Lighting, Light Fixtures, Ceiling and Exhaust Fans - Wiring problem

I have a ceiling fixture in the basement with no wall switch. I am replacing the fixture with one with a pull chain. There are 2 black wires that were connected together and 2 white wires that were connected together. I believe the power is coming from another box abut 12 feet away. The problem is that one of the white wires is very, very short. Yes I inadvertantly cut off too much. My question is can you safely attach another length of white wire to the shorter piece? If so how? Would I twist the 2 together and cover the 'splice' with electrical tape? Also, do I actually need to attach the 2 black and the 2 whites or can I just attach the new fixture to the 1 black and 1 white and cap off the other 2.
:wall: Any help would be appreciated.

Most likely since there is no wall mounted switch the power comes into the box and another cable carries the power on to another box. If you only connect one set of wires part of the circuit may go dead.

Yes, you can add a short lenght of wireto the piece that is too short. No, you just don't wrap the connection with electrical tape. You need to use an approved connector. Most commonly called wire nuts, they twist on to hold the splice securely.

On as side note the Code requires that there be 3" of conductor past the open face of the box for the smaller device size boxes.

You may want to check your local library for a basic wiring book to familiarize yourself with using the wire connectors.