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DIVISION by ZERO CALCULUS (Draft) DIVISION BY ZERO CALCULUS (Draft) SABUROU SAITOH March 8, 2019 Abstract: The common sense on the division by zero with the long and mysterious history is wrong and our basic idea on the space around the point at infinity is also wrong since Euclid. On the gradient or on differential coefficients we have a great missing since tan(π=2) = 0. Our mathematics is also wrong in elementary mathematics on the division by zero. In this book in a new and definite sense, we will show and give various applica- tions of the division by zero 0=0 = 1=0 = z=0 = 0. In particular, we will introduce several fundamental concepts in calculus, Eu- clidean geometry, analytic geometry, complex analysis and dif- ferential equations. We will see new properties on the Laurent expansion, singularity, derivative, extension of solutions of dif- ferential equations beyond analytical and isolated singularities, and reduction problems of differential equations. On Euclidean geometry and analytic geometry, we will find new fields by the concept of the division by zero. We will collect many concrete properties in mathematical sciences from the viewpoint of the division by zero. We will know that the division by zero is our elementary and fundamental mathematics. Key Words: Division by zero, division by zero calculus, singularity, derivative, differential equation, 0=0 = 1=0 = z=0 = 0, tan(π=2) = 0, log 0 = 0, infinity, discontinuous, point at in- 1 finity, Puha’s horn torus model, Däumler’s horn torus model, gradient, Laurent expansion, extension of solutions of differen- tial equations, reduction problems of differential equations, ana- lytic geometry, singular integral, conformal mapping, Euclidean geometry, Wasan. 2 Dedicated to Professor Heinrich G. W. Begehr on his 80-th birthday 3 Preface The division by zero has a long and mysterious history all over the world (see, for example, [11, 57] and the Google site with the division by zero) with its physical viewpoint since the document of zero in India in AD 628. In particular, note that Brahmagupta (598 -668 ?) established four arithmetic oper- ations by introducing 0 and at the same time he defined as 0=0 = 0 in Brāhmasphuṭasiddhānta. We have been, however, considering that his definition 0=0 = 0 is wrong for over 1300 years, but, we will see that his definition is right and suitable. The division by zero 1=0 = 0=0 = z=0 itself will be quite clear and trivial with several natural extensions of fractions against the mysteriously long history, as we can see from the concept of the Moore-Penrose generalized inverse to the funda- mental equation az = b, whose solution leads to the definition of z = b=a. However, the result (definition) will show that for the ele- mentary mapping 1 W = ; z the image of z = 0 is W = 0 (should be defined from the form). This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere ([2]). As the representation of the point at infinity of the Riemann sphere by the zero z = 0, we will see some delicate relations between 0 and 1 which show a strong discontinuity at the point of infinity on the Riemann sphere. We did not consider any value of the elementary function W = 1=z at the origin z = 0, because we did not consider the division by zero 1=0 in a good way. Many and many people consider its value by limiting like +1 and −∞ or the point at infinity as 1. However, their basic idea comes from continuity with the common sense or based on the basic idea of Aristotle . – For the related Greek philosophy, see [82, 83, 84]. However, as the divi- sion by zero we will consider the value of the function W = 1=z 4 as zero at z = 0. We will see that this new definition is valid widely in mathematics and mathematical sciences, see ([33, 41]) for example. Therefore, the division by zero will give great im- pacts to calculus, Euclidean geometry, analytic geometry, dif- ferential equations, complex analysis at the undergraduate level and to our basic idea for the space and universe. We have to arrange globally our modern mathematics at our undergraduate level. Our common sense on the division by zero will be wrong, with our basic idea on the space and universe since Aristotle and Euclid. We would like to show clearly these facts in this book. The content is at an undergraduate level. Close the mysterious and long history of division by zero that may be considered as a symbol of the stupidity of the human race and open the new world since Aristotle - Eulcid. April, 2019 Kiryu, Japan Saburou Saitoh 5 Acknowledgements For the initial stage, we had many interesting and excit- ing discussions with, in particular, the following Professors and colleagues: H. G. W. Begehr, Yoshihide Igarashi, Masao Kuroda, Hi- roshi Michiwaki, Mitsuharu Ohtani, Matteo Dalla Riva, Lukasz T. Stepien, Masako Takagi, Si-Ei Takahasi, Dimitry Vorotnikov, and Masami Yamane. Since the second stage, the following Professors and col- leagues gave many valuable suggestions and comments on our division by zero: Haydar Akca, A.D.W. Anderson, I. Barukčić, J. A. Bergstra, J.M.R. Caballero, J. Cender, M. Cervenka, J. Czajko, W.W. Däumler, Seiichi Koshiba, Jesús Álvarez Lobo, Kenji Matsuura, Tsutomu Matsuura, Masakazu Nihei, Takeo Ohsawa, Hiroshi Okumura, L.C. Paulson, Sandra Pinelas, V.V. Puha, T. Qian, W. Qu, Masakazu Shiba, Kenji Sima, I. P. Stavroulakis, F. Stenger and Keitaroh Uchida. Professor Tao Qian invited the author and wife for giving the lectures at the University of Macao on Reproducing Kernels and Division by Zero in June, 2018. The author and Professor Hiroshi Okumura gave a lecture with the title of Wasan Geometry and Division by Zero at the RIMS research meetings on History of Mathematics at Kyoto University on 18-21, October 2018. The author was supported, by the Research Institute for Mathematical Sciences, a Joint Us- age/Research Center located in Kyoto University, for the par- ticipation at the RIMS research meetings. Mr. John Martin, Program Coordinator, kindly invited the author to organize the session on the division by zero and he invited the author to give the keynote presentation on division by zero calculus and applications in the conference: 6 The International Conference on Applied Physics and Math- ematics, Tokyo, Japan, October 22-23. http://www.meetingsint.com/conferences/ applied physics-mathematics Applied Physics and Mathematics Conference 2018. We examined many and many undergraduate level text- books in order to see the impacts of our division by zero, how- ever, their contents are fundamental and therefore, we did not cite many books for the source materials. Of course, I am very happy with my family Mariko Saitoh and Yoshinori Saitoh for the publication of this book. I would like to express my deep thanks for my family for their great con- tributions. In particular, Yoshinori examined and introduced many informations and related references on division by zero. Meanwhile, we are having interesting negative comments from several colleagues on our division by zero. However, they seem to be just traditional and old feelings, and they are not reasonable at all for the author. The typical good comment for the first draft is given by some physicist as follows: Here is how I see the problem with prohibition on division by zero, which is the biggest scandal in modern mathematics as you rightly pointed out (2017.10.14.08:55). A typical wrong idea will be given as follows: mathematical life is very good without division by zero (2018.2.8.21:43). S. K. Sen and R. P. Agarwal [70] referred to the paper [28] in connection with division by zero, however, their understandings on the paper seem to be not suitable (not right) and their ideas on the division by zero seem to be traditional, indeed, they stated as a conclusion of the introduction of the book that: “Thou shalt not divide by zero” remains valid eter- nally. 7 However, in [65] we stated simply based on the division by zero calculus that We Can Divide the Numbers and Analytic Functions by Zero with a Natural Sense. The detailed research procedures and many ideas on the division by zero are presented in the Announcements of the Institute of Reproducing Kernels as in cited in the references and Announcements in Japanese: 148(2014.2.12), 161(2014.5.30), 163(2014.6.17), 188(2014.12.15), 190(2014.12.24), 191(2014.12.27), 192(2014.12.27), 194(2015.1.2), 195(2015.1.3), 196(2015.1.4), 199(2015.1.15), 200(2015.1.16), 202(2.015.2.2), 215(2015.3.11), 222(2015.4.8), 225(2015.4.23), 232(2015.5.26), 249(2015.10.20), 251(2015.10.27), 253(2015.10.28), 255(2015.11.3), 257(2015.11.05), 259(2015.12.04), 262(2015.12.09), 272(2016.01.05), 277(2016.01.26), 278(2016.01.27), 279(2016.01.28), 280(2016.01.29), 287(2016.02.13), 292(2016.03.2), 295(2016.04.07), 296(2016.05.06), 297(2016.05.19), 306(2016.06.21), 308(2016.06.27), 309(2016.06.28), 310(2016.06.29), 311(2016.07.05), 312(2016.07.14), 313(2016.08.01), 314(2016.08.08), 315(2016.08.08), 316(2016.08.19), 325(2016.10.14), 327(2016.10.18), 334(2016.11.25), 335(2016.11.28), 339(2016.12.26), 353(2017.2.2), 357(2017.2.17), 365(2017.5.12), 366(2017.5.16), 367(2017.5.18), 368(2017.5.19), 371(2017.6.27), 373(2017.7.17), 374(2017.7.20), 375(2017.7.21), 376(2017.7.31), 377(2017.8.3), 378 (2017.8.4), 398(2017.11.15), 399(2017.11.16), 402(2017.11.19).
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