I did math and phsyics for undergrad, but there was no overlap - two separate departments. My BSc in math was nearly all pure/theoretical courses, as was the content of my undergrad thesis.
My MSc is in applied mathematics, but it was probably 50/50 theoretical courses and computer stuff. I love pure math, but decided to get the applied degree because I didn't to limit my job options to being a math professor. That's just not for me - I'd go insane being around those types on a daily basis. I'm just slightly too "normal" for it.
Pretty much any job one would get with a degree in math that is not teaching requires one to know how to code well (in a functional/computational sense). And much of such coding is based on numerical methods, computation theory, and a buch of other "approximating" stuff. So you gotta compromise a bit. Probably will do the same thing if/when I go for my PhD. Though keep in mind that applied mathematics needs to always be defended by pure mathematics, so an eager applied student will want to justify via theoretical mathematics why the applied stuff is allowed to be as it is. I got into many arguments with my master's thesis advisor about just how much of the theoretical stuff I could include in my paper.
My sister and her boyfriend both are seeking PhDs in pure mathematics (and I admit both are smarter than I am). So I am constantly around it, since I spend a lot of time with them, and they seem to never leave nerd mode.
But ya, I don't think I am a superfreak or anything. I am gifted with logical thinking skills and math has always came naturally to me. But something like playing an instrument - I took piano for 4 years and probably couldn't even play Marry had a Little Lamb. To me, writing and playing music blows my mind away. It is in a sense kind of relative.