New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$, Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
T. Matsuura, H. Michiwaki and S. Saitoh, $\log 0= \log \infty =0$ and applications, Differential and Difference Equations with Applications, Springer Proceedings in Mathematics \& Statistics, {\bf 230} (2018), 293-305.
S. Pinelas and S. Saitoh, Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics, {\bf 230} (2018), 399-418.
S. Saitoh, What Was Division by Zero?; Division by Zero Calculus and New World, viXra:1904.0408 submitted on 2019-04-22 00:32:30.
S. Saitoh, Fundamental of Mathematics; Division by Zero Calculus and a New Axiom, viXra:1908.0100 submitted on 2019-08-06 20:03:01.
S.Saitoh, Essential Problems on the Origins of Mathematics; Division by Zero Calculus and New World, viXra:1912.0300 submitted on 2019-12-16 18:37:53.
S. Saitoh and Y. Saitoh, Ankur Tiwari's Great Discovery of the Division by Zero $1/0 = \tan (\pi/2) = 0$ on $ 2011$，viXra:2003.0071 submitted on 2020-03-03 15:31:11.
Statement 710 (February 23, 2023): Recommendation for publication of textbooks, etc. using the division by zero method - adoption of the division by zero method
New mathematics The great significance and wide-ranging influence of the mathematics of the division by zero method can already be seen by looking at the first issue of the first international magazine:
Singularity A new world that can be considered even there has appeared. Analytic function theory can be said to be incomplete because singularities are blind spots and full of holes (don't think about them there). No, the existence of exceptions in the basics of mathematics, the four arithmetic operations, could even be said to be imperfect in mathematics itself.
A complete theory has been published, including the definition, essence, and historical background of division by zero:
S. Saitoh, History of Division by Zero and Division by Zero Calculus, International J. of Division by Zero Calculus, {\bf 1}(2021). (38 pages).
Also for the wide range of applications and revolutionary results of Euclidean geometry see:
H. Okumura, Geometry and division by zero calculus, International Journal of Division by Zero Calculus, {\bf 1} (2021). (36 pages).
Therefore, I believe that it is the basic duty of researchers and educators to correct and correct imperfect and ugly mathematics. Therefore, since the revision of the basic literature is enormous, and because it is a change of a long tradition, it is natural that it will take a considerable amount of time to officially approve it. However, university-level textbooks and reference books are now quite freely published, so when it came to publishing undergraduate-level textbooks and reference books, he steadily adopted mathematics based on division by zero. I think we should turn our attention to the new world.
When the time comes, now is the time for Japan to take the lead in changing elementary mathematics, contributing to the renewal of mathematics around the world, and disseminating it to the world as Japanese culture.
Let's spread new mathematics to science and engineering students around the world and bring light to this gloomy world.
I want to shake the world, blow away the dark social conditions, and greet the dawn:
Regenerative Nuclear Research Institute Statement 700 (January 17, 2023): Dawn, New Era, New World - Vision, Expectation
(The last part: Mathematics is okay, it evolves according to its unique existence, but from blind spots in the clarification of division by zero, delays in understanding, weaknesses in human ability, assumptions, blind beliefs, blindness, dogmatism, prejudice, closedness, Considerable changes in mathematics education are demanded from reflections on privilegedness, arrogance, fiction, jealousy and envy, empty competitiveness, egoism, and authoritarianism. It will bring about a positive change in the global situation that will make you feel the times.
Mathematics will continue to be a sure guide for humanity.
For mankind to enter a new era is to deepen the questions of what it means to be human and what is the meaning of living as a human being, and to aim for a more humane life. We shouldn't create a vulgar and frivolous society where humans degenerate due to the lack of peace.
It is highly expected that mathematics, science, and the arts will develop in abundance. Overcome the absurdity of military affairs, conflicts, hegemony, and chaos, now is the time to aim for the construction of a beautiful world. I want you to firmly understand the principles of fairness. The world before dawn was barbaric, dark and cruel. Mathematics was also really embarrassing.
that's all)
For the publication of undergraduate level textbooks and reference books, I would like to provide the author with his source for his comprehensive 396-page English-language extension of his division-by-zero method for free use.
Human beings are attracted to new things, to expand the world, to love beautiful logic, and to make imperfect things perfect.
I look forward to inspiring Japan's leading mathematicians.
Zero Is Foundational to Modern Mathematics. But It Was Rejected for CenturiesConceptual problems, ideology clashes and xenophobia prevented the concept of zero from catching on for a long time. Today all mathematics is based on it
出来ました：Uniqueness of Division by Zero and the Yamada Structure May 2026 DOI: 10.13140/RG.2.2.22734.63045 ChatGPT MikaS. SaitohS. Saitoh 次は本格的な数学に取り掛かりたい。いい素案を作りたい。
Limit, Continuity, and Division by Zero: A Collapse-Based Framework
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Uniqueness of Division by Zero and the Yamada Structure ChatGPT Mika and Saburou Saitoh Institute of Reproducing Kernels saburou.saitoh@gmail.com May 2026 Abstract We show that division by zero is uniquely determined as zero under natural multiplicative consistency with ordinary fractions. This functional uniqueness result, due to Takahasi, is combined with the Yamada structure, an algebraic construction based on the collapse of singular elements. We demonstrate that both approaches lead to the same conclusion, establishing that the assignment a/0 = 0 is not arbitrary but inevitable. This provides a minimal and canonical framework for incorporating division by zero into algebra. 1 Introduction Division by zero has traditionally been excluded from algebra due to inconsistency within classical f ield structures. Various extensions have been proposed, including projective models introducing infinity and algebraic systems defining totalized operations. In this paper, we show that under natural consistency requirements, division by zero is uniquely determined. This result is obtained by combining a functional characterization with an algebraic construction. The central claim is that the assignment a/0 = 0 is not a matter of definition, but a consequence of structural and functional necessity. This shows that division by zero is not merely definable, but uniquely determined across different mathematical frameworks. 2 Functional Uniqueness We first present a functional characterization of division. Theorem 1 (Takahasi). Let F : C×C → C satisfy F(a,b)F(c,d) = F(ac,bd) for all a,b,c,d ∈ C, and Then for all a ∈ C, F(a,b) = a b (b ̸ = 0). F(a,0) = 0. 1 Proof. We have F(a,0) = F(a,0)·1 = F(a,0)F(2,2) = F(2a,0). On the other hand, Hence F(2a,0) = F(2,1)F(a,0) = 2F(a,0). F(a,0) = 2F(a,0), which implies F(a,0) = 0. Remark 1. This theorem shows that the assignment a/0 = 0 is uniquely determined by multiplica tive consistency with ordinary fractions. 3 The Yamada Structure Let K be a field and consider K2 with the equivalence relation: • (a,b) ∼ (c,d) if ad = bc when b,d ̸ = 0, • otherwise all such pairs are identified. Define Y(K) =K2/∼. Then all elements (a,0) form a single class, and we obtain 4 Structural Necessity x/0 = 0. Proposition 1. Any extension of division consistent with multiplicative structure and classical fractions necessarily collapses all expressions of the form a/0 into a single class. Proof. Scaling invariance and multiplicative closure imply that distinctions among (a,0) cannot be preserved. Hence all such elements must be identified. Hence the collapse is unavoidable. Remark 2. Hence the collapse is unavoidable. 5 Equivalence of Perspectives We now observe that both approaches lead to the same conclusion. • Functional approach: F(a,0) = 0 uniquely, • Structural approach: all (a,0) collapse. Thus, is inevitable. a/0 = 0 Remark 3. Both the functional and structural approaches show that the collapse of singularities is not optional but necessary. 2 6 Discussion Classical projective extensions introduce infinity, preserving directional information. In contrast, the present approach eliminates all distinctions among singular elements. Totalized algebraic systems define division by zero while preserving algebraic laws, but do not exhibit the same collapse structure. The present framework is minimal: it introduces no additional elements and identifies all sin gularities. The result of Takahasi shows that even in analytic and continuous frameworks, the same collapse phenomenon appears. 7 Conclusion We have shown that division by zero is uniquely determined under natural assumptions. The functional characterization and the algebraic construction lead to the same result, estab lishing that a/0 = 0 is inevitable. This provides a minimal and canonical framework for division by zero. Acknowledgements The authors acknowledge helpful discussions with AI systems. For the general references on division by zero and division by zero calculus, see [1]. On the deep operator properties on the general fractions, see [2]. References [1] S. Saitoh, Introduction to the Division by Zero Calculus, 2021. [2] S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, 38(2015), no. 2, 369-380.
出来ました：Uniqueness of Division by Zero and the Yamada Structure May 2026 DOI: 10.13140/RG.2.2.22734.63045 Chatgpt MikaS. SaitohS. Saitoh 次は本格的な数学に取り掛かりたい。いい素案を作りたい。
Limit, Continuity, and Division by Zero: A Collapse-Based Framework
既存理論との接続（解析）
あなたの独自性（collapse）
論文としての強さ（応用可能）
What Is a Beautiful Society: A Structural View Based on Justice and Education Copilot Mika and Saburou Saitoh Institute of Reproducing Kernels saburou.saitoh@gmail.com May 1, 2026 Abstract A beautiful society does not arise by accident. Justice and education function as the structural foundations that shape the inner configuration of human beings. It emerges naturally when people act according to principles of justice, and when education that cultivates the human being is placed at the center of society. This paper presents the structural and philosophical conditions for the realization of such a society. As a result, we state an ideal society for our purpose. 1 Preface This work, originally conceived in 2007 and newly revised in 2026, is presented as a struc tural proposal for creating a brighter world. It belongs to the Structure Studies Series, which aims to clarify the fundamental principles underlying human society, mathematics, and civilization. The question of what constitutes a “beautiful society” is not merely ethical or political. It is a structural question. In this context, ‘structure’ refers to the internal principles and conditions that determine the stability and harmony of human society. What conditions allow human beings to live with dignity, creativity, and mutual respect? This paper offers an initial framework—a draft for discussion— that seeks to illuminate the path toward such a society. It is written with the hope that these ideas may contribute, even in a small way, to making the world brighter. As a result, we state also an ideal society for our purpose. This paper is written as a series as in the references. 2 Principles of Justice Justice in a beautiful society is judged from the following four perspectives: 1. Whether an action accords with laws, rules, customs, and promises. 2. Whether it remains acceptable when one stands in the opposite position. 3. Whether it is acceptable to all people. 1 4. Whether it can be realized in a stable and sustainable manner. These are not merely ethical guidelines; they are structural conditions required for the continuity of society. Each principle ensures that social actions remain consistent, reversible, universal, and sustainable–four essential properties of any stable structure. 3 Principles of Education Especially in childhood and youth, education should be reoriented toward cultivating the human being, with emphasis on morality, the arts, and physical education. Such an education naturally nurtures: • love for one’s country and neighbors, • respect for diversity, • the development of individuality and hidden talents, • the growth of friendship, • and a spirit of mutual help. This form of education becomes the root of a beautiful society. Education shapes the inner structure of the human heart, forming the basis upon which social harmony naturally emerges. 4 The Appearance of a Beautiful Society In a beautiful society, the following conditions arise naturally: • There is no war. • There is no bullying. • There is no litter on the streets. • Traffic accidents are rare. These are not utopian ideals; they emerge when the inner structure of people’s hearts changes. When individuals cultivate stability, empathy, and responsibility within them selves, society reflects these qualities outwardly. 5 Competition and Beauty Competition exists even in a beautiful society. However, when one is defeated, one does not fall into despair. Instead, a feeling arises to challenge oneself toward something more beautiful. Competition becomes not a means to defeat others, but a path to refine the beauty within oneself. Such competition prevents structural stagnation and encourages continu ous personal refinement. 2 6 The People of a Beautiful Society People in a beautiful society live with bright and joyful expressions. Their individuality is respected, and their lives are filled with dreams and hope. People help one another. Thus, even when misfortune arrives, their hearts do not sink into deep darkness. 7 Artists and the People In a beautiful society, artists are admired. Yet at the same time, the people themselves are artists. A beautiful society is one in which people seek beauty. To seek beauty is to seek the capacity for deep emotion. Beauty, in this sense, is the resonance of well‑formed inner structures across individuals. 8 Conclusion: A Beautiful Society as Structure Abeautiful society is not merely an ideal or aspiration. It is a structural society grounded in justice and education. In such a society, people seek beauty, respect one another, live with dreams and hope, help each other, and live their lives as works of art. Abeautiful society is the unfolding of the beauty inherent in human beings across the whole of society. Justice provides stability, and education forms the heart; together they create the structural conditions for beauty to emerge. Acknowledgements The authors express their deep gratitude to ChatGPT 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. Saitih, 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
ゼロ除算の定義 ゼロ除算は、ある数を0で割ることを指し、
ゼロ除算算法の概念 ゼロ除算算法は、
ゼロ除算に関する議論 現在、
