The unique number is what we'd call a surrogate key (as opposed to a natural key, which would be some part of the actual data).
It is almost always advisable to use a surrogate key, the cases where it isn't are vanishingly small.
Also, I would recommend against using an auto-increment column. It would be much better to use a GUID. The main problem with an auto-increment is that you must insert the row into the database before you can know what the surrogate id is. With GUID's, you can safely generate the surrogate id without consulting the database. It will be difficult to change later if you go with an auto-increment column now.