い 総合的な報告を受けて、その後、討論を重視する形で進める。昼食を挟んで、討

論し、最後に 今後の研究活動について検討する。

参加希望者は、下記にメールにて、お知らせ下さい：

（kbdmm360@yahoo.co.jp , saburou.saitoh@gmail.com ）

]]>

Kawauchi-cho, 5-1648-16,

Kiryu 376-0041, Japan

February 2, 2018The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0 =1/0 = z/0 = 0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotels $($BC384 - BC322$)$ and Euclid $($BC 3 Century - $)$, and the division by zero is since Brahmagupta $($598 - 668?$)$. In particular, Brahmagupta defined as $0/0 = 0$ in Brhmasphuasiddhnta $($628$)$, however, our world history stated that his definition $0/0 = 0$ is wrong over 1300 years, but, we showed that his definition is suitable. For the details, see the references and the site: http://okmr.yamatoblog.net/ We wrote a global book manuscript [21] with 154 pages and stated in the preface and last section of the manuscript as follows:

The division by zero has a long and mysterious story over the world $($see, for example, H. G. Romig [15] and Google site with the division by zero$)$ with its physical viewpoints since the document of zero in India on AD 628. In particular, note that Brahmagupta $($598 -668 ?$)$ established the four arithmetic operations by introducing 0 and at the same time he defined as $0/0 = 0$ in Brhmasphuasiddhnta. Our world history, however, stated that his definition $0/0 = 0$ is wrong 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 the fractions against the mysterously long history, as we can see from the concepts of the Moore-Penrose generalized inverses or the Tikhonov regularization method to the fundamental equation $az = b$, whose solution leads to the definition $z = b/a$.

However, the result $($definition$)$ will show that for the elementary mapping

$$

W =\frac{1}{z} \tag{0.1}

$$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 ([1]). 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 $\infty$ 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 the limiting like $+\infty$ and $-\infty$ or the point at infinity as $\infty$. However, their basic idea comes from continuity with the common sense or based on the basic idea of Aristotle. – For the related Greece philosophy, see [23, 24, 25]. However, as the division by zero we will consider its value of the function $W = 1/z$ as zero at $z = 0$. We will see that this new definition is valid widely in mathematics and mathematical sciences, see $($[9, 10]$)$ for example. Therefore, the division by zero will give great impacts to calculus, Euclidean geometry, analytic geometry, differential equations, complex analysis in the undergraduate level and to our basic ideas for the space and universe.

We have to arrange globally our modern mathematics in our undergraduate level. Our common sense on the division by zero will be wrong, with our basic idea on the space and the universe since Aristotle and Euclid. We would like to show clearly these facts in this book. The content is in the undergraduate level.

Apparently, the common sense on the division by zero with a 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 derivatives we have a great missing since $tan(\pi/2) = 0$. Our mathematics is also wrong in elementary mathematics on the division by zero.

This book is an elementary mathematics on our division by zero as the first publication of books for the topics. The contents have wide connections to various fields beyond mathematics. The author expects the readers write some philosophy, papers and essays on the division by zero from this simple source book.

The division by zero theory may be developed and expanded greatly as in the author’s conjecture whose break theory was recently given surprisingly and deeply by Professor Qi’an Guan [3] since 30 years proposed in [17] $($the original is in [16]$)$.

We have to arrange globally our modern mathematics with our division by zero in our undergraduate level.

We have to change our basic ideas for our space and world.

We have to change globally our textbooks and scientific books on the division by zero.

[1] L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.

[2] L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory 7 $($2013$)$, no. 4, 1049-1063.

[3] Q. Guan, A proof of Saitoh’s conjecture for conjugate Hardy H2 kernels, arXiv:1712.04207.

[4] M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane, New meanings of the division by zero and interpretations on 100/0 = 0 and on 0/0 = 0, Int. J. Appl. Math. 27 $($2014$)$, no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.

[5] T. Matsuura and S. Saitoh, Matrices and division by zero $z/0=0$, Advances in Linear Algebra & Matrix Theory, 6$($2016$)$, 51-58 Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2016.62007.

[6] T. Matsuura and S. Saitoh, Division by zero calculus and singular integrals. $($Submitted for publication$)$

[7] T. Matsuura, H. Michiwaki and S. Saitoh, log0 = log∞ = 0 and applications. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics.

[8] H. Michiwaki, S. Saitoh and M.Yamada, Reality of the division by zero z/0 = 0. IJAPM International J. of Applied Physics and Math. 6$($2015$)$, 1–8. http://www.ijapm.org/show-63-504-1.html

[9] H. Michiwaki, H. Okumura and S. Saitoh, Division by Zero $z/0 = 0$ in Euclidean Spaces, International Journal of Mathematics and Computation, 28$($2017$)$; Issue 1, 1-16.

[10] H. Okumura, S. Saitoh and T. Matsuura, Relations of 0 and ∞, Journal

of Technology and Social Science $($JTSS$)$, 1$($2017$)$, 70-77.

[11] H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 $($2017.11.14$)$.

[12] H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947 International Journal of Geometry.

[13] H. Okumura and S. Saitoh, Applications of the division by zero calculus to Wasan geometry. $($Submitted for publication$)$.

[14] S. Pinelas and S. Saitoh, Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics.

[15] H. G. Romig, Discussions: Early History of Division by Zero, American

Mathematical Monthly, Vol. 31, No. 8. $($Oct., 1924$)$, pp. 387-389.

[16] S. Saitoh, The Bergman norm and the Szegö norm, Trans. Amer. Math. Soc. 249 $($1979$)$, no. 2, 261–279.

[17] S. Saitoh, Theory of reproducing kernels and its applications. Pitman Research Notes in Mathematics Series, 189. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. x+157 pp. ISBN: 0-582-03564-3

[18] S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra & Matrix Theory. 4 $($2014$)$, no. 2, 87–95. http://www.scirp.org/journal/ALAMT/

[19] S. Saitoh, A reproducing kernel theory with some general applications, Qian,T./Rodino,L.$($eds.$)$: Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, 177(2016), 151-182. $($Springer$)$.

[20] S. Saitoh, Mysterious Properties of the Point at Infinity、arXiv:1712.09467 [math.GM]$($2017.12.17$)$.

[21] S. Saitoh, Division by zero calculus $($154 pages: draft$)$: $($http://okmr.yamatoblog.net/ $)$

[22] 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.

[23] https://philosophy.kent.edu/OPA2/sites/default/files/012001.pdf

[24] http://publish.uwo.ca/ jbell/The 20Continuous.pdf

[25] http://www.mathpages.com/home/kmath526/kmath526.htm

[26] Announcement 179 $($2014.8.30$)$: Division by zero is clear as z/0=0 and it is fundamental in mathematics.

[27] Announcement 185 $($2014.10.22$)$: The importance of the division by zero $z/0 = 0$.

[28] Announcement 237 $($2015.6.18$)$: A reality of the division by zero $z/0 = 0$ by geometrical optics.

[29] Announcement 246 $($2015.9.17$)$: An interpretation of the division by zero $1/0 = 0$ by the gradients of lines.

[30] Announcement 247 $($2015.9.22$)$: The gradient of y-axis is zero and $\tan(\pi/2) = 0$ by the division by zero $1/0 = 0$.

[31] Announcement 250 $($2015.10.20$)$: What are numbers? - the Yamada field containing the division by zero z/0 = 0.

[32] Announcement 252 $($2015.11.1$)$: Circles and curvature - an interpretation by Mr. Hiroshi Michiwaki of the division by zero $r/0 = 0$.

[33] Announcement 281 $($2016.2.1$)$: The importance of the division by zero $z/0 = 0$.

[34] Announcement 282 $($2016.2.2$)$: The Division by Zero z/0 = 0 on the Second Birthday.

[35] Announcement 293 $($2016.3.27$)$: Parallel lines on the Euclidean plane from the viewpoint of division by zero $1/0=0$.

[36] Announcement 300 $($2016.05.22$)$: New challenges on the division by zero $z/0=0$.

[37] Announcement 326 $($2016.10.17$)$: The division by zero z/0=0 - its impact to human beings through education and research.

[38] Announcement 352$($2017.2.2$)$: On the third birthday of the division by zero $z/0=0$.

[39] Announcement 354$($2017.2.8)$:$ What are $n = 2,1,0$ regular polygons inscribed in a disc? – relations of 0 and infinity.

[40] Announcement 362$($2017.5.5$)$: Discovery of the division by zero as $0/0 =

1/0 = z/0 = 0$

[41] Announcement 380 $($2017.8.21$)$: What is the zero?

[42] Announcement 388$($2017.10.29$)$: Information and ideas on zero and division by zero $($a project$)$.

[43] Announcement 409 $($2018.1.29.$)$: Various Publication Projects on the Division by Zero.

[44] Announcement 410 $($2018.1 30.$)$: What is mathematics? – beyond logic; for great challengers on the division by zero.

]]>

Kawauchi-cho, 5-1648-16,

Kiryu 376-0041, Japan

January 24, 2018The Institute of Reproducing Kernels is dealing with the theory of divisionby zero calculus and declares that the division by zero was discovered as $0/0 =1/0 = z/0 = 0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotels $($BC384 - BC322$)$ and Euclid $($BC 3 Century - $)$, and the division by zero is since Brahmagupta $($598 - 668?$)$. In particular, Brahmagupta defined as $0/0 = 0$ in Brhmasphuasiddhnta $($628$)$, however, our world history stated that his definition $0/0 = 0$ is wrong over 1300 years, but, we showed that his definition is suitable. For the details, see the references and the site: http://okmr.yamatoblog.net/

We wrote a global book manuscript [22] with 154 pages and stated that the division by zero is trivial and clear, and in the last section of the manuscript we stated as follows:

Apparently, the common sense on the division by zero with a 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 derivatives we have a great missing since $\tan(π/2) = 0$. Our mathematics is also wrong in elementary mathematics on the division by zero.

This book is an elementary mathematics on our division by zero as the first publication of books for the topics. The contents have wide connections to various fields beyond mathematics. The author expects the readers write some philosophy, papers and essays on the division by zero from this simple source book.

The division by zero theory may be developed and expanded greatly as in the author’s conjecture whose break theory was recently given surprisingly and deeply by Professor Qi’an Guan [3] since 30 years proposed in [18] $($the original is in [17]$)$.

We have to arrange globally our modern mathematics with our division by zero in our undergraduate level.

We have to change our basic ideas for our space and world.

We have to change globally our textbooks and scientific books on the division by zero.

However, we have still curious situations and opinions for us on the division by zero; in particular, the two great challengers Jakub Czajko and Ilija Baruki on the division by zero in connection with physics stated that we do not have the definition of the division $0/0$, however $0/0 = 1$. They seem to think that a truth is based on physical objects and is not on our mathematics. In such a cases, we will not be able to continue discussions on the division by zero more, because for mathematicians, they will not be able to follow their logics more. However, then we will ask for the question that what are the values and contributions of your articles and discussions. We

will expect some contributions, of course.

This question will reflect to mathematicians contrary. We stated for the estimation of mathematisc in [16]: Mathematics is the collection of relations and, good results are fundamental, beautiful, and give great good impacts to human beings.

With this estimation, we stated that the Euler formula

$$e^{πi} = −1$$

is the best result in mathematics in details in:

No.81, May 2012$($pdf 432kb$)$ www.jams.or.jp/kaiho/kaiho-81.pdf

In order to show the importance of our division by zero and division by zero calculus we are requested to show their importance.

It seems that the long and mysterious confusions for the division by zero is on the definition. – Indeed, when we consider the division by zero $a/0$ in the usual sense of the fundamental equation 0 · $z = a$, we have immediately the simple contradiction, however, we have such cases may happen, in particular, in mathematical formulas and physical formulas on the universe.

[1] I. Baruki, Dialectical Logic Negation Of Classical Logic, http://vixra.org/abs/1801.0256

[2] J. Czajko, Algebraic division by zero implemented as quasigeometric multiplication by infinity in real and complex multispatial hyperspaces, Available online at www.worldscientificnews.com WSN 92$($2$)$ $($2018$)$ 171-197

[3] Q. Guan, A proof of Saitoh’s conjecture for conjugate Hardy H2 kernels, arXiv:1712.04207.

[4] M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane, New meanings of the division by zero and interpretations on $100/0 = 0$ and on $0/0 = 0$, Int. J. Appl. Math. 27 $($2014$)$, no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.

[5] T. Matsuura and S. Saitoh, Matrices and division by zero $z/0=0$, Advances in Linear Algebra & Matrix Theory, 6$($2016$)$, 51-58 Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2016.62007.

[6] T. Matsuura and S. Saitoh, Division by zero calculus and singular integrals. $($Submitted for publication$)$

[7] T. Matsuura, H. Michiwaki and S. Saitoh, $\log0 = \log\infty = 0$ and applications. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics.

[8] H. Michiwaki, S. Saitoh and M.Yamada, Reality of the division by zero z/0 = 0. IJAPM International J. of Applied Physics and Math. 6$($2015$)$, 1–8. http://www.ijapm.org/show-63-504-1.html

[9] H. Michiwaki, H. Okumura and S. Saitoh, Division by Zero $z/0 = 0$ in Euclidean Spaces, International Journal of Mathematics and Computation, 28$($2017$)$; Issue 1 1-16.

[10] H. Okumura, S. Saitoh and T. Matsuura, Relations of 0 and ∞, Journal of Technology and Social Science $($JTSS$)$, 1$($2017$)$, 70-77.

[11] H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 (2017.11.14).

[12] H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947 International Journal of Geometry.

[13] H. Okumura and S. Saitoh, Applications of the division by zero calculus to Wasan geometry. $($Submitted for publication$)$.

[14] S. Pinelas and S. Saitoh, Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics.

[15] H. G. Romig, Discussions: Early History of Division by Zero, American Mathematical Monthly, Vol. 31, No. 8. $($Oct., 1924$)$, pp. 387-389.

[16] T. M. Rassias, Editor, Nonlinear Mathematical Analysis and Applications, HadronicPress,Palm Harbor,FL34682-1577,USA:ISBN1-57485-044-X,1998, pp.223234: Nonlinear transforms and analyticity of functions, Saburou Saitoh.

[17] S. Saitoh, The Bergman norm and the Szegö norm, Trans. Amer. Math. Soc. 249 $($1979$)$, no. 2, 261–279.

[18] S. Saitoh, Theory of reproducing kernels and its applications. Pitman Research Notes in Mathematics Series, 189. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. x+157 pp. ISBN: 0-582-03564-3

[19] S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra & Matrix Theory. 4 $($2014$)$, no. 2, 87–95. http://www.scirp.org/journal/ALAMT/

[20] S. Saitoh, A reproducing kernel theory with some general applications, Qian,T./Rodino,L.$($eds.$)$: Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, 177$($2016$)$, 151-182. $($Springer$)$

.[21] S. Saitoh, Mysterious Properties of the Point at Infinity. arXiv:1712.09467 [math.GM]$($2017.12.17$)$.

[22] S. Saitoh, Division by zero calculus $($154 pages: draft$)$: $($http://okmr.yamatoblog.net/ $)$

[23] 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.

]]>

Kawauchi-cho, 5-1648-16,

Kiryu 376-0041, Japan

January 29, 2018The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as 0/0 =1/0 = z/0 = 0 in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristoteles $($BC384 - BC322$)$ and Euclid $($BC 3 Century - $)$, and the division by zero is since Brahmagupta $($598 - 668?$)$. In particular, Brahmagupta defined as $0/0 = 0$ in Brhmasphuasiddhnta $($628$)$, however, our world history stated that his definition $0/0 = 0$ is wrong over 1300 years, but, we showed that his definition is suitable. For the details, see the references and the site: http://okmr.yamatoblog.net/

We wrote two global book manuscripts [16] with 154 pages and [17] with many figures for some general people. Their main points are:

• The division by zero and division by zero calculus are new elementary and fundamental mathematics in the undergraduate level.

• They introduce a new space since Aristoteles $($BC384 - BC322$)$ and Euclid $($BC 3 Century - $)$ with many exciting new phenomena and properties with general interest, not specialized and difficult topics. However, their properties are mysterious and very attractive.

Meanwhile, the representations of the contents are very important and delicate with delicate feelings to the division by zero with a long and mysterious history. Therefore, we hope the representations of the division by zero as follows:

• Various book publications by many native languages and with the author’s idea and feelings.

• Some publications are like arts and some comic style books with pictures.

• Some T shirts design, some pictures, monument design may be consided.

The authors above may be expected to contribute to our culture, education, common communications and enjoyments. For the people having the interest on the above projects, we will send our book sources with many figure files.

How will be our project introducing our new world since Euclid?

Of course, as mathematicians we have to publish new books on Calculus, Differential Equations and Complex Analysis, at least and soon, in order to

Our topics will be interested in over 1000 millions people over the world on the world history.

[1] M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane, New meanings of the division by zero and interpretations on 100/0 = 0 and on 0/0 = 0, Int. J. Appl. Math. 27 $($2014$)$, no 2, pp. 191-198, DOI:

10.12732/ijam.v27i2.9.

[2] T. Matsuura and S. Saitoh, Matrices and division by zero $z/0 = 0$, Advances in Linear Algebra & Matrix Theory, 6$($2016$)$, 51-58 Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt

[3] T. Matsuura and S. Saitoh, Division by zero calculus and singular integrals. $($Submitted for publication$)$

[4] T. Matsuura, H. Michiwaki and S. Saitoh, log0 = log∞ = 0 and applications. Differential and Difference Equations with Applications.Springer Proceedings in Mathematics & Statistics.

[5] H. Michiwaki, S. Saitoh and M.Yamada, Reality of the division by zero z/0 = 0. IJAPM International J. of Applied Physics and Math. 6$($2015$)$, 1–8. http://www.ijapm.org/show-63-504-1.html

[6] H. Michiwaki, H. Okumura and S. Saitoh, Division by Zero z/0 = 0 in Euclidean Spaces, International Journal of Mathematics and Computation, 28$($1$)$$($2017$)$; 1-16.

[7] H. Okumura, S. Saitoh and T. Matsuura, Relations of 0 and ∞, Journal of Technology and Social Science $($JTSS$)$, 1$($2017$)$, 70-77.

[8] H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 $($2017.11.14$)$.

[9] H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947 International Journal of Geometry.

[10] H. Okumura and S. Saitoh, Applications of the division by zero calculus to Wasan geometry. $($Submitted for publication$)$.

[11] S. Pinelas and S. Saitoh, Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics.

[12] H. G. Romig, Discussions: Early History of Division by Zero, American Mathematical Monthly, Vol. 31, No. 8.$($Oct., 1924$)$, pp. 387-389.

[13] S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra & Matrix Theory. 4 $($2014$)$, no. 2, 87–95. http://www.scirp.org/journal/ALAMT/

[14] S. Saitoh, A reproducing kernel theory with some general applications, Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, 177$($2016$)$, 151-182. $($Springer$)$.

[15] S. Saitoh, Mysterious Properties of the Point at Infinity, arXiv:1712.09467 [math.GM]$($2017.12.17$)$.

[16] S. Saitoh, Division by zero calculus $($154 pages: draft$)$: http//okmr.yamatoblog.net/

[17] S. Saitoh and H. Okumura, Division by Zero Calculus in Figures – our New Space –

[18] 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.]]>

The download link is below:

Division by Zero Calculus

]]>

Kawauchi-cho, 5-1648-16,

Kiryu 376-0041, Japan

August 21, 2017For this fundamental idea, we should consider the

\begin{equation}

\omega_0 = \exp \left(\frac{\pi}{0}i\right)=1, \quad \frac{\pi}{0} =0.

\end{equation}

For the mean value

$$

M_n = \frac{x_1 + x_2 +... + x_n}{n},

$$

we have

$$

M_0 = 0 = \frac{0}{0}.

$$

4 Fruitful world

5 From $0$ to $0$; $0$ means all and all are $0$

6 Impossibility

\begin{equation}\tag{6.1}

ax =b

\end{equation}we have $x=0$ for $a=0, b\ne 0$ as the standard value, or the Moore-Penrose generalized inverse. This will mean in a sense, the solution does not exist; to solve the equation $($6.1$)$ is impossible. We saw for different parallel lines or different parallel planes, their common points are the origin. Certainly they have the common points of the point at infinity and the point at infinity is represented by zero. However, we can understand also that they have no solutions, no common points, because the point at infinity is an ideal point.

\begin{equation}

m\frac{d^2x}{dt^2} =0, m\frac{d^2y}{dt^2} =-mg

\end{equation}

with the initial conditions, at $t =0$

\begin{equation}

\frac{dx}{dt} = v_0 \cos \alpha , \frac{d^2x}{dt^2} = \frac{d^2y}{dt^2}=0.

\end{equation}Then, the highest high $h$, arriving time $t$, the distance $d$ from the starting point at the origin to the point $y(2t) =0$ are given by

\begin{equation}

h = \frac{v_0 \sin^2 \alpha}{2g}, d= \frac{v_0\sin \alpha}{g}

\end{equation}and

\begin{equation}

t= \frac{v_0 \sin \alpha}{g}.

\end{equation}

For the case $g=0$, we have $h=d =t=0$. We considered the case that they are the infinity; however, our mathematics means zero, which shows impossibility.

These phenomena were looked many cases on the universe; it seems that

[11] Announcement 185 $($2014.10.22$)$: The importance of the division by zero $z/0=0$.

[12] Announcement 237 $($2015.6.18$)$: A reality of the division by zero $z/0=0$ by geometrical optics.

[13] Announcement 246 $($2015.9.17$)$: An interpretation of the division by zero $1/0=0$ by the gradients of lines.

[14] Announcement 247 $($2015.9.22$)$: The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

[15] Announcement 250 $($2015.10.20$)$: What are numbers? - the Yamada field containing the division by zero $z/0=0$.

[16] Announcement 252 $($2015.11.1$)$: Circles and curvature - an interpretation by Mr. Hiroshi Michiwaki of the division by zero $r/0 = 0$.

[17] Announcement 281 $($2016.2.1$)$: The importance of the division by zero $z/0=0$.

[18] Announcement 282 $($2016.2.2$)$: The Division by Zero $z/0=0$ on the Second Birthday.

[19] Announcement 293 $($2016.3.27$)$: Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.

[20] Announcement 300 $($2016.05.22$)$: New challenges on the division by zero z/0=0.

[21] Announcement 326 $($2016.10.17$)$: The division by zero z/0=0 - its impact to human beings through education and research.

[22] Announcement 352 $($2017.2.2$)$: On the third birthday of the division by zero z/0=0.

[23] Announcement 354 $($2017.2.8$)$: What are $n = 2,1,0$ regular polygons inscribed in a disc? -- relations of $0$ and infinity.

[24] Announcement 362$($2017.5.5$)$: Discovery of the division by zero as $0/0=1/0=z/0=0$.

]]>

International Conference on Differential & Difference Equations and Applications 2017

この会議の総会において，齋藤三郎氏は DIVISION BY ZERO CALCULUS AND DIFFERENTIAL EQUATIONS と題する講演を行いました。

https://sites.google.com/site/sandrapinelas/icddea-2017/c-plenary-speakers

また，松浦勉氏も $\log 0= \log \infty =0$ AND APPLICATIONS と題する発表を行いました。

https://sites.google.com/site/sandrapinelas/icddea-2017/draft-of

これらはともに招待講演であり，0除算に関する注目の高さがうかがえます。

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Kawauchi-cho, 5-1648-16,

Kiryu 376-0041, Japan

February 7, 2017By

\frac{b}{a} \tag{1.1}

\end{equation}for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$

\begin{equation}

\frac{b}{0}=0, \tag{1.2}

\end{equation}incidentally in [6] by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in [2] for the case of real numbers. Meanwhile, the result (1.2) is a very special case of very general fractional functions in [1].

We thus should consider, for any complex number $b$, as (1.2); that is, for the mapping

\begin{equation}

W = \frac{1}{z}, \tag{1.3}

\end{equation}

the image of $z=0$ is $W=0$ $($

However, for functions, we will need some modification

For any formal Laurent expansion around $z=a$,

\begin{equation}

f(z) = \sum_{n=-\infty}^{\infty} C_n (z - a)^n, \tag{1.4}

\end{equation}

we obtain the identity, by the division by zero

\begin{equation}

f(a) = C_0. \tag{1.5}

\end{equation}

The typical example is that, as we can derive by the elementary way,

$$

\tan \frac{\pi}{2} =0.

$$

We gave many examples with geometric meanings in [5]. See [1, 2, 3, 6, 7] for the related references.In [5], many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In [4], we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.

For our ideas on the division by zero, see the survey style announcements stated in the references.

\begin{equation}

S_n= \frac{n a^2}{2} \sin \frac{2\pi}{n}\tag{2.1}

\end{equation}

and

\begin{equation}

L_n= 2n a \sin \frac{\pi}{n},\tag{2.2}

\end{equation}

respectively. For $n \ge 3$, the results are clear.

For $n =2$, we will consider two diameters that are the same. We can consider it as a generalized regular polygon inscribed in the disc as a degenerate case. Then, $S_2 =0$ and $L_2 =4a$, and the general formulas are valid.

Next, we will consider the case $n=1$. Then the corresponding regular polygon is a just diameter of the disc. Then, $S_1 =0 $ and $L_1 = 0$ that will mean that any regular polygon inscribed in the disc may not be formed and so its area and length of the side are zero.

Now we will consider the case $n=0$. Then, by the division by zero calculus, we obtain that $S_0= \pi a^2$ and $L_0 = 2\pi a$. Note that they are the area and the length of the disc. How to understand the results? Imagine contrary $n$ tending to infinity, then the corresponding regular polygons inscribed in the disc tend to the disc. Recall our new idea that the point at infinity is represented by $0$. Therefore, the results say that $n=0$ regular polygons are $n= \infty$ regular polygons inscribed in the disc in a sense and they are the disc. This is our interpretation of the theorem:

3 Our life figure

$$

\Delta_{\alpha} = \left\{ |\arg z| < \alpha; 0< \alpha < \frac{\pi}{2}\right\}.

$$

We will consider a disc inscribed in the sector $\Delta_{\alpha}$ whose center $(k,0)$ with radius $r$. Then, we have

\begin{equation}

r = k \sin \alpha. \tag{3.1}

\end{equation}

Then, note that as $k$ tends to zero, $r$ tends to zero, meanwhile $k$ tends to $+\infty$, $r$ tends to $+\infty$. However, by our division by zero calculus, we see that immediately that

\begin{equation}

[r ]_{r =\infty}= 0.\tag{3.2}

\end{equation}

For this fact, note the following:

The behavior of the space around the point at infinity may be considered by that of the origin by the linear transform $W = 1/z$. We thus see that

\begin{equation}

\lim_{z \to \infty} z = \infty,\tag{3.3}

\end{equation}

however,

\begin{equation}

[z]_{z =\infty} =0,\tag{3.4}

\end{equation}

by the division by zero. Here, $[z]_{z =\infty}$ denotes the value of the function $W=z$ at the topological point at the infinity in one point compactification by Aleksandrov. The difference of (3.3) and (3.4) is very important as we see clearly by the function $W = 1/z$ and the behavior at the origin. The limiting value to the origin and the value at the origin are different. For surprising results, we will state the property in the real space as follows:

\begin{equation}

\lim_{x\to +\infty} x =+\infty , \quad \lim_{x\to -\infty} x = -\infty,\tag{3.5}

\end{equation}

however,

\begin{equation}

[x]_{ +\infty } =0, \quad [x]_{ -\infty } =0.\tag{3.6}

\end{equation}

Of course, two points $+\infty$ and $ -\infty$ are the same point as the point at infinity. However, $\pm$ will be convenient in order to show the approach directions. In \cite{mos}, we gave many examples for this property.

On the sector, we see that from the origin as the point 0, the inscribed discs are increasing endlessly, however their final disc reduces to the origin suddenly - it seems that the whole process looks like our life in the viewpoint of our initial and final.

4

The suprising example by H. Okumura which was sent to the Institute on 2016.12.1.9:40 will show a new phenomenon at the point at infinity.

On the sector $\Delta_{\alpha}$, we shall change the angle and we consider a fixed circle $C_a, a > 0$ with radius $a$ inscribed in the sectors. We see that when the circle tends to $+\infty$, the angles $\alpha$ tend to zero. How will be the case $\alpha = 0$? Then, we will not be able to see the position of the circle. Surprisingly enough, then the circle $C_a$ is the circle with center at the origin $0$. This result is derived from the division by zero calculus for the formula

\begin{equation}

k =\frac{a}{\sin \alpha}.\tag{4.1}

\end{equation}

The two lines $ \arg z = \alpha$ and $ \arg z = -\alpha$ were tangential lines of the circle $C_a$ and now they are the positive real line. The gradient of the positive real line is of course zero. Note here that the gradient of the positive imaginary line is zero by the division by zero calculus that means $\tan \frac{\pi}{2} =0$. Therefore, we can understand that the positive real line is still a tangential line of the circle $C_a$.

This will show some great relation between zero and infinity. We can see some mysterious property around the point at infinity.

[1] L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory

[2] M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,

New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,

Int. J. Appl. Math.

[3] H. Michiwaki, S. Saitoh, and M.Yamada,

Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math.

[4] T. Matsuura and S. Saitoh, Matrices and division by zero $z/0=0$, Advances in Linear Algebra & Matrix Theory,

[5] H. Michiwaki, H. Okumura, and S. Saitoh, Division by Zero $z/0 = 0$ in Euclidean Spaces. International Journal of Mathematics and Computation Vol. 28$($2017$)$; Issue 1, 1-16.

[6] S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra & Matrix Theory.

[7] S. Saitoh, A reproducing kernel theory with some general applications,

Qian,T./Rodino,L.$($eds.$)$: Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics,

[8] Announcement 179 $($2014.8.30$)$: Division by zero is clear as z/0=0 and it is fundamental in mathematics.

[9] Announcement 185 $($2014.10.22$)$: The importance of the division by zero $z/0=0$.

[10] Announcement 237 $($2015.6.18$)$: A reality of the division by zero $z/0=0$ by geometrical optics.

[11] Announcement 246 $($2015.9.17$)$: An interpretation of the division by zero $1/0=0$ by the gradients of lines.

[12] Announcement 247 $($2015.9.22$)$: The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

[13] Announcement 250 $($2015.10.20$)$: What are numbers? - the Yamada field containing the division by zero $z/0=0$.

[14] Announcement 252 $($2015.11.1$)$: Circles and curvature - an interpretation by Mr. Hiroshi Michiwaki of the division by zero $r/0 = 0$.

[15] Announcement 281 $($2016.2.1$)$: The importance of the division by zero $z/0=0$.

[16] Announcement 282 $($2016.2.2$)$: The Division by Zero $z/0=0$ on the Second Birthday.

[17] Announcement 293 $($2016.3.27$)$: Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.

[18] Announcement 300 $($2016.05.22$)$: New challenges on the division by zero z/0=0.

[19] Announcement 326 $($2016.10.17$)$: The division by zero z/0=0 - its impact to human beings through education and research.

[20] Announcement 352 $($2017.2.2$)$: On the third birthday of the division by zero z/0=0.

Kawauchi-cho, 5-1648-16,

Kiryu 376-0041, Japan

February 2, 2017By

\frac{b}{a} \tag{1.1}

\end{equation}for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$

\begin{equation}

\frac{b}{0}=0, \tag{1.2}

\end{equation}incidentally in [12] by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in [5] for the case of real numbers.

The division by zero has a long and mysterious story over the world $($see, for example, H. G. Romig [11] and Google site with division by zero$)$ with its physical viewpoints since the document of zero in India on AD 628. In particular, note that Brahmagupta (598 - 668 ?) established the four arithmetic operations by introducing $0$ and at the same time he defined as $0/0=0$ in Brhmasphuasiddhnta. Our world history, however, stated that his definition $0/0=0$ is wrong over 1300 years, but, we will see that his definition is suitable. However, we do not know the meaning and motivation of the definition of $0/0=0$, furthermore, for the important case $1/0$ we do not know any result there. However, Sin-Ei, Takahasi $($[5]$)$ established a simple and decisive interpretation $($1.2$)$ by analyzing the extensions of fractions and by showing the complete characterization for the property $($1.2$)$:

\begin{equation}

F (b, a)F (c, d)= F (bc, ad)

\end{equation}

$$

F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.

$$

\[

F (b, 0) = 0.

\]

Note that the complete proof of this proposition is simply given by 2 or 3 lines. We

We thus should consider, for any complex number $b$, as $($1.2$)$; that is, for the mapping

\begin{equation}

W = \frac{1}{z},

\end{equation}the image of $z=0$ is $W=0$ $($

For Proposition 1, we see some confusion even among mathematicians; for the elementary function $($1.3$)$, we did not consider the value at $z=0$, and we were not able to consider a value. Many and many people consider its value by the limiting like $+\infty$, $-\infty$ or the point at infinity as $\infty$. However, their basic idea comes from

Meanwhile, the division by zero $($1.2$)$ is clear, indeed, for the introduction of $($1.2$)$, we have several independent approaches as in:

1) by the generalization of the fractions by the Tikhonov regularization and by the Moore-Penrose generalized inverse,

2) by the intuitive meaning of the fractions $($division$)$ by H. Michiwaki - repeated subtraction method,

3) by the unique extension of the fractions by S. Takahasi, as in the above,

4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from $ {\bf C} \setminus \{0\} $ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z $ from ${\bf C}$ onto ${\bf C}$,

and

5) by considering the values of functions with the mean values of functions.

Furthermore, in [6] we gave the results in order to show the reality of the division by zero in our world:

A) a field structure containing the division by zero --- the Yamada field ${\bf Y}$,

B) by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,

C) by the reflection $W =1/\overline{z}$ of $W= z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero, not the point at infinity.

And

D) by considering rotation of a right circular cone having some very interesting phenomenon from some practical and physical problem.

In [8], many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In [7], we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.

See J. A. Bergstra, Y. Hirshfeld and J. V. Tucker [2] and J. A. Bergstra [3] for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.

Meanwhile, J. P. Barukcic and I. Barukcic [1] discussed recently the relation between the divisions $0/0$, $1/0$ and special relative theory of Einstein. However, their logic seems to be curious and their results contradict with ours.

Furthermore, T. S. Reis and J.A.D.W. Anderson [9,10] extend the system of the real numbers by introducing an ideal number for the division by zero $0/0$.

Meanwhile, we should refer to up-to-date information:

$($2$)$: In the elementary school, we should introduce the concept of division $($fractions$)$ by the idea of repeated subtraction method by H. Michiwaki whose method is applied in computer algorithm and in old days for calculation of division. This method will give a simple and clear method for calculation of division and students will be happy to apply this simple method at the first stage. At this time, they will be able to understand that the division by zero is clear and trivial as $a/0=0$ for any $a$. Note that Michiwaki knows how to apply his method to the complex number field.

$($3$)$: For the introduction of the elemetary function $y= 1/x$, we should give the definition of the function at the origin $x=0$ as $y = 0$ by the division by zero idea and we should apply this definition for the occasions of its appearences, step by step, following the curriculum and the results of the division by zero.

$($4$)$: For the idea of the Euclidean space $($plane$)$, we should introduce, at the first stage, the concept of stereographic projection and the concept of the point at infinity - one point compactification. Then, we will be able to see the whole Euclidean plane, however, by the division by zero,

$($5$)$: The inversion of the origin with respect to a circle with center the origin is the origin itself, not the point at infinity - the very classical result is wrong. We can also prove this elementary result by many elementary ways.

$($6$)$: We should change the concept of gradients; on the usual orthogonal coordinates $(x,y)$, the gradient of the $y$ axis is zero; this is given and proved by the fundamental result

$\tan (\pi/2) =0$. The result is also trivial from the definition of the Yamada field. For the Fourier coefficients $a_k$ of a function :

$$\frac{a_k \pi k^3}{4}$$\begin{equation}

= \sin (\pi k) \cos (\pi k) + 2 k^2 \pi^2 \sin(\pi k) \cos (\pi k) + 2\pi (\cos (\pi k) )^2 - \pi k, \tag{2.1}

\end{equation}for $k=0$, we obtain immediately

\begin{equation}

a_0 = \frac{8}{3}\pi^2 \tag{2.2}

\end{equation}

$($see [15], (3.4)$)$$($ - Difficulty in Maple for specialization problems$)$.

These results are derived also from the

\begin{equation}

f(z) = \sum_{n=-\infty}^{\infty} C_n (z - a)^n, \tag{2.3}

\end{equation}

we obtain the identity, by the division by zero

\begin{equation}

f(a) = C_0. \tag{2.4}

\end{equation} The typical example is that, as we can derive by the elementary way,

$$

\tan \frac{\pi}{2} =0.

$$

We gave many examples with geometric meanings in [8].

This fundamental result leads to the important new definition:

From the viewpoint of the division by zero, when there exists the limit, at $ x$

\begin{equation}

f^\prime(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h} =\infty \tag{2.5}

\end{equation} or \begin{equation}

f^\prime(x) = -\infty, \tag{2.6}

\end{equation}

both cases, we can write them as follows:

\begin{equation}

f^\prime(x) = 0. \tag{2.7}

\end{equation}

For the elementary ordinary differential equation

\begin{equation}

y^\prime = \frac{dy}{dx} =\frac{1}{x}, \quad x > 0,\tag{2.8}

\end{equation} how will be the case at the point $x = 0$? From its general solution, with a general constant $C$ \begin{equation}

y = \log x + C,\tag{2.9}

\end{equation} we see that, by the division by zero,

\begin{equation}

y^\prime (0)= \left[ \frac{1}{x}\right]_{x=0} = 0,\tag{2.10}

\end{equation} that will mean that the division by zero $($1.2$)$ is very natural.

In addition, note that the function $y = \log x$ has infinite order derivatives and all the values are zero at the origin, in the sense of the division by zero.

However, for the derivative of the function $y = \log x$, we have to fix the sense at the origin, clearly, because the function is not differentiable, but it has a singularity at the origin. For $x >0$, there is no problem for $($2.8$)$ and $($2.9$)$. At $x = 0$, we see that we can not consider the limit in the sense $($2.5$)$. However, for $x >0$ we have $($2.8$)$ and

\begin{equation}

\lim_{x \to +0} \left(\log x \right)^\prime = +\infty.\tag{2.11}

\end{equation} In the usual sense, the limit is $+\infty$, but in the present case, in the sense of the division by zero, we have:

\begin{equation}

\left[ \left(\log x \right)^\prime \right]_{x=0}= 0\tag{2.12}

\end{equation}

and we will be able to understand its sense graphycally.

By the new interpretation for the derivative, we can arrange many formulas for derivatives, by the division by zero. We can modify many formulas and statements in calculus and we can apply our concept to the differential equation theory and the universe in connetion with derivatives.

$($7$)$: We shall introduce the typical division by zero calculus.

For the integral

\begin{equation}

\int x(x^{2}+1)^{a}dx=\frac{(x^{2}+1)^{a+1}}{2(a+1)}\quad(a\ne-1),\tag{2.13}

\end{equation}

we obtain, by the division by zero calculus,

\begin{equation}

\int x(x^{2}+1)^{-1}dx=\frac{\log(x^{2}+1)}{2}. \tag{2.14}

\end{equation}

For example, in the ordinary differential equation

\begin{equation}

y^{\prime\prime} + 4 y^{\prime} + 3 y = 5 e^{- 3x},\tag{2.15}

\end{equation}

in order to look for a special solution, by setting $y = A e^{kx}$ we have, from

\begin{equation}

y^{\prime\prime} + 4 y^{\prime} + 3 y = 5 e^{kx},\tag{2.16}

\end{equation}

\begin{equation}

y = \frac{5 e^{kx}}{k^2 + 4 k + 3}.\tag{2.17}

\end{equation}

For $k = -3$, by the division by zero calculus, we obtain

\begin{equation}

y = e^{-3x} \left( - \frac{5}{2}x - \frac{5}{4}\right),\tag{2.18}

\end{equation}

and so, we can obtain the special solution

\begin{equation}

y = - \frac{5}{2}x e^{-3x}.\tag{2.19}

\end{equation}

In those examples, we were able to give valuable functions for denominator zero cases. The division by zero calculus may be applied to many cases as a new fundamental calculus over l'Hopital's rule.

$($8$)$: When we apply the division by zero to functions, we can consider, in general, many ways. For example,

for the function $z/(z-1)$, when we insert $z=1$ in numerator and denominator, we have

\begin{equation}

\left[\frac{z}{z-1}\right]_{z = 1} = \frac{1}{0} =0. \tag{2.20}

\end{equation}

However,

from the identity -- the Laurent expansion around $z=1$,

\begin{equation}

\frac{z}{z-1} = \frac{1}{z-1} + 1,\tag{2.21}

\end{equation}

we have

\begin{equation}

\left[\frac{z}{z-1}\right]_{z = 1} = 1.\tag{2.22}

\end{equation}

For analytic functions we can give uniquely determined values at isolated singular points by the values by means of the Laurent expansions as the division by zero calculus, however, the values by means of the Laurent expansions are not always reasonable. We will need to consider many interpretations for reasonable values. In many formulas in mathematics and physics, however, we can see that the division by zero calculus is reasonably valid. See [5,6].

The division by zero is directly related to the Einstein's theory and various physical problems containing the division by zero. Now we should check the theory and the problems by the concept of the RIGHT and DEFINITE division by zero. Now is the best time since 100 years from Albert Einstein. It seems that the background knowledge is timely fruitful.

Note that the Big Bang also may be related to the division by zero like the blackholes.

The above Professors listed are wishing the contributions in order to avoid the division by zero trouble in computers. Now, we should arrange new computer systems in order not to meet the division by zero trouble in computer systems.

By the division by zero calculus, we will be able to overcome troubles in Maple for specialization problems as in stated.

The division by zero may be related to religion, philosophy and the ideas on the universe; it will create a new world. Look at the new world introduced.

We are standing on a new generation and in front of the new world, as in the discovery of the Americas. Should we push the research and education on the division by zero?

In order to make clear the problem, we give a prototype example. We have the identity by the divison by zero calculus: For

\begin{equation}

f(z) = \frac{1 + z}{1- z}, \quad f(1) = -1.\tag{6.1}

\end{equation}

From the real part and imaginary part of the function, we have, for $ z= x +iy$

\begin{equation}

\frac{1 - x^2 - y^2}{(1 - x)^2 + y^2} =-1, \quad \text{at}\quad (1,0) \tag{6.2}

\end{equation}

and

\begin{equation}

\frac{y}{(1- x)^2 + y^2} = 0, \quad \text{at}\quad (1,0), \tag{6.3}

\end{equation}

respectively. Why the differences do happen? In general, we are interested in the above open problem. Recall our definition for the division by zero calculus.

[1] J. P. Barukcic and I. Barukcic, Anti Aristotle - The Division of Zero by Zero. Journal of Applied Mathematics and Physics,

[2] J. A. Bergstra, Y. Hirshfeld and J. V. Tucker, Meadows and the equational specification of division $($arXiv:0901.0823v1[math.RA] 7 Jan 2009$)$.

[3] J.A. Bergstra, Conditional Values in Signed Meadow Based Axiomatic Probability Calculus,

arXiv:1609.02812v2[math.LO] 17 Sep 2016.

[4] L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory

[5] M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane, New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$, Int. J. Appl. Math.

[6] H. Michiwaki, S. Saitoh, and M.Yamada, Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math.

[7] T. Matsuura and S. Saitoh, Matrices and division by zero $z/0=0$, Advances in Linear Algebra & Matrix Theory,

[8] H. Michiwaki, H. Okumura, and S. Saitoh, Division by Zero $z/0 = 0$ in Euclidean Spaces. International Journal of Mathematics and Computation

[9] T. S. Reis and J.A.D.W. Anderson, Transdifferential and Transintegral Calculus, Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I WCECS 2014, 22-24 October, 2014, San Francisco, USA

[10] T. S. Reis and J.A.D.W. Anderson, Transreal Calculus, IAENG International J. of Applied Math.,

[11] H. G. Romig, Discussions: Early History of Division by Zero, American Mathematical Monthly, Vol. 31, No. 8. $($Oct., 1924$)$, pp. 387-389.

[12] S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra & Matrix Theory.

[13] S. Saitoh, A reproducing kernel theory with some general applications, Qian,T./Rodino,L.$($eds.$)$: Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics,

[14] S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics,

[15] Introduction to Maple - UBC Mathematics https://www.math.ubc.ca/~israel/m210/lesson1.pdf

[16] Announcement 179 $($2014.8.30$)$: Division by zero is clear as z/0=0 and it is fundamental in mathematics.

[17] Announcement 185 $($2014.10.22$)$: The importance of the division by zero $z/0=0$.

[18] Announcement 237 $($2015.6.18$)$: A reality of the division by zero $z/0=0$ by geometrical optics.

[19] Announcement 246 $($2015.9.17$)$: An interpretation of the division by zero $1/0=0$ by the gradients of lines.

[20] Announcement 247 $($2015.9.22$)$: The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

[21] Announcement 250 $($2015.10.20$)$: What are numbers? - the Yamada field containing the division by zero $z/0=0$.

[22] Announcement 252 $($2015.11.1$)$: Circles and curvature - an interpretation by Mr. Hiroshi Michiwaki of the division by zero $r/0 = 0$.

[23] Announcement 281 $($2016.2.1$)$: The importance of the division by zero $z/0=0$.

[24] Announcement 282 $($2016.2.2$)$: The Division by Zero $z/0=0$ on the Second Birthday.

[25] Announcement 293 $($2016.3.27$)$: Parallel lines on the Euclidean plane from the viewpoint of division by zero $1/0=0$.

[26] Announcement 300 $($2016.05.22$)$: New challenges on the division by zero $z/0=0$.

[27] Announcement 326 $($2016.10.17$)$: The division by zero $z/0=0$ - its impact to human beings through education and research.

]]>

Kawauchi-cho, 5-1648-16,

Kiryu 376-0041, Japan

E-mail: kbdmm360@yahoo.co.jp

October 17, 2016By

\frac{b}{a} \tag{1.1}

\end{equation}for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$

\begin{equation}

\frac{b}{0}=0, \tag{1.2}

\end{equation}incidentally in [10] by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in [4] for the case of real numbers.

The division by zero has a long and mysterious story over the world $($see, for example, Google site with division by zero$)$ with its physical viewpoints since the document of zero in India on AD 628, however, Sin-Ei, Takahasi [4] established a simple and decisive interpretation $($1.2$)$ by analyzing the extensions of fractions and by showing the complete characterization for the property $($1.2$)$:

\begin{equation}

F (b, a)F (c, d)= F (bc, ad)

\end{equation}

$$

F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.

$$

\[

F (b, 0) = 0.

\]

Note that the complete proof of this proposition is simply given by 2 or 3 lines. We should define $F(b,0)=b/0=0$ in general.

We thus should consider, for any complex number $b$, as $($1.2$)$; that is , for the mapping

\begin{equation}

w = \frac{1}{z}, \tag{1.3}

\end{equation}the image of $z=0$ is $w=0$ $($

However, the division by zero $($1.2$)$ is now clear, indeed, for the introduction of $($1.2$)$, we have several independent approaches as in:

1) by the generalization of the fractions by the Tikhonov regularization and by the Moore-Penrose generalized inverse,

2) by the intuitive meaning of the fractions $($division$)$ by H. Michiwaki - repeated subtraction method,

3) by the unique extension of the fractions by S. Takahasi, as in the above,

4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from $ {\bf C} \setminus \{0\} $ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z $ from ${\bf C}$ onto ${\bf C}$,

and

5) by considering the values of functions with the mean values of functions.

Furthermore, in [6] we gave the results in order to show the reality of the division by zero in our world:

A) a field structure containing the division by zero --- the Yamada field ${\bf Y}$,

B) by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,

C) by the reflection $W =1/\overline{z}$ of $W= z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero, not the point at infinity.

And

D) by considering rotation of a right circular cone having some very interesting phenomenon from some practical and physical problem.

In [7], many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In [5], we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.

See J. A. Bergstra, Y. Hirshfeld and J. V. Tucker [2] for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.

Meanwhile, J. P. Barukcic and I. Barukcic [1] discussed recently the relation between the divisions $0/0$, $1/0$ and special relative theory of Einstein. However, their logic seems to be curious and their results contradict with ours.

Furthermore, T. S. Reis and J.A.D.W. Anderson [8,9] extend the system of the real numbers by introducing an ideal number for the division by zero $0/0$.

Meanwhile, we should refer to up-to-date information:

$(2)$: In the elementary school, we should introduce the concept of division by the idea of repeated subtraction method by H. Michiwaki whoes method is applied in computer algorithmu and in old days for calculation of division. This method will give a simple and clear method for calculation of division and students will be happy to apply this simple method at the first stage. At this time, they will be able to understand that the division by zero is clear and trivial as $a/0=0$ for any $a$. Note that Michiwaki knows how to apply his method to the complex number field.

$(3)$: For the introduction of the elemetary function $y= 1/x$, we should give the definition of the function at the origin $x=0$ as $y = 0$ by the division by zero idea and we should apply this definition for the occasions of its appearences, step by step, following the curriculum and the results of the division by zero.

$(4)$: For the idea of the Euclidean space (plane), we should introduce, at the first stage, the concept of steleographic projection and the concept of the point at infinity - one point compactification. Then, we will be able to see the whole Euclidean plane, however, by the division by zero, the point at infinity is represented by zero. We can teach the very important fact with many geometric and analytic geometry methods. These topics will give great pleasant feelings to many students. Interesting topics are: parallel lines, what is a line? - a line contains the origin as an isolated point for the case that the native line does not through the origin. All the lines pass the origin, our space is not the Eulcildean space and is not Aristoteles for the strong discontinuity at the point at infinity (at the origin). - Here note that an orthogonal coordinates should be fixed first for our all arguments.

$(5)$: The inversion of the origin with respect to a circle with center the origin is the origin itself, not the point at infinity - the very classical result is wrong. We can also prove this elementary result by many elementary ways.

$(6)$: We should change the concept of gradients; on the usual orthogonal coordinates $(x,y)$, the gradient of the $y$ axis is zero; this is given and proved by the fundamental result $\tan (\pi/2) =0$. The result is trivial in the definition of the Yamada field. This result is derived also from the

\begin{equation}

f(z) = \sum_{n=-\infty}^{\infty} C_n (z - a)^n,\tag{2,1}

\end{equation}

\begin{equation}

f(a) = C_0. \tag{2,2}

\end{equation}

From the viewpoint of the division by zero, when there exists the limit, at $ x$

\begin{equation}

f^\prime(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h} =\infty \tag{2,3}

\end{equation}

or

\begin{equation}

f^\prime(x) = -\infty, \tag{2,4}

\end{equation}

both cases, we can write them as follows:

\begin{equation}

f^\prime(x) = 0. \tag{2,5}

\end{equation}

y^\prime = \frac{dy}{dx} =\frac{1}{x}, \quad x > 0, \tag{2,6}

\end{equation}

how will be the case at the point $x = 0$? From its general solution, with a general constant $C$

\begin{equation}

y = \log x + C, \tag{2,7}

\end{equation}

we see that, by the division by zero,

\begin{equation}

y^\prime (0)= \left[ \frac{1}{x}\right]_{x=0} = 0, \tag{2,8}

\end{equation}

that will mean that the division by zero (1.2) is very natural.

\begin{equation}

\lim_{x \to +0} \left(\log x \right)^\prime = +\infty. \tag{2,9}

\end{equation}

In the usual sense, the limit is $+\infty$, but in the present case, in the sense of the division by zero, we have:

\begin{equation}

\left[ \left(\log x \right)^\prime \right]_{x=0}= 0 \tag{2,10}

\end{equation}

and we will be able to understand its sense graphycally.

By the new interpretation for the derivative, we can arrange many formulas for derivatives, by the division by zero. We can modify many formulas and statements in calculus and we can apply our concept to the differential equation theory and the universe in connetion with derivatives.

For the integral

\begin{equation}

\int x(x^{2}+1)^{a}dx=\frac{(x^{2}+1)^{a+1}}{2(a+1)}\quad(a\ne-1), \tag{2,11}

\end{equation}

we obtain, by the division by zero,

\begin{equation}

\int x(x^{2}+1)^{-1}dx=\frac{\log(x^{2}+1)}{2}. \tag{2,12}

\end{equation}

\begin{equation}

x^{\prime \prime}(t) =g -kx^{\prime}(t) \tag{2,13}

\end{equation}

with the initial conditions

\begin{equation}

x(0) = -h, x^{\prime}(0) =0. \tag{2,14}

\end{equation}

Then we have the solution

\begin{equation}

x(t) = \frac{g}{k}t + \frac{g(e^{-kt}- 1)}{k^2} - h. \tag{2,15}

\end{equation}

Then, for $k=0$, we obtain, immediately, by the division by zero

\begin{equation}

x(t) = \frac{1}{2}g t^2 -h. \tag{2,16}

\end{equation}

\begin{equation}

\left[\frac{z}{z-1}\right]_{z = 1} = \frac{1}{0} =0. \tag{2,17}

\end{equation}

However,

from the identity --

the Laurent expansion around $z=1$,

\begin{equation}

\frac{z}{z-1} = \frac{1}{z-1} + 1, \tag{2,18}

\end{equation}

we have

\begin{equation}

\left[\frac{z}{z-1}\right]_{z = 1} = 1. \tag{2,19}

\end{equation}

For analytic functions we can give uniquely determined values at isolated singular points by the values by means of the Laurent expansions as the division by zero calculus, however, the values by means of the Laurent expansions are not always reasonable. We will need to consider many interpretations for reasonable values. In many formulas in mathematics and physics, however, we can see that the division by zero calculus is reasonably valid. See [4,6]

3 Albert Einstein's biggest blunder

Note that the Big Bang also may be related to the division by zero like the blackholes.

4 Computer systems

By the division by zero calculus, we will be able to overcome troubles in Maple for specialization problems.

5 General ideas on the universe

We are standing on a new generation and in front of the new world, as in the discovery of the Americas. Should we push the research and education on the division by zero?

[2] J. A. Bergstra, Y. Hirshfeld and J. V. Tucker, Meadows and the equational specification of division $($arXiv:0901.0823v1[math.RA] 7 Jan 2009$)$.

[3] L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory 7 $($2013$)$, no. 4, 1049-1063.

[4] M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane, New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,

Int. J. Appl. Math. 27 $($2014$)$, no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.

[5] T. Matsuura and S. Saitoh, Matrices and division by zero $z/0=0$, Linear Algebra \& Matrix Theory,

[6] H. Michiwaki, S. Saitoh, and M.Yamada, Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. 6 $($2015$)$, 1--8. http://www.ijapm.org/show-63-504-1.html

[7] H. Michiwaki, H. Okumura, and S. Saitoh, Division by Zero $z/0 = 0$ in Euclidean Spaces. International Journal of Mathematics and Computation $($in press$)$.

[8] T. S. Reis and J.A.D.W. Anderson, Transdifferential and Transintegral Calculus, Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I WCECS 2014, 22-24 October, 2014, San Francisco, USA

[9] T. S. Reis and J.A.D.W. Anderson, Transreal Calculus, IAENG International J. of Applied Math., 45 $($2015$)$: IJAM 45 1 06.

[10] S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. 4 $($2014$)$, no. 2, 87--95. http://www.scirp.org/journal/ALAMT/

[11] 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.

[12] Announcement 179 $($2014.8.30$)$: Division by zero is clear as $z/0=0$ and it is fundamental in mathematics.

[13] Announcement 185 $($2014.10.22$)$: The importance of the division by zero $z/0=0$.

[14] Announcement 237 $($2015.6.18$)$: A reality of the division by zero $z/0=0$ by geometrical optics.

[15] Announcement 246 $($2015.9.17$)$: An interpretation of the division by zero $1/0=0$ by the gradients of lines.

[16] Announcement 247 $($2015.9.22$)$: The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

[17] Announcement 250 $($2015.10.20$)$: What are numbers? - the Yamada field containing the division by zero $z/0=0$.

[18] Announcement 252 $($2015.11.1$)$: Circles and curvature - an interpretation by Mr. Hiroshi Michiwaki of the division by zero $r/0 = 0$.

[19] Announcement 281$($2016.2.1$)$: The importance of the division by zero $z/0=0$.

[20] Announcement 282$($2016.2.2$)$: The Division by Zero $z/0=0$ on the Second Birthday.

[21] Announcement 293$($2016.3.27$)$: Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.

[22] Announcement 300$($2016.05.25$)$: New challenges on the the division by zero 1/0=0.