# 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 352: On the third birthday of the division by zero $z/0=0$

Institute of Reproducing Kernels
Kawauchi-cho, 5-1648-16,
Kiryu 376-0041, Japan
February 2, 2017

Abstract:In this announcement, for its importance we would like to state the situation on the division by zero and propose basic new challenges to education and research on our wrong world history of the division by zero.

1 Introduction
By a natural extension of the fractions

\frac{b}{a} \tag{1.1}
for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$

\frac{b}{0}=0, \tag{1.2}
incidentally in [12] 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 [5] for the case of real numbers.
The division by zero has a long and mysterious story over the world $($see, for example, H. G. Romig [11] and Google site with division by zero$)$ with its physical viewpoints since the document of zero in India on AD 628.  In particular, note that Brahmagupta (598 -  668 ?) established the four arithmetic operations by introducing $0$ and at the same time he defined as $0/0=0$ in Brhmasphuasiddhnta. Our world history, however, stated that his definition $0/0=0$ is wrong over 1300 years, but, we will see that his definition is suitable. However, we do not know the meaning and motivation of  the definition of $0/0=0$, furthermore, for the important case $1/0$ we do not know any result there. However, Sin-Ei, Takahasi  $($[5]$)$ established a simple and decisive interpretation $($1.2$)$ by analyzing the extensions of fractions and by showing the complete characterization for the property $($1.2$)$:

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

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 $b \in {\bf C }$
$F (b, 0) = 0.$
Note that the complete proof of this proposition is simply given by  2 or 3 lines.  We should define $F(b,0)=b/0=0$ in general.

We thus should consider, for any complex number $b$, as  $($1.2$)$; that is, for the mapping

W = \frac{1}{z},
the image of $z=0$ is $W=0$ $($should be defined$)$. 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 ideas for the space and universe.

For Proposition 1, we see some confusion even among mathematicians;  for the elementary function $($1.3$)$, we did not consider the value at $z=0$, and we were not able to consider a value. Many and many people consider its value by the limiting like $+\infty$, $-\infty$ or the point at infinity as $\infty$. However, their basic idea comes from continuity with the common sense or based on the basic idea of Aristotle. However, by the division by zero $($1.2$)$ we will consider its value of the function $W = \frac{1}{z}$ as zero at $z=0$. We would like to consider the value so. We will see that this new definition is valid widely in mathematics and mathematical sciences. However, for functions, we will need some modification  by the idea of the division by zero calculus as in stated in the sequel.

Meanwhile, the division by zero $($1.2$)$ is clear, indeed, for the introduction of $($1.2$)$, we have several independent approaches as in:

1) by the generalization of the fractions by the Tikhonov regularization and by the Moore-Penrose generalized inverse,
2) by the intuitive meaning of the fractions $($division$)$ by H. Michiwaki - repeated subtraction method,
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 [6] 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 $W =1/\overline{z}$ of $W= z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero, not the point at infinity.
And
D) by considering rotation of a right circular cone having some very interesting phenomenon  from some practical and physical problem.
In [8],  many division by zero results in Euclidean spaces are given and  the basic idea at the point at infinity should be changed. In [7], we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.
See  J. A. Bergstra, Y. Hirshfeld and J. V. Tucker [2] and J. A. Bergstra [3] for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.
Meanwhile,  J. P.  Barukcic and I.  Barukcic [1] discussed recently the relation between the divisions $0/0$, $1/0$ and special relative theory of Einstein. However, their logic seems to be curious and their results contradict with ours.
Furthermore,  T. S. Reis and J.A.D.W. Anderson [9,10] extend the system of the real numbers by introducing an ideal number for the division by zero $0/0$.
Meanwhile, we should refer to up-to-date information:
Kurt Arbenz  https://www.researchgate.net/publication/272022137   Riemann Hypothesis Addendum -   Breakthrough.
Here, we recall Albert Einstein's words on mathematics: Blackholes are where God divided by zero. I don't believe in mathematics. 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: Gamow, G., My World Line $($Viking, New York$)$. p 44, 1970.
Apparently, the division by zero is a great missing in our mathematics and the result $($1.2$)$ is definitely determined as our basic mathematics, as we see from Proposition 1.  Note  its very general assumptions and  many fundamental evidences in our world in [5, 6, 8 ,13]. The results will give great impacts  on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and  physical problems.
The mysterious history of the division by zero over one thousand years is a great shame of  mathematicians and human race on the world history, like the Ptolemaic system $($geocentric theory$)$. The division by zero will become a typical  symbol of foolish human race with long and unceasing struggles. Future people will realize this fact as a definite common sense.
We should check and fill our mathematics, globally and beautifully, from the viewpoint of the division by zero. Our mathematics will be more perfect and beautiful,  and will give great impacts to our basic ideas on the universe.
For our ideas on the division by zero, see the survey style announcements.

2 Basic Materials of Mathematics

$($1$)$: First, we should declare that the divison by zero is possible in the natural and uniquley determined sense and its importance.
$($2$)$: In the elementary school, we should introduce the concept of division $($fractions$)$ by the idea of repeated subtraction method by H. Michiwaki whose method is applied in computer algorithm and in old days for calculation of division. This method will give a simple and clear method for calculation of division and students will be happy to apply this simple method at the first stage. At this time, they will be able to understand that the division by zero is clear and trivial as $a/0=0$ for any $a$. Note that Michiwaki knows how to apply his method to the complex number field.
$($3$)$: For the introduction of the elemetary function $y= 1/x$, we should give the definition of the function at the origin $x=0$ as $y = 0$ by the division by zero idea and we should apply this definition for the occasions of its appearences, step by step, following the curriculum and the results of the division by zero.
$($4$)$: For the idea of the Euclidean space $($plane$)$, we should introduce, at the first stage, the concept of stereographic projection and the concept of the point at infinity  -  one point compactification. Then, we will be able to see the whole Euclidean plane, however, by the division by zero,  the point at infinity is represented by zero, not by $\infty$. We can teach  the very important fact with many geometric and analytic geometry methods. These topics will give great pleasant feelings to many students. Interesting topics are: parallel lines, what is a line? - a line contains the origin as an isolated point for the case that the native line does not through the origin. All the lines pass the origin, our space is not the Eulcildean space and is not Aristoteles for the strong discontinuity at the point at infinity $($at the origin$)$. - Here note that an orthogonal coordinate system should be fixed first for our all arguments.
$($5$)$: The inversion of the origin with respect to a circle with center the origin is the origin itself, not the point at infinity - the very classical result is wrong. We can also prove this elementary result by many elementary ways.
$($6$)$: We should change the concept of gradients; on the usual orthogonal coordinates $(x,y)$,  the gradient of the $y$ axis is zero; this is given and proved by the fundamental result
$\tan (\pi/2) =0$. The result is also trivial from the definition of the Yamada field. For the Fourier coefficients $a_k$ of a function :
$$\frac{a_k \pi k^3}{4}$$
= \sin (\pi k) \cos (\pi k) + 2 k^2 \pi^2 \sin(\pi k) \cos (\pi k) + 2\pi (\cos (\pi k) )^2 - \pi k, \tag{2.1}
for $k=0$, we obtain immediately

a_0  = \frac{8}{3}\pi^2 \tag{2.2}

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