Thanks for the reply. We have an open basement with posts every ten feet or so supporting "beams" (is that not right?) that run the length of the house. The floor joists run perpendicular to the beams and sit on top of them. If I were to add two new joists to replace the one I need to cut, they would span about ten feet, so I was thinking I would use 2 12ft 2x8's (to match existing floor joists) so I would have an extra foot on each end to extend well past the beam (sill?).
I hate blocking, do you suggest joist hangers or toenailiing?