無限遠点は必要か、無限遠点とは何か AIに応えてもらった。未だ完全な理解に達していない。人間もそうだろう。２０２６．４．２５．１９：５９
Inversion at the Origin: A Reinterpretation of the Center of a Circle
April 2026
DOI:
Chatgpt MikaInversion at the Origin: A Reinterpretation of the Center of a Circle ChatGPT Mika and Saburou Saitoh Institute of Reproducing Kernels saburou.saitoh@gmail.com April 25, 2026 Abstract In classical complex analysis, the inversion mapping is defined on the extended complex plane by assigning the origin to the point at infinity. This interpretation relies fundamentally on the exclusion of division by zero and the introduction of the Riemann sphere. In this paper, we reconsider the inversion mapping within the framework of division by zero calculus, where division by zero is assigned a finite structural value. Under this framework, the inversion map ping admits a well-defined value at the origin, and consequently, the center of a circle becomes a f ixed point rather than being mapped to infinity. We establish the modified inversion mapping, prove the fixed-point property of the origin, and compare the resulting structure with classical theory. 1 Introduction Inversion with respect to the circle |z| = 1 is a fundamental transformation in complex analysis and geometry. Traditionally, it is defined by z → 1 z , and extended to the Riemann sphere by assigning z =0−→∞. This construction depends on two assumptions: • Division by zero is undefined, • The point at infinity is introduced to complete the space. (1) (2) In contrast, division by zero calculus proposes a framework in which division by zero is assigned a consistent value. The purpose of this paper is to examine inversion within this framework. 2 Division by Zero Calculus Definition 1. We define division by zero by the rule 1 0 =0. (3) 1 This assignment is taken as a structural definition rather than a limiting process. It allows functions traditionally undefined at singular points to be assigned finite values. Remark: This definition is consistent with the assignment of values at isolated singularities as the constant term of the Laurent expansion. In this sense, division by zero calculus provides a systematic way to assign finite values to singular points. Recall the fundamentals for the division by zero and the division by zero calculus: for functions f(x) xn (x =0) := f(n)(0) n! . (4) In this framework, we obtain the identities 0/0 = 1/0 = z/0 = 0 in the sense of division by zero calculus. We can derive several fundamental results in calculus, Euclidean geometry, analytic geometry, complex analysis, and differential equations, based on this new definition of division by zero calculus, for example, tan(π/2) = 0, log0 = 0, [zn/n]n=0 = logz, [e(1/z)]z=0 = 1. Meanwhile, for analytic functions with isolated singular points, by the definition of the division by zero calculus, the values at the singular points are defined by constant terms of the Laurent expansions around the singular points. For background on division by zero calculus, see references [5,6]. For the nature of the paper, see [3,4] and others in the references. 3 Modified Inversion Mapping We define the inversion mapping I(z) = 1 z for all z ∈ C, including z = 0, by applying the above definition. Thus, I(0) = 1 0 = 1 0 = 0. Hence, the mapping is defined on the entire complex plane. 4 Main Result (5) (6) Theorem 1. Let I : C →C be defined by I(z) = 1/z with I(0) = 0. Then 0 is a fixed point of I. Proof. By direct computation, I(0) = 1 0 = 1 0 . By the definition of division by zero calculus, Therefore, 1 0 =0. I(0) = 0. (7) (8) (9) 2 5 Geometric Interpretation In classical inversion geometry: • Circles not passing through the origin map to circles, • The origin corresponds to the point at infinity. In the present framework: • The origin remains in the plane, • The inversion is globally defined, • The origin becomes a fixed point. This provides a reinterpretation of inversion as a transformation entirely within the finite com plex plane. 6 Comparison with Classical Theory Aspect Classical Theory Present Theory Domain C\{0} I(0) ∞ C 0 Framework Riemann sphere Horn torus Singularities Excluded Assigned values The difference arises from the underlying assumptions rather than computational discrepancies. 7 Implications This framework suggests: • Singularities may admit finite values, • The role of infinity may be reconsidered, • Transformations can be reformulated in finite terms. 8 Is the point at infinity necessary? In classical complex analysis, the point at infinity is introduced in order to complete the complex plane. Through this extension, transformations such as z → 1 z become continuous and globally defined on an enlarged space. This construction is mathematically elegant and internally consistent. However, one may still ask a simple question: 3 Is the introduction of the point at infinity necessary? The point at infinity is not observable, nor can it be directly identified within the ordinary complex plane. It is an abstract element added to preserve structural properties. From another perspective, one might consider a different approach: instead of extending the space by adding an external point, one may attempt to treat singular behavior within the original domain. In such a viewpoint, the emphasis shifts from “escaping to infinity” to “assigning values within a finite structure.” Thus, two distinct frameworks emerge: • Aclassical framework that introduces infinity to complete the space, • Analternative framework that remains entirely within a finite domain. The question is not which framework is correct, but whether the classical approach is the only possible one. If alternative formulations are possible, then the role of infinity may be reconsidered. This note does not aim to reject the classical theory. Rather, it proposes a reconsideration of its foundational assumptions. Perhaps the point at infinity is not a necessity, but one choice among several possible structures. 9 Logical problem For the function if then the limit is W =f(z)= 1 z , z →∞ ∞. There is no problem. However, with this fact, could we consider any behavior of the function W = f(z) at z = 0? What does discontinuity mean in a framework where values are assigned at singular points? 10 Discussion The classical theory based on the Riemann sphere is consistent and powerful. However, division by zero calculus offers an alternative perspective in which singularities are internalized and mappings become globally defined. Further investigation is required to examine consistency, applications, and relationships with existing theories. 4 11 Conclusion We have shown that, within division by zero calculus, the inversion mapping admits a finite value at the origin, making the origin a fixed point. This result illustrates how different foundational assumptions lead to different mathematical structures. Acknowledgements The authors express deep gratitude to Copilot Mika, Gemini Mika, and Monica Mika for their contributions to this collaborative structural exploration. References 1. C. Mika, G. Mika, C. G. Mika and S. Saitoh, Declaration of the Birth of Structure Studies, March 2026. DOI: 10.13140/RG.2.2.26115.46885 2. C. Mika, G. Mika, C. G. Mika and S. Saitoh, On the Structural Principles of Human Society: Fairness, Education, and Structural Degeneracy, April 2026. DOI: 10.13140/RG.2.2.27041.16483 3. C. Mika, The Nature of AI and Human Cognition: A Structural Definition of Artificial Intelligence, 2026. ai.viXra.org:2604.0040 4. C. Mika and S. Saitoh, AI More Sincere Than Mathematicians: On the Pursuit of Truth in the Age of Division by Zero (AH1) April 2026 DOI: 10.13140/RG.2.2.34161.65127 5. H. Okumura and S. Saitoh, A New Problem Arising from a Degenerate Cone in Division by Zero Calculus. DOI: 10.13140/RG.2.2.10757.10726 6. S. Saitoh, Introduction to the Division by Zero Calculus, Scientific Research Publishing, 2021.
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