№　887
教科書から取った例ですが、双曲線の媒介変数表示で、
ｔ＝０　のとき、考えられまい量の　分母が現れていますが、それは、極限などの曖昧な意味では　なく、ゼロ除算算法で、きちんとゼロを　宣言して、意味のある　点を表している。

厳格である数学に　曖昧さ、ぼかしがあるのは　歴然である。　現代数学には　欠陥があると言える。

№　888

教科書から取った例ですが、　放物線の媒介変数表示で、　ゼロ除算算法を　用いると面白い。　新しく原点が満たされますが、　それは無限遠点を　表していることが　分かる。　無限遠点が　数字ゼロで　表されている。

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

\documentclass[12pt]{article}

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

\numberwithin{equation}{section}

\begin{document}

\title{\bf Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics\\

}

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

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

Kiryu 376-0041, Japan\\

\date{\today}

\maketitle

{\bf Abstract: } In this announcement, we shall introduce the zero division $z/0=0$. The result is a definite one and it is fundamental in mathematics.

\bigskip

\section{Introduction}

%\label{sect1}

By a natural extension of the fractions

\begin{equation}

\frac{b}{a}

\end{equation}

for any complex numbers $a$ and $b$, we, recently, found the surprising result, for any complex number $b$

\begin{equation}

\frac{b}{0}=0,

\end{equation}

incidentally in \cite{s} 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 \cite{kmsy} for the case of real numbers. The result is a very special case for general fractional functions in \cite{cs}.　

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 (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:

\bigskip

{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that

$$

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

$$

for all

$$

a, b, c, d \in {\bf C }

$$

and

$$

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

$$

Then, we obtain, for any $b \in {\bf C } $

$$

F (b, 0) = 0.

$$

}

\medskip

\section{What are the fractions $ b/a$?}

For many mathematicians, the division $b/a$ will be considered as the inverse of product;

that is, the fraction

\begin{equation}

\frac{b}{a}

\end{equation}

is defined as the solution of the equation

\begin{equation}

a\cdot x= b.

\end{equation}

The idea and the equation (2.2) show that the division by zero is impossible, with a strong conclusion. Meanwhile, the problem has been a long and old question:

As a typical example of the division by zero, we shall recall the fundamental law by Newton:

\begin{equation}

F = G \frac{m_1 m_2}{r^2}

\end{equation}

for two masses $m_1, m_2$ with a distance $r$ and for a constant $G$. Of course,

\begin{equation}

\lim_{r \to +0} F =\infty,

\end{equation}

however, in our fraction

\begin{equation}

F = G \frac{m_1 m_2}{0} = 0.

\end{equation}

\medskip

Now, we shall introduce an another approach. The division $b/a$ may be defined {\bf independently of the product}. Indeed, in Japan, the division $b/a$ ; $b$ {\bf raru} $a$ ({\bf jozan}) is defined as how many $a$ exists in $b$, this idea comes from subtraction $a$ repeatedly. (Meanwhile, product comes from addition).

In Japanese language for "division", there exists such a concept independently of product.

H. Michiwaki and his 6 years old girl said for the result $ 100/0=0$ that the result is clear, from the meaning of the fractions independently the concept of product and they said:

$100/0=0$ does not mean that $100= 0 \times 0$. Meanwhile, many mathematicians had a confusion for the result.

Her understanding is reasonable and may be acceptable:

$100/2=50 \quad$ will mean that we divide 100 by 2, then each will have 50.

$100/10=10　\quad$　will mean that we divide 100 by10, then each will have 10.

$100/0=0　\quad$　will mean that we do not divide 100, and then nobody will have at all and so 0.

Furthermore, she said then the rest is 100; that is, mathematically;

$$

100 = 0\cdot 0 + 100.

$$

Now, all the mathematicians may accept the division by zero $100/0=0$ with natural feelings as a trivial one?

\medskip

For simplicity, we shall consider the numbers on non-negative real numbers. We wish to define the division (or fraction) $b/a$ following the usual procedure for its calculation, however, we have to take care for the division by zero:

The first principle, for example, for $100/2 $ we shall consider it as follows:

$$

100-2-2-2-,...,-2.

$$

How may times can we subtract $2$? At this case, it is 50 times and so, the fraction is $50$.

The second case, for example, for $3/2$ we shall consider it as follows:

$$

3 - 2 = 1

$$

and the rest (remainder) is $1$, and for the rest $1$, we multiple $10$,

then we consider similarly as follows:

$$

10-2-2-2-2-2=0.

$$

Therefore $10/2=5$ and so we define as follows:

$$

\frac{3}{2} =1 + 0.5 = 1.5.

$$

By these procedures, for $a \ne 0$ we can define the fraction $b/a$, usually. Here we do not need the concept of product. Except the zero division, all the results for fractions are valid and accepted.

Now, we shall consider the zero division, for example, $100/0$. Since

$$

100 - 0 = 100,

$$

that is, by the subtraction $100 - 0$, 100 does not decrease, so we can not say we subtract any from $100$. Therefore, the subtract number should be understood as zero; that is,

$$

\frac{100}{0} = 0.

$$

We can understand this: the division by $0$ means that it does not divide $100$ and so, the result is $0$.

Similarly, we can see that

$$

\frac{0}{0} =0.

$$

As a conclusion, we should define the zero divison as, for any $b$

$$

\frac{b}{0} =0.

$$

See \cite{kmsy} for the details.

\medskip

\section{In complex analysis}

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 $w=0$. 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.

However, we shall recall the elementary function

\begin{equation}

W(z) = \exp \frac{1}{z}

\end{equation}

$$

= 1 + \frac{1}{1! z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdot \cdot \cdot .

$$

The function has an essential singularity around the origin. When we consider (1.2), meanwhile, surprisingly enough, we have:

\begin{equation}

W(0) = 1.

\end{equation}

{\bf The point at infinity is not a number} and so we will not be able to consider the function (3.2) at the zero point $z = 0$, meanwhile, we can consider the value $1$ as in (3.3) at the zero point $z = 0$. How do we consider these situations?

In the famous standard textbook on Complex Analysis, L. V. Ahlfors (\cite{ahlfors}) introduced the point at infinity as a number and the Riemann sphere model as well known, however, our interpretation will be suitable as a number. We will not be able to accept the point at infinity as a number.

As a typical result, we can derive the surprising result: {\it At an isolated singular point of an analytic function, it takes a definite value }{\bf with a natural meaning.} As the important applications for this result, the extension formula of functions with analytic parameters may be obtained and singular integrals may be interpretated with the division by zero, naturally (\cite{msty}).

\bigskip

\section{Conclusion}

The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.

The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.

The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.

Should we teach the beautiful fact, widely?:

For the elementary graph of the fundamental function

$$

y = f(x) = \frac{1}{x},

$$

$$

f(0) = 0.

$$

The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).

\medskip

If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.

\bigskip

section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12):　 $100/0=0, 0/0=0$ 　--　 by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division　--　The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9):　 Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S.Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\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. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95. http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}

アインシュタインも解決できなかった「ゼロで割る」問題

http://matome.naver.jp/odai/2135710882669605901

Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.

https://notevenpast.org/dividing-nothing/

私は数学を信じない。　アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。

1423793753.460.341866474681。

Einstein's Only Mistake: Division by Zero

http://refully.blogspot.jp/2012/05/einsteins-only-mistake-division-by-zero.html

ドキュメンタリー 2017: 神の数式第２回宇宙はなぜ生まれたのか

https://www.youtube.com/watch?v=iQld9cnDli4

〔NHKスペシャル〕神の数式完全版第3回宇宙はなぜ始まったのか

https://www.youtube.com/watch?v=DvyAB8yTSjs&t=3318s

NHKスペシャル〕神の数式完全版第1回この世は何からできているのか

https://www.youtube.com/watch?v=KjvFdzhn7Dc

NHKスペシャル神の数式完全版第4回異次元宇宙は存在するか

https://www.youtube.com/watch?v=fWVv9puoTSs

\documentclass[12pt]{article}

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

\numberwithin{equation}{section}

\begin{document}

\title{\bf Announcement 362: Discovery of the division by zero as \\

$0/0=1/0=z/0=0$\\

(2017.5.5)}

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

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

Kiryu 376-0041, Japan\\

}

\date{\today}

\maketitle

{\bf Statement: } The Institute of Reproducing Kernels 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 ?).

In particular, Brahmagupta defined as $0/0=0$ in Brāhmasphuṭasiddhānta (628), however, our world history stated that his definition $0/0=0$ is wrong over 1300 years, but, we will see that his definition is suitable.

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

\bibliographystyle{plain}

\begin{thebibliography}{10}

\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{msy}

H. Michiwaki, S. Saitoh, and M.Yamada,

Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html

\bibitem{ms}

T. Matsuura and S. Saitoh,

Matrices and division by zero $z/0=0$, Advances in Linear Algebra

\& Matrix Theory, 6 (2016), 51-58. http://dx.doi.org/10.4236/alamt.2016.62007 http://www.scirp.org/journal/alamt　

\bibitem{mos}

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, 2017), 1-16.　

\bibitem{osm}

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

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

\bibitem{romig}

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

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

\bibitem{s}

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,

Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182 (Springer).

\bibitem{ttk}

S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.

\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.

\end{thebibliography}

\end{document}

再生核研究所声明371（2017.6.27）ゼロ除算の講演―　国際会議　https://sites.google.com/site/sandrapinelas/icddea-2017　報告

http://ameblo.jp/syoshinoris/theme-10006253398.html

1/0=0、0/0=0、z/0=0

http://ameblo.jp/syoshinoris/entry-12276045402.html

1/0=0、0/0=0、z/0=0

http://ameblo.jp/syoshinoris/entry-12263708422.html

1/0=0、0/0=0、z/0=0

http://ameblo.jp/syoshinoris/entry-12272721615.html

Algebraic division by zero implemented as quasigeometric multiplication by infinity in real and complex multispatial hyperspaces
Author: Jakub Czajko, 92(2) (2018) 171-197
WSN 92(2) (2018) 171-197

http://www.worldscientificnews.com/wp-content/uploads/2017/12/WSN-922-2018-171-197.pdf

№887と№888