Limitations of the Argand Diagram Paradigm in Thermodynamic Phenomena: Development of New Lemmas in Complex Variable Theory

Abstract

The complex number and other similar expressions are commonly represented using the Argand diagram, a geometric representation introduced by Argand. Based on this representation, the Cartesian coordinate system is extended into what is known as the Argand plane, where complex numbers are visualized as points in a two-dimensional plane. This interpretation, however, reveals certain conceptual limitations in the traditional understanding of the Argand diagram, particularly when considering the geometric interpretation of planes in relation to algebraic sign conventions. In this context, complex numbers of the forms and are located in the first, fourth, second, and third quadrants respectively. This raises an important question regarding whether these representations correspond to a single unified plane or whether meaningful distinctions exist among them in terms of mathematical and physical interpretation. Despite the widespread acceptance of the Argand plane as a valid representation of complex numbers, limited attention has been given to such ambiguities in its geometric interpretation. While it effectively supports the notion that is a valid numerical entity and that complex numbers can be represented similarly to real numbers on coordinate axes, certain conceptual gaps remain unresolved. This paper addresses these issues by introducing a set of seven newly developed lemmas aimed at providing deeper insight into the behavior and interpretation of complex numbers across different quadrants of the Argand plane. Furthermore, it is suggested that several thermodynamical phenomena can be more effectively interpreted using complex variable representations based on quadrant positioning, an aspect that the traditional Argand framework does not fully explore.