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# 愛は0なのでゼロ除算でしたw引用ツイート

2020年1月30日(木) 8:57

もずんこ@PZejc39aFrb6Gp8僕の場合、愛は0なのでゼロ除算でしたw引用ツイートもずんこ@PZejc39aFrb6Gp8 · 16時間この曲永遠に分子教えてくれなくて草

ゼロ除算は面白いので、参照して下さい：

ゼロ除算はできます。ゼロ除算の本当の意味を知らない人たちはできないという。

どのようにできたのでしょうか：５W1H：私達の結果を見てください：

まだ計算機は　ゼロ除算ができないですね。できるようになったら　直ぐに教えてください。　日本が先端を切って　できたと宣言して欲しい。

ゼロ除算は　新しい発見であり、毎日のように新しい局面が現れています。

どのような結果が出るのでしょうか。　詳しく知りたい。　Isabelle/HOL 　正確に私たちの結果を出していますが、　プログラムを組んで出しています。

もうすぐに世界の計算機はゼロ除算ができるようになるでしょう。　日本が先取して世界に宣言して欲しい。既にできる数学は完成しています。

それは有名な話しでありふれていないでしょうか。　人間は飽きるものですね。新奇な凄い世界が拓かれています。　ご覧あれ：できない、ダメでは　数学にならず、創造的な研究はできず、　空回りになってしまいます。　できないことを可能に考えるのが　数学の　神聖な歴史　です。ゼロの意味を考える必要があります。ゼロには数字の、算数の意味はありますが、同時に　ゼロ回やるはやらないこと意味し、２－２＝０は　同じものを引いたら無くなってしまう、ないことを意味します。

ゼロで割るも　それは割らないことを意味し、割られた数は存在しません。

ゼロです。きちんとした意味が　やっと2000年以上の迷信から覚めた。

2000年の盲点を　発見して、新しい世界が　拓かれます

この内容は、世界の一億人以上の人が分かって、感動し、新しい世界を　見ることができるだろう。

ご覧あれ：

まずは、発見するや、これは世界史上の事件と　直ちに判断して、詳しい記録を取ってきている。声明１４８が　ゼロ除算に関する声明の最初で、その付録にその作製過程が生き生きと書かれている。関数の場合と数値の場合は本質的に異なるが、そのことは最初の論文にも既に述べられていたが、ゼロ除算とゼロ除算算法の違いの認識は　明瞭ではなかったと言える。人間の認識とは不思議なもので、分かったや理解した　には深さがあり、表現も微妙に変化してくる。　6年間も楽しみながら研究してきたが　ゼロ除算が　すっかり分かったと感じるようになったのは　つい最近のことであると感じている。それは　ゼロ除算算法は　解析関数で考えられると考えてきたのが、実は　微分可能な関数で定義されると認識したからである：

viXra:2001.0091 submitted on 2020-01-06 17:52:07,

Division by Zero Calculus for Differentiable Functions L'Hôpital's Theorem
Versions

それですっきり分かったと　感じられるようになった。ゼロ除算は　何にもかも当たり前である。

For a function $y = f(x)$ which is $n$ order differentiable at $x =a$, we will
{\bf define} the value of the function, for $n >0$

$$\frac{f(x)}{(x -a)^n}$$

at the point $x=a$ by the value

$$\frac{f^{(n)}(a)}{n!}.$$

For the important case of $n=1$,

\frac{f(x)}{x -a} |_{x=a} = f^\prime(a).

これで、分母がゼロになる場合の数学の公式において　いわば考えてはならない特異点での値が定義され、新しい世界が拓かれる。この公式の初等数学、

ユークリッド幾何学、代数学、解析学への基本的な大きな影響についてはここでは述べない。　むしろ、将来を楽しく展望しよう。

ゼロ除算における割り算とは、普通の意味ではなく、その意味の上記のような捉え方で、ゼロの意味の新しい発見であったと言える。

１．ゼロで割れないのは、おかしい。

ゼロで割ることは、割らないことであり、割られて対応する数はなく、ゼロで割れば　ゼロであること。ゼロには　しないこと、無い状態や　できないことを表す　性質があること。

２．最も基本的な反比例の、直角双曲線関数　y=1/x　の　原点での値が分からない。

その値は　ゼロで、大きな意味を有する。

3．x, y　座標平面で　y　軸の勾配が分からない。　平面上に垂直に立った柱の勾配が分からない。

それらは、傾きゼロであること。　微分係数がゼロ、tan (\pi/2) =0.　大きな意味がある。

これらの先は本格的な研究を必要として　今後の研究課題である。特に新しく現れた空間のモデル horn torus　上の解析学は　MATHEMATICA等の数式処理が可能である手段で武装しなければ立ち入ることのできない　雄大な分野と考えられる：

viXra:1904.0052 submitted on 2019-04-03 20:31:13,

D\"aumler's Horn Torus Model and\\ Division by Zero \\ - Absolute Function
Theory -\\ New World

また、アインシュタインの最大の懸案であったとされるゼロ除算の解明にしたがって　物理学への大きな影響を期待している。

また数学教育界の数学の教科書の修正構想の推進を期待している。

以　上

\documentclass[12pt]{article}

\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}

\numberwithin{equation}{section}

\begin{document}

\title{\bf Announcement 540: The 6th birthday of the division by zero $z/0=0$ \\

(2020.2.2)}

\author{{\it Institute of Reproducing Kernels}\\

Kawauchi-cho, 5-1648-16,\\

Kiryu 376-0041, Japan\\

{\bf kbdmm360@yahoo.co.jp}\\

}

\date{\today}

\maketitle

The 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 Aristotelēs (BC384 - BC322) and Euclid (BC 3 Century - ), and the division by zero is since Brahmagupta (598 - 668 ?).

For the details, see the references and the site: http://okmr.yamatoblog.net/

We have a global book manuscript with 300 pages and for some delay of the book publication, we are publishing the materials in the series of viXra as in stated in the last part.

We stated in the preface and last section of the manuscript as follows:

\bigskip

{\bf Preface}

\medskip

The division by zero has a long and mysterious history all over the world (see, for example, \index{Romig, H. G.} \cite{boyer, romig} \index{Boyer, C. B.} 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 \index{Brahmagupta} Brahmagupta (598 -668 ?) established four arithmetic operations by introducing $0$ and at the same time he defined as $0/0=0$ in

Brāhmasphuṭasiddhānta. \index{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 \index{Moore-Penrose} \index{Tikhonov regularization} to the fundamental equation $az=b$, whose solution leads to the definition of $z =b/a$.

However, the result (definition) will show that

for the elementary mapping

$$W = \frac{1}{z},$$

the image of $z=0$ is $W=0$ ({\bf 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 \index{Riemann sphere} (\cite{ahlfors}). As the representation of the \index{point at infinity} point at infinity of the \index{Riemann sphere} Riemann sphere by the

zero $z = 0$, we will see some delicate relations between $0$ and $\infty$ which show a strong \index{discontinuity}

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 $+\infty$ and $- \infty$ or the

point at infinity as $\infty$. However, their basic idea comes from {\bf continuity} with the common sense or

based on the basic idea of Aristotele. \index{Aristotele} --

For the related Greek philosophy, see \cite{a,b,c}. However, as the division by zero we will consider the 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 (\cite{mos,osm}) for example. Therefore, the division by zero will give great impacts to calculus, Euclidean geometry, analytic geometry, differential 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 Aristotele 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 Aristotelēs - Euclid.

\bigskip

\bigskip

\bigskip

{\bf Conclusion}

\medskip

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 elementary 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 to 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 \index{Guan, Q.}Qi'an Guan \cite{guan} since 40 years proposed in \cite{s88} (the original is in \cite {s79}).

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.

Our division by zero research group wonders why our elementary results may still not be accepted by some wide world.

\medskip

%We hope that:

%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.

% \medskip

From the funny history of the division by zero, we will be able to realize that

\medskip

human beings are full of prejudice and prejudice, and are narrow-minded, essentially.

\medskip

It seems that the long history of the division by zero is our shame and our mathematics in the elementary level has basic missings. Meanwhile, we have still great confusions and wrong ideas on the division by zero. Therefore, we would like to ask for the good corrections for the wrong ideas and some official approval for our division by zero as our basic duties.

\bibliographystyle{plain}

\begin{thebibliography}{10}

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\bibitem{boyer}

C. B. Boyer, An early reference to division by zero, The Journal of the American Mathematical Monthly, {\bf 50} (1943), (8), 487- 491. Retrieved March 6, 2018, from the JSTOR database.

\bibitem{cs}

L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.

\bibitem{guan}

Q. Guan, A proof of Saitoh's conjecture for conjugate Hardy H2 kernels, J. Math. Soc. Japan,
{\bf 71}, No. 4 (2019), 1173–-1179.

doi: 10.2969/jmsj/80668066

\bibitem{kmsy}

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. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.

\bibitem{ms16}

T. Matsuura and S. Saitoh,

Matrices and division by zero $z/0=0$,

Advances in Linear Algebra \& Matrix Theory, {\bf 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.

\bibitem{mms18}

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.

\bibitem{msy}

H. Michiwaki, S. Saitoh and

Reality of the division by zero $z/0=0$. IJAPM
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\bibitem{mos}

H. Michiwaki, H. Okumura and S. Saitoh,

Division by Zero $z/0 = 0$ in Euclidean Spaces,

International Journal of Mathematics and Computation, {\bf 2}8(2017); Issue 1, 1-16.

\bibitem{oku18}

H. Okumura, Is It Really Impossible To Divide By Zero? Biostat Biometrics Open Acc J. 2018; 7(1): 555703.

DOI: 10.19080/BBOJ.2018.07.555703.

\bibitem{o}

H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947 International
Journal of Geometry. {\bf 7}(2018), No. 1, 17-20.

\bibitem{ok1812}

H. Okumura,

An Analogue to Pappus Chain theorem with Division by Zero, Forum Geom., {\bf 18} (2018), 409--412.

\bibitem{ok18}

H. Okumura, Solution to 2017-1 Problem 4 with division by

zero, Sangaku Journal of Mathematics, {\bf 2} (2018), 27-30.

\bibitem{okumurafield}

H. Okumura, To Divide by Zero is to Multiply by Zero, viXra: 1811.0283 submitted on 2018-11-18 20:46:54.

\bibitem{okumura19a}

H. Okumura, A Remark of the Definition of $0/0=0$ by Brahmagupta, viXra:1902.0221 submitted on 2019-02-12 23:41:31.

\bibitem{okumura20}

H. Okumura, A Chain of Circles Touching a Circle and Its Tangent and Division by Zero,

viXra:2001.0034 submitted on 2020-01-03 01:08:58.

\bibitem{okumura20b}

H. Okumura, Pappus Chain and Division by Zero, viXra:2001.0123 replaced on 2020-01-08 06:57:36.

\bibitem{osm}

H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and
$\infty$,

Journal of Technology and Social Science (JTSS), {\bf 1}(2017), 70-77.

\bibitem{os}

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

\bibitem{os18april}

H. Okumura and S. Saitoh,

Harmonic Mean and Division by Zero,

Dedicated to Professor Josip Pe\v{c}ari\'{c} on the occasion of his 70th birthday, Forum Geometricorum, {\bf
18} (2018), 155—159.

\bibitem{os18}

H. Okumura and S. Saitoh,

Remarks for The Twin Circles of Archimedes in a Skewed Arbelos by H. Okumura and M. Watanabe, Forum Geometricorum, {\bf 18}(2018), 97-100.

\bibitem{os18e}

H. Okumura and S. Saitoh,

Applications of the division by zero calculus to Wasan geometry.

GLOBAL JOURNAL OF ADVANCED RESEARCH ON CLASSICAL AND MODERN GEOMETRIES” (GJARCMG), {\bf 7}(2018), 2,
44--49.

\bibitem{os1811}

H. Okumura and S. Saitoh,

Wasan Geometry and Division by Zero Calculus,

Sangaku Journal of Mathematics (SJM), {\bf 2 }(2018),
57--73.

\bibitem{ps18}

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.

\bibitem{romig}

H. G. Romig, Discussions: Early History of Division by Zero,

American Mathematical Monthly, {\bf 3}1, No. 8. (Oct., 1924), 387-389.

\bibitem{s79}

S. Saitoh, The Bergman norm and the Szeg\"{o} norm, Trans. Amer. Math. Soc., {\bf 249} (1979), no. 2, 261-279.

\bibitem{s88}

S. Saitoh, Theory of reproducing kernels and its applications. Pitman Research Notes in Mathematics Series, {\bf 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.

\bibitem{s14}

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

\bibitem{s16}

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

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\bibitem{s17}

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\bibitem{s18}

S. Saitoh, Division by zero calculus (draft): http//okmr.yamatoblog.net/

\bibitem{ttk}

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\bibitem{a}

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

\bibitem{b}

http://publish.uwo.ca/~jbell/The 20Continuous.pdf

\bibitem{c}

http://www.mathpages.com/home/kmath526/kmath526.htm

\bibitem{ann179}

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

\bibitem{ann185}

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

\bibitem{ann237}

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

\bibitem{ann246}

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

\bibitem{ann247}

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

\bibitem{ann250}

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

\bibitem{ann252}

Announcement 252 (2015.11.1): Circles and

curvature - an interpretation by Mr.

Hiroshi Michiwaki of the division by

zero $r/0 = 0$.

\bibitem{ann281}

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

\bibitem{ann282}

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

\bibitem{ann293}

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

\bibitem{ann300}

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

\bibitem{ann326}

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

\bibitem{ann352}

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

\bibitem{ann354}

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

\bibitem{362}

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

\bibitem{380}

Announcement 380 (2017.8.21):
What is the zero?

\bibitem{388}

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

\bibitem{409}

Announcement 409 (2018.1.29.): Various Publication Projects
on the Division by Zero.

\bibitem{410}

Announcement 410 (2018.1 30.):
What is mathematics? -- beyond logic; for great challengers on the division by zero.

\bibitem{412}

Announcement 412 (2018.2.2.): The 4th birthday of the division by zero $z/0=0$.

\bibitem{433}

Announcement 433 (2018.7.16.):
Puha's Horn Torus Model for the Riemann Sphere From the Viewpoint of Division by Zero.

\bibitem{448}

Announcement 448 (2018.8.20):
Division by Zero;

Funny History and New World.

\bibitem{454}

Announcement 454 (2018.9.29): The International Conference on Applied Physics and Mathematics, Tokyo, Japan, October 22-23.

\bibitem{460}

Announcement 460 (2018.11.06): Change the Poor Idea to the Definite Results For the Division by Zero - For the Leading Mathematicians.

\bibitem{461}

Announcement 461 (2018.11.10): An essence of division by zero and a new axiom.

\end{thebibliography}

Saburou Saitoh:

\medskip

[30] viXra:2001.0586 submitted on 2020-01-27 16:28:38,

Division by Zero Calculus, Derivatives and Laurent's Expansion

[29] viXra:2001.0091 submitted on 2020-01-06 17:52:07,

Division by Zero Calculus for Differentiable Functions L'Hôpital's Theorem Versions.

[28] viXra:1912.0300 submitted on 2019-12-16 18:37:53,

Essential Problems on the Origins of Mathematics; Division by Zero Calculus and New World.

[27] viXra:1911.0115 submitted on 2019-11-06 18:56:03,

General Order Differentials and Division by Zero Calculus.

[26] viXra:1910.0477 submitted on 2019-10-23 19:21:11,

Remainder Theorem and the Division by Zero Calculus.

[25] viXra:1910.0414 submitted on 2019-10-21 19:54:08,

Divergence Series and Integrals From the Viewpoint of the Division by Zero Calculus.

[24] viXra:1909.0658 submitted on 2019-09-30 21:12:36,

On the Value of the Function $\exp {(ax)}/f(a)$ at $a=0$ for $f(a)=0$.

[23] viXra:1909.0517 submitted on 2019-09-24 16:12:58,

What Was Division by Zero?; Division by Zero Calculus and New World (Compact Version).

[22] viXra:1909.0200 submitted on 2019-09-09 16:24:17,

Okumura's Disc Series Can Beyond the Crucial Point of D\"aumler-Puha's Horn Torus Models for the Riemann Sphere.

[21] viXra:1908.0100 submitted on 2019-08-06 20:03:01,

Fundamental of Mathematics; Division by Zero Calculus and a New Axiom.

[20] viXra:1907.0437 submitted on 2019-07-23 20:48:54,

Values of the Riemann Zeta Function by Means of Division by Zero Calculus,

Hiroshi Okumura, Saburou Saitoh.

[19] viXra:1906.0569 submitted on 2019-06-30 18:51:51,

Division by Zero Calculus in Equations and Inequalities.

[18] viXra:1906.0185 submitted on 2019-06-11 20:12:46,

Division by Zero Calculus in Multiply Dimensions and Open Problems (An Extension).

[16] viXra:1905.0407 submitted on 2019-05-22 03:20:09,

Division by Zero Calculus and Pompe's Theorem, Tsutomu Matsuura, Hiroshi Okumura, Saburou Saitoh

[14] viXra:1904.0408 submitted on 2019-04-22 00:32:30,

What Was Division by Zero?; Division by Zero Calculus and New World.

[13] viXra:1904.0052 submitted on 2019-04-03 20:31:13,

D\"aumler's Horn Torus Model and Division by Zero - Absolute Function Theory - New World.

[12] viXra:1904.0028 submitted on 2019-04-02 20:08:20,

Division by Zero and Bh\={a}skara's Example.

[11] viXra:1903.0566 submitted on 2019-03-31 15:59:03,

Division by Zero Calculus in Trigonometric Functions.

[10] viXra:1903.0488 submitted on 2019-03-27 21:04:39,

Division by Zero Calculus in Complex Analysis.

[9] viXra:1903.0432 submitted on 2019-03-24 23:28:16,

Division by Zero Calculus and Singular Integrals.

[8] viXra:1903.0371 submitted on 2019-03-20 23:56:47,

Division by Zero Calculus in Multiply Dimensions and Open Problems.

[7] viXra:1903.0184 submitted on 2019-03-10 20:57:02,

Who Did Derive First the Division by Zero $1/0$ and the Division by Zero Calculus $\tan(\pi/2)=0, \log 0=0$ as the Outputs of a Computer?

[6] viXra:1902.0432 submitted on 2019-02-25 22:16:51,

Zero Expresses Non-possibility.

[5] viXra:1902.0240 submitted on 2019-02-13 22:57:25,

Zero and Infinity; Their Interrelation by Means of Division by Zero.

[4] viXra:1902.0223 submitted on 2019-02-12 18:39:18,

Horn Torus Models for the Riemann Sphere and Division by Zero, Wolfgang W. D\"aumler, Hiroshi Okumura, Vyacheslav V. Puha, Saburou Saitoh.

[3] viXra:1902.0204 submitted on 2019-02-11 18:46:02,

A Meaning and Interpretation of Minus Areas of Figures by Means of Division by Zero.

[2] viXra:1902.0187 submitted on 2019-02-10 22:40:35,

The Simple and Typical Physical Examples of the Division by Zero 1/0=0 by Ctes\'ibio (BC. 286-222) and e. Torricelli (1608 1646).

[1] viXra:1902.0058 submitted on 2019-02-03 22:47:53,

We Can Divide the Numbers and Analytic Functions by Zero\\ with a Natural Sense.

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