 # 0除算に関するアナウンスメントを公開しています

The purpose of this site is to publish announcements on the theory of division by zero

## Announcement 281: The importance of the division by zero $z/0=0$

Institute of Reproducing Kernels
Kawauchi-cho, 5-1648-16,
Kiryu 376-0041, Japan
E-mail: kbdmm360@yahoo.co.jp
January 11, 2016

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

Introduction
By a natural extension of the fractions
\begin{equation}
\frac{b}{a} \tag{0.1}
\end{equation}for any complex numbers $a$ and $b$, we found the result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0, \tag{0.2}
\end{equation}incidentally in  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  for the case of real numbers. The result is a very special case for general fractional functions in .
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  $($$)$ established a simple and decisive interpretation (0.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (0.2). His result will show that our mathematics says that the result (0.2) should be accepted as a natural one:

Proposition. Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
\begin{equation}
F (b, a)F (c, d)= F (bc, ad)
\end{equation}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 $b \in {\bf C }$
$$F (b, 0) = 0.$$

We thus should consider, for any complex number $b$, as  $($0.2$)$; 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. Therefore, the division by zero will give great impacts to complex analysis and to our idea for the space and universe.
However, the division by zero $($0.2$)$ is now clear, indeed, for the introduction of $($0.2$)$, we have several independent approaches as in:
1) by the generalization of the fractions by the Tikhonov regularization or by the Moore-Penrose generalized inverse,
2) by the intuitive meaning of the fractions $($division$)$ by H. Michiwaki,
3) by the unique extension of the fractions by S. Takahasi, as in the above,
4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from ${\bf C} \setminus \{0\}$ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z$ from  ${\bf C}$ onto ${\bf C}$,
and
5) by considering the values of functions with the mean values of functions.

Furthermore, in $($$)$ we gave the results in order to show the reality of the division by zero in our world:
A) a field structure  containing the division by zero --- the Yamada field ${\bf Y}$,
B)  by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,
C) by the reflection $1/\overline{z}$ of $z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero,
and
D) by considering rotation of a right circle cone having some very interesting phenomenon  from some practical and physical problem --- EM radius.

See also  for the relationship between fields and the division by zero, and the importance of the division by zero for computer science.
Meanwhile, Professors J. P.  Barukcic and I.  Barukcic $($$)$ discussed recently the relation between the division $0/0$ and special relative theory of Einstein.
Furthermore,  Reis and Anderson $($[7,8]$)$ extends the system of the real numbers by defining division by zero.
For our results, see the survey style announcements 179,185,237,246, 247,250 and 252 of the Institute of Reproducing Kernels$($[11,12,13,14,15,16,17]$)$.
At this moment, the following theorem may be looked as the fundamental theorem of the division by zero:

Theorem  $($$)$.  Any analytic function takes a definite value  at an  isolated singular point  with a natural meaning.

The following corollary shows  how to determine the value of an analytic function at the singular point; that is,  the value is determined from the regular part of the Laurent expansion:

Corollary
. For an isolated singular point $a$ of an analytic function $f(z)$, we have the Cauchy integral formula
$$f(a) = \frac{1}{2\pi i} \int_{\gamma} f(z) \frac{dz}{z - a},$$where $\gamma$ is a rectifiable simple Jordan closed curve that surrounds one time the point $a$
on a regular region of the function $f(z)$.

The essential meaning of this theorem and corollary is given by that:  the values of functions may be understood in the sense of the mean values of analytic functions.
We will state the importance of the division by zero $z/0=0$.

1
On AD 628, the zero was appeared in India in the document, and the zero division $z/0=0$ was discovered on February 2, 2014, definitely with the clear definition and motivation. The uniqueness and the natural interpretation were given in [10,4], respectively. Several physical interpretations of the division by zero were given in .

2
By the introduction of the division by zero $z/0=0$, four arithmetic operations; that is,
addition, subtraction, multiplication, and division are always possible; note that for division, we were not able to divide by zero. There was one exceptional case for the division by zero.
Even the Yamada field containing the division by zero was established in $($$)$.

3
For the Euclidean $($B.C. 3 Century $)$ geometry, two non-Euclidean geometries were appeared about 2 hundred years ago, and in particular, in the elliptic type non-Euclidean geometry, the point at infinity was introduced by the stereoprojection of the Euclidean plane to the sphere and the concept is a standard one in complex analysis around over one hundred years. And then we have considered as $1/0= \infty$. However, surprisingly enough, the division by zero means that $1/0=0$.

4
We will recall the fundamental law by Newton:
\begin{equation}
F = G\frac{m_1 m_2}{r^2} \tag{4.1}
\end{equation}for two masses $m_1, m_2$ with a distance $r$ and a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty, \tag{4.2}
\end{equation}however, we obtain the important interpretation:
\begin{equation}
F = 0 = G \frac{m_1 m_2}{0}. \tag{4.3}
\end{equation}Of course, here, we can consider the above interpretation for the mathematical formula $($4.1$)$ as the new interpretation $($4.3$)$. We can find many physical formulas with the division by zero.
See the following article for Einstein and Newton:
Impact of 'Division by Zero' in Einstein's Static Universe and ...
www.researchgate.net/.../242574738 Impact of 'Division by Zero' in Einstein's Static Universe and Newton's Equations in Classical Mechanics on ResearchGate, the professional network for.
In particular:  George Gamow $($1904-1968$)$ Russian-born American nuclear physicist and cosmologist remarked that "it is well known to students of high school algebra" that division by zero is not valid; and Einstein admitted it as the biggest blunder of his life : 1. Gamow, G., My World Line (Viking, New York). p 44, 1970

5
In complex analysis, linear fractional functions
$$W = \frac{az + b}{cz + d}, \quad ad -bc \ne 0,$$
map the extended complex plane onto the extended complex plane containing the point at infinity, one to one, conformally, beautifully. This beautiful property is changed as the beautiful formula that linear fractional functions map the whole complex plane onto the whole complex plane, one to one, however, at one point of the singular point, the linear fractional functions have strong discontinuity.
The division by zero excludes the infinity from the numbers.

6
We did, essentially, not consider the division by zero, and so the property of the division by zero; that is, at the isolated singular points of analytic functions, to consider the analytic functions are new mathematics and new research topics, essentially.

7
The impact to complex analysis is unclear, we, however, obtain a typical new theorem:
Theorem : Any analytic function takes a definite value at an isolated singular point with a natural meaning. The definite value is given by the first coefficient of the regular part in the Laurent expansion around the isolated singular point.
This will be the fundamental theorem on the division by zero in Complex Analysis and we have many applications for the Sato hyperfunction theory, generating functions theory and singular integral theory $($$)$.

8
In particular, the division by zero gives new interpretations on the finite part of Hadamard for singular integrals and the Cauchy's principal values. The division by zero will represent discontinuity properties on the universe.

9
Even for middle high school students, the division by zero may be accepted as the beautiful result with great pleasures:
For the elementary function
$$y = f(x) = \frac{1}{x},$$we have $f(0) = 0$; that is, $1/0=0$.

10
We can introduce the division by zero $100/0=0,0/0=0$ with the simple and natural definition for the division by the Hiroshi Michiwachi method  in the elementary school. The division by zero will request the change of all the related books and scientific books.

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 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 $($ Announcement 166$)$. If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the introduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.

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 and English:

Announcement 148 $($2014.2.12$)$:　 $100/0=0, 0/0=0$ 　--　 by a natural extension of fractions -- A wish of the God
Announcement 154 $($2014.4.22$)$: A new world: division by zero, a curious world, a new idea
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?

(11/03)
(10/03)
(09/21)
(09/16)
(07/26)