New Number System: An Extension of Binary Arithmetic

Abstract. This paper introduces $\mathbb{NS}_4$, a novel number system that extends traditional binary arithmetic by incorporating algorithmic mappings for addition, carry-over, and multiplication operations. We establish a mathematical framework demonstrating that $\mathbb{NS}_4$ forms a commutative ring, thereby extending beyond conventional number systems like complex numbers. The proposed system offers efficient representation and manipulation of numerical structures, with potential applications in computational mathematics and digital logic design.