools to help keep track of what space we’re landing in when we perform these sorts of operations.

On your second point, I don’t know exactly what you’re asking, but I know that whenever I’m working through a differential equation or any sort of calculation, I frequently check my units to make sure I’m not adding quantities of unlike units (i.e., of different vector spaces, to stay consistent), that I have arguments of transcendental/trigonometric functions that are unitless, etc. If you’re ever in a predicament in which you’re violating some of these rules of sorts, that’s a major red flag that something has gone wrong in your computation.