Neutrosophic Interval-Indeterminate Numbers of the Form a + bI: A Comparative Analysis with Interval and Affine Arithmetic, with Multi-Source and Hesitant Extensions

Abstract

Uncertainty representation in applied mathematics has evolved through classical interval arithmetic, affine arithmetic, and neutrosophic theory; yet no systematic comparison exists between neutrosophic interval-indeterminate numbers a + bI and affine arithmetic, despite their parallel development since 1993. This paper formalizes the comparison: in one dimension we prove range equivalence (Theorem 1); we propose a multi-source neutrosophic extension (MSNN) that achieves N-dimensional parity with affine arithmetic while preserving the semantic transparency of the a + bI notation (Theorem 2); and we introduce a hesitant extension to model expert disagreement. A computational benchmark on fifteen algebraic expressions is provided with runtime, range width, and overestimation percentages. The hesitant extension N_H = a + b · H(I) captures epistemic states produced by expert disagreement that neither interval arithmetic nor native affine arithmetic can represent. We conclude that neutrosophic and affine arithmetic occupy complementary positions: affine for engineering precision with multi-source tracking; neutrosophic for semantic transparency, broader qualitative scope, and natural hesitant-evidence consolidation. A reference Python implementation is released under MIT License at https://github.com/mleyvaz/neutrosophic-affine, enabling full reproduction of theorems, examples, benchmarks, and figures.