Pure Mathematics for Theoretical Computer Science PMTCS 2995-3162 10.54216/PMTCS https://www.americaspg.com/journals/show/4083 2023 2023 Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan Takaaki Takaaki Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050, KPK, Pakistan Arif Mehmood Department of Information Technology, Management Technical College, Southern Technical University, Basrah, 61004, Iraq Arkan A. Ghaib A variety of uncertainty-handling frameworks—such as Fuzzy Sets,1 Hyperfuzzy Sets,2 Bipolar Fuzzy Sets,3 Neutrosophic Sets,4 Vague Set,5 Hesitant Fuzzy Sets,6, 7 Picture Fuzzy Sets,8 Soft Sets,9, 10 Rough Sets,11 and Plithogenic Sets12, 13—have been extensively studied for modeling and reasoning under vagueness and imprecision. A fuzzy set extends classical set theory by assigning each element a membership value in the unit interval [0, 1], thereby capturing partial inclusion.1 Neutrosophic Sets further generalize this idea by introducing three independent membership functions—truth, indeterminacy, and falsity—each mapping into [0, 1]. Many of these frameworks have been enriched by incorporating offset concepts, which permit membership degrees to take values beyond the unit interval. Similarly, D-numbers extend Dempster–Shafer belief functions by assigning to each subset B ⊆ X a mass D(B) ∈ [0, 1] with P B D(B) ≤ 1, thus accommodating incomplete uncertainty.14 In this work, we introduce and formally define four new constructs: D-OffNumber, D-OverNumber, D-UnderNumber, and Neutrosophic D-Number, and we investigate their mathematical foundations, structural properties, and interrelationships. The present study focuses exclusively on theoretical development, leaving potential applications—such as their integration into decision-making frameworks—for future research. 2025 2025 48 61 10.54216/PMTCS.050105 https://www.americaspg.com/articleinfo/40/show/4083