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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.

Right now I'm studying EM waves and I'm just solving Maxwell's Eqs to see certain things that must follow for an EM wave. All of this is done with letters like ##\epsilon_o## and standard arithmetical rules. Never do we worry about units. Man, I don't even know what the units for ##\epsilon_o## are. Nobody seems to care! Of course they are important, though… But this isn't my point.

I need a profound reason or something that could prove to me that units 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 and that whatever applies to dimensionless equations applies to dimensionful ones.

I don't know if my question even makes sense, so thank you if you've read this far.

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.

Right now I'm studying EM waves and I'm just solving Maxwell's Eqs to see certain things that must follow for an EM wave. All of this is done with letters like ##\epsilon_o## and standard arithmetical rules. Never do we worry about units. Man, I don't even know what the units for ##\epsilon_o## are. Nobody seems to care! Of course they are important, though… But this isn't my point.

I need a profound reason or something that could prove to me that units 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 and that whatever applies to dimensionless equations applies to dimensionful ones.

I don't know if my question even makes sense, so thank you if you've read this far.

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