Could I deal in floats but use some math magic to make the top left of the screen 0.f,0.f and the bottom right 1920.f,1080f?
That's straightforward enough, I think. Assuming you're using shaders in OpenGL, you to use a projection matrix like the one documented here: http://msdn.microsoft.com/en-gb/library/windows/desktop/dd373965%28v=vs.85%29.aspx. If you're using fixed function OpenGL, then you can just use the glOrtho function directly.
The problem is that you might need to think about how to deal with different aspect ratios. Sometimes working to a 14:9 virtual resolution is best, then cut off bits or black bar it to cope with 4:3 and 16:9 - depends on your game really.