再生核研究所

Dear Colleagues:

On the division by zero 1/0, I think we were able to obtain the definite results, mathematically. By the uniqueness theorem by Professor Takahasi on the division by zero, we will be able to confirm the results. We can give several important physical interpretations for the results, however, initially, I would like to give our Announcement 185:

With best regards,

Sincerely yours,

Saburou Saitoh

2015.9.26.13:00

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\title{\bf Announcement 185 : The importance of the division by zero $z/0=0$}

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

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

Kiryu 376-0041, Japan\\

E-mail: kbdmm360@yahoo.co.jp\\}

\date{\today}

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

\bigskip

{\bf Introduction}

\bigskip

%\label{sect1}

By {\bf 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

Furthermore, note that Hiroshi Michiwaki gave the important interpretation of the division by zero $z/0=0$ by the intuitive meaning of the division, independently of the concept of the product (see \cite{ann}).

We shall state the importance of the division by zero $z/0=0$.

\bigskip

\section{}

On AD 628, the zero was appeared in India, and the zero division $z/0=0$ was discovered on Feburary 2, 2014, definitely with the clear definition and motievation. The uniquess and the natural interpretation were given in \cite{taka, ttk,kmsy} and \cite{ann}, respectively. Several physical interpretations of the division by zero were given in \cite{kmsy}.

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