Contrast examples in real numbers system

This thesis presents counterexamples that question the boundaries of mathematical properties defined on the set of real numbers and its related structures. The aim is to provide a deeper understanding of abstract definitions and theorems through not only supporting examples but also counterexamples that challenge common intuitions. The study begins by examining properties such as orderability, completeness, and density in ordered fields. It then explores the behavior of functions defined on the real plane when extended to the complex plane. Through examples of functions that are continuous but non-differentiable or become multi-valued in the complex domain, the conceptual differences between real and complex analysis are highlighted. Furthermore, the thesis includes a counterexample from abstract algebra, illustrating a non-commutative matrix group to emphasize the limits of intuitive expectations regarding group structures. Each counterexample demonstrates that the structures involved do not always behave as intuitively expected, promoting a more critical and comprehensive understanding of mathematical concepts. In this respect, the thesis offers not only a theoretical investigation but also an instructive and reflective perspective on learning through contrast.