This paper presents mathematical foundations of the Overlap Resolution Number System (ORNS) which is based on signed Continuous Valued Digits (CVD's). ORNS is a redundant Number System employing residue arithmetic. In contrast to the implementation of arithmetic by binary or multiple-valued logic circuits, arithmetic operations in this novel number system are performed by analog digit manipulation circuitry. The redundancy in an ensemble of Continuous Valued Digits that comprises a number provides tolerance to implementation imprecisions. Processing with these analog digits is performed by carry-free arithmetic structures with systematic circuit level redundancy.