\title{\bf Announcement 461: An essence of division by zero and a new axiom} \author{{\it Institute of Reproducing Kernels}\\

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\title{\bf  Announcement 461:  An essence of division by zero and a new axiom}

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

kbdmm360@yahoo.co.jp

 }

\date{2018.11.10}

\maketitle

 In order to see an essence of our division by zero calculus, we will state a simple survey.

As the number system,  division by zero is realized as the {\bf Yamada field} with the definition of the general fractions $a/b$ containing the case $b=0$, and its various meanings and applications are given. In particular, see \cite{msy} and see also the references.

The field structure is, of course, fundamental in the algebraic structure.

However, apart from  various motivations and any  background, we will give the definition of the  division by zero calculus as follows:

\medskip

 For any  \index{Laurent expansion}Laurent expansion around $z=a$,

\begin{equation} \label{dvc5.1}

f(z) = \sum_{n=-\infty}^{-1}  C_n (z - a)^n + C_0 + \sum_{n=1}^{\infty} C_n (z - a)^n

\end{equation}

we define the division by zero calculus

\begin{equation}\label{dvc5.2}

f(a) =  C_0.

\end{equation}

For the correspondence \eqref{dvc5.2} for the function $f(z)$, we will call it {\bf the division by zero calculus}. By considering  derivatives in \eqref{dvc5.1}, we {\bf  define} any order derivatives of the function $f$ at the singular point $a$ as

$$

f^{(n)}(a) = n! C_n.

$$

\medskip

The division by zero calculus seems to be strange firstly, however, by its various applications and results, we will see that the concept is fundamental in our elementary mathematics, globally. See the references.

For its importance, the division by zero calculus  may be looked as a {\bf  new axiom.}

\medskip

Firstly, for the fundamental function $W= F(z) = 1/z$, we have, surprisingly

$$

 F(0) = 0.

$$

We see its great impacts to our basic idea for the space and in our Euclidean space.

  From the form, we should consider that

\begin{equation}

\frac{1}{0} =0.

\end{equation}

Note that this representation and identity is not any result, but it is only the definition of

$\frac{1}{0}$. Of course, it is not the usual definition as the solution of the equation $0 \cdot z =1$. Here, we are stating that the division by zero calculus and the form of the elementary function lead us to the identity (0.3).

\medskip

\bigskip

{\bf \Large Could we divide the numbers and functions by zero?}

\medskip

For this old and general question, we will give a simple answer.

For any analytic function

$f(z)$ around the origin $z=0$ that is permitted to have any singularity at $z=0$ (of course, any constant function is permitted),

we can consider the value, by the division by zero calculus

\begin{equation}

\frac{f(z)}{z^n}

\end{equation}

at the point $z=0$, for any positive integer $n$. This will mean that from the form

we can consider it as follows:

\begin{equation}

\frac{f(z)}{z^n}\mid_{x=0}.

\end{equation}

\bigskip

For example,

$$

\frac{e^{x}}{x^n}\mid_{x=0} = \frac{1}{n!}.

$$

\medskip

{\bf \Huge In this sense, we can divide the numbers and analytic functions by zero.}

\bibliographystyle{plain}

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

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\bibitem{mos}

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\bibitem{osm}

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\end{thebibliography}

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