All my life the approach has been as follows:
In math class I learn the rules and almost always deal with purely numerical problems.
In physics class I apply the things learned in math class but this time our quantities have units. Now, once the equation is well put and has dimensional homogeneity the problem is magically reduced to a purely numerical problem like the ones dealt with in math class. You might say that units do not actually vanish and it's actually just for practical reasons that teachers choose not to write them down, but I would like to see an explanation as to why this is OK? (Ignoring the units).
For instance, in my mathematical physics class I learn about vector calculus and vector analysis. First I learn it abstractly in a mathematically rigorous manner mostly dealing with no numbers because we derive the equations and theorems via the manipulation of letters. Now, I'm supposed to apply the exact same theorems and the same reasoning for problems that are dimensionful and this tears me apart for some unknown reason.
I need a profound reason or something that could prove to me that units are shouldn't bother me at all. I kind of see dimensionless and dimensional math as somehow separate, so I need a good argument that tells me this shouldn't be my way of seeing things.
I don't know if my question even makes sense, so thank you if you've read this far.
Best Answer
We can think of units in a formal sense: they can be multiplied and divided, but different units cannot be added. In one way, this is like working in a space of real numbers where each axis represents a power of a dimension, those axes are closed under addition, and that multiplication sends us to a different axis (i.e. you can do meters + meters, but not meters + meters-squared). So in a sense, you could form a basis from the fundamental units and their powers -- actually, this is pretty much what is happening in the Buckingham Pi theorem! Of course, we have a unitless basis element as well.
Nevertheless, it's not really worth worrying about. As I said, we needn't worry about $\mathbb{C}^5$ if we're trying to compute $22 \times 58$. So as far as profound reasons go, that's about as profound as I can make it -- the terms dimension and unit are used for precisely that reason. Operating on them allows us to work in a dimension with a unit basis where certain very-familiar rules hold.
Another analogy: if you walk 5 meters in one direction, you walk 5 meters. If you walk 5 meters in a different direction, you walk 5 meters. It may be eventually important to know whether you're walking east or south, but for the scope of simply wanting to know how far you've walked, we need not even care that east or south exist!