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\title{\bf Announcement 258: A new viewpoint of the division by zero $z/0=0$ from area and the point at infinity
}

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

\date{November 26, 2015}

\maketitle
{\bf Abstract: } In this announcement, we will state a reality of the division by zero $z/0=0$ from the viewpoint of area and the point at infinity. We will be able to see a great impact for the idea of our space.

\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 $b$, the division by zero
\begin{equation}
\frac{b}{0}=0,
\end{equation}
is clear and trivial. See (\cite{msy}) for the recent results. See also the survey style announcements 179,185,237,246,247,250 and 252 of the Institute of Reproducing Kernels (\cite{ann179,ann185,ann237,ann246,ann247,ann250,ann252}). The division by zero is not only mathematical problems, but also it will give great impacts to human beings and the idea on the universe. The Institute of Reproducing Kernels is presenting various opinions in Announcements (many in Japanese) on the universe.

In this Announcement, we will refer to a new viewpoint of the division by zero in the Euclidean space from area and the point at infinity. In our common level, the results will be very surprized for many peopule.

\section{The point at infinity}

We will be able to see the whole Euclidean plane by the stereographic projection onto the Riemann sphere. The behavior of the space around the point at infinity may be considered by that around the origin by the linear transform $W = 1/z$(\cite{ahlfors}). We thus see that

\begin{equation}
\lim_{z \to \infty} z = \infty,
\end{equation}
however,
\begin{equation}
[z]_{z =\infty} =0,
\end{equation}
by the division by zero. The difference of (1.1) and (1.2) is very important as we see clearly from the function $1/z$ and the behavior at the origin. The limiting value to the origin and the value at the origin are different. For the 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,
\end{equation}
however,
\begin{equation}
[x]_{ +\infty } =0, \quad [x]_{ -\infty } =0.
\end{equation}

\section{Interpretation by area}

In orde to see some realization of the properties of (1.3) and (1.4), we will consider the triangle with the basic edge (side) $a$ and high $h$. Then, the area $S$ of the triangle is given
by
\begin{equation}
S = \frac{1}{2} ah.
\end{equation}
By fixing the high $h$ and the line containing the side $a$, we will consider the limit $a \to +\infty$. Then, of course,
\begin{equation}
\lim_{a \to +\infty} S = +\infty.
\end{equation}
However, we will see that
\begin{equation}
[S]_{a=\infty} =0,
\end{equation}
just like the division by zero, because, when $a=\infty$, the triangle is broken,
we cannot consider the area of the triangle. Here, the notation $a=\infty$ is not good, however, its meaning is clear; it will mean the case of the parallel lines of the line containing the side $a$ and the line through the fixed vertex of the triangles when we consider $a$ tends to $+\infty$.

The strong discontinuity of the division by zero is appeared as the broken of the triangles.
These phenomena may be looked in many situations as the unverse one.
We can consider similar problems for many types volumes. However, the simplest cases are
disc and sphere (ball) with radius $1/R$. When $R \to +0$, the areas and volumes tend to $+\infty$, however, when $R=0$, they are zero, because they become the half-plane and half-space, respectively.

\bigskip

\bibliographystyle{plain}
\begin{thebibliography}{10}

\bibitem{ahlfors}
Ahlfors, L. V. (1966). {\it Complex Analysis}. McGraw-Hill Book Company.

\bibitem{bht}
Bergstra, J. A., Hirshfeld Y., \& Tucker, J. V. (2009).
{\it Meadows and the equational specification of division} (arXiv:0901.0823v1[math.RA] 7 Jan) .

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

\bibitem{kmsy}
Kuroda, M., Michiwaki, H., Saitoh, S.,\& Yamane, M. (2014).
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
{\it Int. J. Appl. Math. Vol. 27, No 2 }, 191-198, DOI: 10.12732/ijam.v27i2.9.

\bibitem{msy}
Michiwaki H., Saitoh S., \& Yamada M. (2015).
Reality of the division by zero $z/0=0$. IJAPM (International J. of Applied Physics and Math. (to appear).

\bibitem{mst}
Michiwaki, H., Saitoh, S., \& Takagi, M.
A new concept for the point at infinity and the division by zero z/0=0
(manuscript).

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

\bibitem{taka}
Takahasi, S.-E. (2014).
{On the identities $100/0=0$ and $ 0/0=0$.}
(note)

\bibitem{ttk}
Takahasi, S.-E., Tsukada, M., \& Kobayashi, Y. (2015).
{\it Classification of continuous fractional binary operations on the real and complex fields. } Tokyo Journal of Mathematics {\bf 8}, no.2(in press).

\bibitem{ann179}
Division by zero is clear as z/0=0 and it is fundamental in mathematics. {\it Announcement 179 (2014.8.30).}

\bibitem{ann185}
The importance of the division by zero $z/0=0$. {\it Announcement 185 (2014.10.22)}.

\bibitem{ann237}
A reality of the division by zero $z/0=0$ by geometrical optics. {\it Announcement 237 (2015.6.18)}.

\bibitem{ann246}
An interpretation of the division by zero $1/0=0$ by the gradients of lines. {\it Announcement 246 (2015.9.17)}.

\bibitem{ann247}
The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$. {\it Announcement 247 (2015.9.22)}.

\bibitem{ann250}
What are numbers? - the Yamada field containing the division by zero $z/0=0$. {\it Announcement 250 (2015.10.20)}.

\bibitem{ann252}
Circles and curvature - an interpretation by Mr. Hiroshi Michiwaki of the division by
zero $r/0 = 0$. {\it Announcement 252 (2015.11.1)}.

\end{thebibliography}

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