Also--2's compliment? You must be joking. They don't emphasize knowing how to solder or replace vacuum tubes today, either.
While knowing 2's complement is probably not that useful, it shows an understanding of the internal workings of a computer, such as gates, adders, registers, memory, the clock, etc. Not knowing these things and working heavily with computers is like being a neuroscientist without understanding the basics of how the brain functions.
...and there isn't much left in the field for "work." There may be "research," into things but the average "job" is tedium.
Wow. First of all, there is no "may be"; there is definitely lots of research out there to be done, and some of it is very interesting (some is not, but that's always true of any field). And second of all, there is always work to be done. Computers are here and they're here to stay, meaning that there will always be employment for people who are well-qualified and want to work with them.
To respond to the OP, all I can do is reiterate (and poorly) what other people have said: figure out what you want to do, and then do whatever you have to to get there. If that means working in a call center by day and studying by night, then so be it. I've managed to get into the game industry (which is what I've always wanted to do) as a software engineer , but it took a lot of work, including knowing two's complement and how a hash table works.
The decision doesn't have to be logical; it was unanimous.