Redefining Zero in Addition and Subtraction: From Neutral Element to Absence of Operation

Abstract

Zero is one of the most fundamental concepts in mathematics, serving as a central pivot in arithmetic, algebra, and number systems. However, its traditional definition as a neutral element in addition and subtraction, an absorbing factor in multiplication, and a problematic case in division has produced philosophical and educational inconsistencies for centuries. This paper proposes a new perspective that redefines zero not as a neutral element but as the absence of an operation. According to this view, adding or subtracting zero does not constitute an actual arithmetic operation; rather, no operation takes place at all. This perspective eliminates philosophical ambiguity, simplifies mathematics education, and establishes a more logically consistent framework for arithmetic.