International Conference on Differential & Difference Equations and Applications 2017

この会議の総会において，齋藤三郎氏は DIVISION BY ZERO CALCULUS AND DIFFERENTIAL EQUATIONS と題する講演を行いました。

https://sites.google.com/site/sandrapinelas/icddea-2017/c-plenary-speakers

また，松浦勉氏も $\log 0= \log \infty =0$ AND APPLICATIONS と題する発表を行いました。

https://sites.google.com/site/sandrapinelas/icddea-2017/draft-of

これらはともに招待講演であり，0除算に関する注目の高さがうかがえます。

]]>

Kawauchi-cho, 5-1648-16,

Kiryu 376-0041, Japan

February 7, 2017By

\frac{b}{a} \tag{1.1}

\end{equation}for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$

\begin{equation}

\frac{b}{0}=0, \tag{1.2}

\end{equation}incidentally in [6] 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 [2] for the case of real numbers. Meanwhile, the result (1.2) is a very special case of very general fractional functions in [1].

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

\begin{equation}

W = \frac{1}{z}, \tag{1.3}

\end{equation}

the image of $z=0$ is $W=0$ $($

However, for functions, we will need some modification

For any formal Laurent expansion around $z=a$,

\begin{equation}

f(z) = \sum_{n=-\infty}^{\infty} C_n (z - a)^n, \tag{1.4}

\end{equation}

we obtain the identity, by the division by zero

\begin{equation}

f(a) = C_0. \tag{1.5}

\end{equation}

The typical example is that, as we can derive by the elementary way,

$$

\tan \frac{\pi}{2} =0.

$$

We gave many examples with geometric meanings in [5]. See [1, 2, 3, 6, 7] for the related references.In [5], many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In [4], 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.

For our ideas on the division by zero, see the survey style announcements stated in the references.

\begin{equation}

S_n= \frac{n a^2}{2} \sin \frac{2\pi}{n}\tag{2.1}

\end{equation}

and

\begin{equation}

L_n= 2n a \sin \frac{\pi}{n},\tag{2.2}

\end{equation}

respectively. For $n \ge 3$, the results are clear.

For $n =2$, we will consider two diameters that are the same. We can consider it as a generalized regular polygon inscribed in the disc as a degenerate case. Then, $S_2 =0$ and $L_2 =4a$, and the general formulas are valid.

Next, we will consider the case $n=1$. Then the corresponding regular polygon is a just diameter of the disc. Then, $S_1 =0 $ and $L_1 = 0$ that will mean that any regular polygon inscribed in the disc may not be formed and so its area and length of the side are zero.

Now we will consider the case $n=0$. Then, by the division by zero calculus, we obtain that $S_0= \pi a^2$ and $L_0 = 2\pi a$. Note that they are the area and the length of the disc. How to understand the results? Imagine contrary $n$ tending to infinity, then the corresponding regular polygons inscribed in the disc tend to the disc. Recall our new idea that the point at infinity is represented by $0$. Therefore, the results say that $n=0$ regular polygons are $n= \infty$ regular polygons inscribed in the disc in a sense and they are the disc. This is our interpretation of the theorem:

3 Our life figure

$$

\Delta_{\alpha} = \left\{ |\arg z| < \alpha; 0< \alpha < \frac{\pi}{2}\right\}.

$$

We will consider a disc inscribed in the sector $\Delta_{\alpha}$ whose center $(k,0)$ with radius $r$. Then, we have

\begin{equation}

r = k \sin \alpha. \tag{3.1}

\end{equation}

Then, note that as $k$ tends to zero, $r$ tends to zero, meanwhile $k$ tends to $+\infty$, $r$ tends to $+\infty$. However, by our division by zero calculus, we see that immediately that

\begin{equation}

[r ]_{r =\infty}= 0.\tag{3.2}

\end{equation}

For this fact, note the following:

The behavior of the space around the point at infinity may be considered by that of the origin by the linear transform $W = 1/z$. We thus see that

\begin{equation}

\lim_{z \to \infty} z = \infty,\tag{3.3}

\end{equation}

however,

\begin{equation}

[z]_{z =\infty} =0,\tag{3.4}

\end{equation}

by the division by zero. Here, $[z]_{z =\infty}$ denotes the value of the function $W=z$ at the topological point at the infinity in one point compactification by Aleksandrov. The difference of (3.3) and (3.4) is very important as we see clearly by the function $W = 1/z$ and the behavior at the origin. The limiting value to the origin and the value at the origin are different. For 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,\tag{3.5}

\end{equation}

however,

\begin{equation}

[x]_{ +\infty } =0, \quad [x]_{ -\infty } =0.\tag{3.6}

\end{equation}

Of course, two points $+\infty$ and $ -\infty$ are the same point as the point at infinity. However, $\pm$ will be convenient in order to show the approach directions. In \cite{mos}, we gave many examples for this property.

On the sector, we see that from the origin as the point 0, the inscribed discs are increasing endlessly, however their final disc reduces to the origin suddenly - it seems that the whole process looks like our life in the viewpoint of our initial and final.

4

The suprising example by H. Okumura which was sent to the Institute on 2016.12.1.9:40 will show a new phenomenon at the point at infinity.

On the sector $\Delta_{\alpha}$, we shall change the angle and we consider a fixed circle $C_a, a > 0$ with radius $a$ inscribed in the sectors. We see that when the circle tends to $+\infty$, the angles $\alpha$ tend to zero. How will be the case $\alpha = 0$? Then, we will not be able to see the position of the circle. Surprisingly enough, then the circle $C_a$ is the circle with center at the origin $0$. This result is derived from the division by zero calculus for the formula

\begin{equation}

k =\frac{a}{\sin \alpha}.\tag{4.1}

\end{equation}

The two lines $ \arg z = \alpha$ and $ \arg z = -\alpha$ were tangential lines of the circle $C_a$ and now they are the positive real line. The gradient of the positive real line is of course zero. Note here that the gradient of the positive imaginary line is zero by the division by zero calculus that means $\tan \frac{\pi}{2} =0$. Therefore, we can understand that the positive real line is still a tangential line of the circle $C_a$.

This will show some great relation between zero and infinity. We can see some mysterious property around the point at infinity.

[1] L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory

[2] M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,

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

Int. J. Appl. Math.

[3] H. Michiwaki, S. Saitoh, and M.Yamada,

Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math.

[4] T. Matsuura and S. Saitoh, Matrices and division by zero $z/0=0$, Advances in Linear Algebra & Matrix Theory,

[5] H. Michiwaki, H. Okumura, and S. Saitoh, Division by Zero $z/0 = 0$ in Euclidean Spaces. International Journal of Mathematics and Computation Vol. 28$($2017$)$; Issue 1, 1-16.

[6] S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra & Matrix Theory.

[7] S. Saitoh, A reproducing kernel theory with some general applications,

Qian,T./Rodino,L.$($eds.$)$: Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics,

[8] Announcement 179 $($2014.8.30$)$: Division by zero is clear as z/0=0 and it is fundamental in mathematics.

[9] Announcement 185 $($2014.10.22$)$: The importance of the division by zero $z/0=0$.

[10] Announcement 237 $($2015.6.18$)$: A reality of the division by zero $z/0=0$ by geometrical optics.

[11] Announcement 246 $($2015.9.17$)$: An interpretation of the division by zero $1/0=0$ by the gradients of lines.

[12] Announcement 247 $($2015.9.22$)$: The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

[13] Announcement 250 $($2015.10.20$)$: What are numbers? - the Yamada field containing the division by zero $z/0=0$.

[14] Announcement 252 $($2015.11.1$)$: Circles and curvature - an interpretation by Mr. Hiroshi Michiwaki of the division by zero $r/0 = 0$.

[15] Announcement 281 $($2016.2.1$)$: The importance of the division by zero $z/0=0$.

[16] Announcement 282 $($2016.2.2$)$: The Division by Zero $z/0=0$ on the Second Birthday.

[17] Announcement 293 $($2016.3.27$)$: Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.

[18] Announcement 300 $($2016.05.22$)$: New challenges on the division by zero z/0=0.

[19] Announcement 326 $($2016.10.17$)$: The division by zero z/0=0 - its impact to human beings through education and research.

[20] Announcement 352 $($2017.2.2$)$: On the third birthday of the division by zero z/0=0.

Kawauchi-cho, 5-1648-16,

Kiryu 376-0041, Japan

February 2, 2017By

\frac{b}{a} \tag{1.1}

\end{equation}for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$

\begin{equation}

\frac{b}{0}=0, \tag{1.2}

\end{equation}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 $($1.2$)$:

\begin{equation}

F (b, a)F (c, d)= F (bc, ad)

\end{equation}

$$

F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.

$$

\[

F (b, 0) = 0.

\]

Note that the complete proof of this proposition is simply given by 2 or 3 lines. We

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

\begin{equation}

W = \frac{1}{z},

\end{equation}the image of $z=0$ is $W=0$ $($

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

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:

$($2$)$: 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,

$($5$)$: 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 :

$$\frac{a_k \pi k^3}{4}$$\begin{equation}

= \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}

\end{equation}for $k=0$, we obtain immediately

\begin{equation}

a_0 = \frac{8}{3}\pi^2 \tag{2.2}

\end{equation}

$($see [15], (3.4)$)$$($ - Difficulty in Maple for specialization problems$)$.

These results are derived also from the

\begin{equation}

f(z) = \sum_{n=-\infty}^{\infty} C_n (z - a)^n, \tag{2.3}

\end{equation}

we obtain the identity, by the division by zero

\begin{equation}

f(a) = C_0. \tag{2.4}

\end{equation} The typical example is that, as we can derive by the elementary way,

$$

\tan \frac{\pi}{2} =0.

$$

We gave many examples with geometric meanings in [8].

This fundamental result leads to the important new definition:

From the viewpoint of the division by zero, when there exists the limit, at $ x$

\begin{equation}

f^\prime(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h} =\infty \tag{2.5}

\end{equation} or \begin{equation}

f^\prime(x) = -\infty, \tag{2.6}

\end{equation}

both cases, we can write them as follows:

\begin{equation}

f^\prime(x) = 0. \tag{2.7}

\end{equation}

For the elementary ordinary differential equation

\begin{equation}

y^\prime = \frac{dy}{dx} =\frac{1}{x}, \quad x > 0,\tag{2.8}

\end{equation} how will be the case at the point $x = 0$? From its general solution, with a general constant $C$ \begin{equation}

y = \log x + C,\tag{2.9}

\end{equation} we see that, by the division by zero,

\begin{equation}

y^\prime (0)= \left[ \frac{1}{x}\right]_{x=0} = 0,\tag{2.10}

\end{equation} that will mean that the division by zero $($1.2$)$ is very natural.

In addition, note that the function $y = \log x$ has infinite order derivatives and all the values are zero at the origin, in the sense of the division by zero.

However, for the derivative of the function $y = \log x$, we have to fix the sense at the origin, clearly, because the function is not differentiable, but it has a singularity at the origin. For $x >0$, there is no problem for $($2.8$)$ and $($2.9$)$. At $x = 0$, we see that we can not consider the limit in the sense $($2.5$)$. However, for $x >0$ we have $($2.8$)$ and

\begin{equation}

\lim_{x \to +0} \left(\log x \right)^\prime = +\infty.\tag{2.11}

\end{equation} In the usual sense, the limit is $+\infty$, but in the present case, in the sense of the division by zero, we have:

\begin{equation}

\left[ \left(\log x \right)^\prime \right]_{x=0}= 0\tag{2.12}

\end{equation}

and we will be able to understand its sense graphycally.

By the new interpretation for the derivative, we can arrange many formulas for derivatives, by the division by zero. We can modify many formulas and statements in calculus and we can apply our concept to the differential equation theory and the universe in connetion with derivatives.

$($7$)$: We shall introduce the typical division by zero calculus.

For the integral

\begin{equation}

\int x(x^{2}+1)^{a}dx=\frac{(x^{2}+1)^{a+1}}{2(a+1)}\quad(a\ne-1),\tag{2.13}

\end{equation}

we obtain, by the division by zero calculus,

\begin{equation}

\int x(x^{2}+1)^{-1}dx=\frac{\log(x^{2}+1)}{2}. \tag{2.14}

\end{equation}

For example, in the ordinary differential equation

\begin{equation}

y^{\prime\prime} + 4 y^{\prime} + 3 y = 5 e^{- 3x},\tag{2.15}

\end{equation}

in order to look for a special solution, by setting $y = A e^{kx}$ we have, from

\begin{equation}

y^{\prime\prime} + 4 y^{\prime} + 3 y = 5 e^{kx},\tag{2.16}

\end{equation}

\begin{equation}

y = \frac{5 e^{kx}}{k^2 + 4 k + 3}.\tag{2.17}

\end{equation}

For $k = -3$, by the division by zero calculus, we obtain

\begin{equation}

y = e^{-3x} \left( - \frac{5}{2}x - \frac{5}{4}\right),\tag{2.18}

\end{equation}

and so, we can obtain the special solution

\begin{equation}

y = - \frac{5}{2}x e^{-3x}.\tag{2.19}

\end{equation}

In those examples, we were able to give valuable functions for denominator zero cases. The division by zero calculus may be applied to many cases as a new fundamental calculus over l'Hopital's rule.

$($8$)$: When we apply the division by zero to functions, we can consider, in general, many ways. For example,

for the function $z/(z-1)$, when we insert $z=1$ in numerator and denominator, we have

\begin{equation}

\left[\frac{z}{z-1}\right]_{z = 1} = \frac{1}{0} =0. \tag{2.20}

\end{equation}

However,

from the identity -- the Laurent expansion around $z=1$,

\begin{equation}

\frac{z}{z-1} = \frac{1}{z-1} + 1,\tag{2.21}

\end{equation}

we have

\begin{equation}

\left[\frac{z}{z-1}\right]_{z = 1} = 1.\tag{2.22}

\end{equation}

For analytic functions we can give uniquely determined values at isolated singular points by the values by means of the Laurent expansions as the division by zero calculus, however, the values by means of the Laurent expansions are not always reasonable. We will need to consider many interpretations for reasonable values. In many formulas in mathematics and physics, however, we can see that the division by zero calculus is reasonably valid. See [5,6].

The division by zero is directly related to the Einstein's theory and various physical problems containing the division by zero. Now we should check the theory and the problems by the concept of the RIGHT and DEFINITE division by zero. Now is the best time since 100 years from Albert Einstein. It seems that the background knowledge is timely fruitful.

Note that the Big Bang also may be related to the division by zero like the blackholes.

The above Professors listed are wishing the contributions in order to avoid the division by zero trouble in computers. Now, we should arrange new computer systems in order not to meet the division by zero trouble in computer systems.

By the division by zero calculus, we will be able to overcome troubles in Maple for specialization problems as in stated.

The division by zero may be related to religion, philosophy and the ideas on the universe; it will create a new world. Look at the new world introduced.

We are standing on a new generation and in front of the new world, as in the discovery of the Americas. Should we push the research and education on the division by zero?

In order to make clear the problem, we give a prototype example. We have the identity by the divison by zero calculus: For

\begin{equation}

f(z) = \frac{1 + z}{1- z}, \quad f(1) = -1.\tag{6.1}

\end{equation}

From the real part and imaginary part of the function, we have, for $ z= x +iy$

\begin{equation}

\frac{1 - x^2 - y^2}{(1 - x)^2 + y^2} =-1, \quad \text{at}\quad (1,0) \tag{6.2}

\end{equation}

and

\begin{equation}

\frac{y}{(1- x)^2 + y^2} = 0, \quad \text{at}\quad (1,0), \tag{6.3}

\end{equation}

respectively. Why the differences do happen? In general, we are interested in the above open problem. Recall our definition for the division by zero calculus.

[1] J. P. Barukcic and I. Barukcic, Anti Aristotle - The Division of Zero by Zero. Journal of Applied Mathematics and Physics,

[2] J. A. Bergstra, Y. Hirshfeld and J. V. Tucker, Meadows and the equational specification of division $($arXiv:0901.0823v1[math.RA] 7 Jan 2009$)$.

[3] J.A. Bergstra, Conditional Values in Signed Meadow Based Axiomatic Probability Calculus,

arXiv:1609.02812v2[math.LO] 17 Sep 2016.

[4] L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory

[5] M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane, New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$, Int. J. Appl. Math.

[6] H. Michiwaki, S. Saitoh, and M.Yamada, Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math.

[7] T. Matsuura and S. Saitoh, Matrices and division by zero $z/0=0$, Advances in Linear Algebra & Matrix Theory,

[8] H. Michiwaki, H. Okumura, and S. Saitoh, Division by Zero $z/0 = 0$ in Euclidean Spaces. International Journal of Mathematics and Computation

[9] T. S. Reis and J.A.D.W. Anderson, Transdifferential and Transintegral Calculus, Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I WCECS 2014, 22-24 October, 2014, San Francisco, USA

[10] T. S. Reis and J.A.D.W. Anderson, Transreal Calculus, IAENG International J. of Applied Math.,

[11] H. G. Romig, Discussions: Early History of Division by Zero, American Mathematical Monthly, Vol. 31, No. 8. $($Oct., 1924$)$, pp. 387-389.

[12] S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra & Matrix Theory.

[13] S. Saitoh, A reproducing kernel theory with some general applications, Qian,T./Rodino,L.$($eds.$)$: Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics,

[14] S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics,

[15] Introduction to Maple - UBC Mathematics https://www.math.ubc.ca/~israel/m210/lesson1.pdf

[16] Announcement 179 $($2014.8.30$)$: Division by zero is clear as z/0=0 and it is fundamental in mathematics.

[17] Announcement 185 $($2014.10.22$)$: The importance of the division by zero $z/0=0$.

[18] Announcement 237 $($2015.6.18$)$: A reality of the division by zero $z/0=0$ by geometrical optics.

[19] Announcement 246 $($2015.9.17$)$: An interpretation of the division by zero $1/0=0$ by the gradients of lines.

[20] Announcement 247 $($2015.9.22$)$: The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

[21] Announcement 250 $($2015.10.20$)$: What are numbers? - the Yamada field containing the division by zero $z/0=0$.

[22] Announcement 252 $($2015.11.1$)$: Circles and curvature - an interpretation by Mr. Hiroshi Michiwaki of the division by zero $r/0 = 0$.

[23] Announcement 281 $($2016.2.1$)$: The importance of the division by zero $z/0=0$.

[24] Announcement 282 $($2016.2.2$)$: The Division by Zero $z/0=0$ on the Second Birthday.

[25] Announcement 293 $($2016.3.27$)$: Parallel lines on the Euclidean plane from the viewpoint of division by zero $1/0=0$.

[26] Announcement 300 $($2016.05.22$)$: New challenges on the division by zero $z/0=0$.

[27] Announcement 326 $($2016.10.17$)$: The division by zero $z/0=0$ - its impact to human beings through education and research.

]]>

Kawauchi-cho, 5-1648-16,

Kiryu 376-0041, Japan

E-mail: kbdmm360@yahoo.co.jp

October 17, 2016By

\frac{b}{a} \tag{1.1}

\end{equation}for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$

\begin{equation}

\frac{b}{0}=0, \tag{1.2}

\end{equation}incidentally in [10] 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 [4] for the case of real numbers.

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 [4] established a simple and decisive interpretation $($1.2$)$ by analyzing the extensions of fractions and by showing the complete characterization for the property $($1.2$)$:

\begin{equation}

F (b, a)F (c, d)= F (bc, ad)

\end{equation}

$$

F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.

$$

\[

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

\begin{equation}

w = \frac{1}{z}, \tag{1.3}

\end{equation}the image of $z=0$ is $w=0$ $($

However, the division by zero $($1.2$)$ is now 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 [7], many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In [5], 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] 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 [8,9] 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:

$(2)$: In the elementary school, we should introduce the concept of division by the idea of repeated subtraction method by H. Michiwaki whoes method is applied in computer algorithmu 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 steleographic 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. 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 coordinates 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 trivial in the definition of the Yamada field. This result is derived also from the

\begin{equation}

f(z) = \sum_{n=-\infty}^{\infty} C_n (z - a)^n,\tag{2,1}

\end{equation}

\begin{equation}

f(a) = C_0. \tag{2,2}

\end{equation}

From the viewpoint of the division by zero, when there exists the limit, at $ x$

\begin{equation}

f^\prime(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h} =\infty \tag{2,3}

\end{equation}

or

\begin{equation}

f^\prime(x) = -\infty, \tag{2,4}

\end{equation}

both cases, we can write them as follows:

\begin{equation}

f^\prime(x) = 0. \tag{2,5}

\end{equation}

y^\prime = \frac{dy}{dx} =\frac{1}{x}, \quad x > 0, \tag{2,6}

\end{equation}

how will be the case at the point $x = 0$? From its general solution, with a general constant $C$

\begin{equation}

y = \log x + C, \tag{2,7}

\end{equation}

we see that, by the division by zero,

\begin{equation}

y^\prime (0)= \left[ \frac{1}{x}\right]_{x=0} = 0, \tag{2,8}

\end{equation}

that will mean that the division by zero (1.2) is very natural.

\begin{equation}

\lim_{x \to +0} \left(\log x \right)^\prime = +\infty. \tag{2,9}

\end{equation}

In the usual sense, the limit is $+\infty$, but in the present case, in the sense of the division by zero, we have:

\begin{equation}

\left[ \left(\log x \right)^\prime \right]_{x=0}= 0 \tag{2,10}

\end{equation}

and we will be able to understand its sense graphycally.

By the new interpretation for the derivative, we can arrange many formulas for derivatives, by the division by zero. We can modify many formulas and statements in calculus and we can apply our concept to the differential equation theory and the universe in connetion with derivatives.

For the integral

\begin{equation}

\int x(x^{2}+1)^{a}dx=\frac{(x^{2}+1)^{a+1}}{2(a+1)}\quad(a\ne-1), \tag{2,11}

\end{equation}

we obtain, by the division by zero,

\begin{equation}

\int x(x^{2}+1)^{-1}dx=\frac{\log(x^{2}+1)}{2}. \tag{2,12}

\end{equation}

\begin{equation}

x^{\prime \prime}(t) =g -kx^{\prime}(t) \tag{2,13}

\end{equation}

with the initial conditions

\begin{equation}

x(0) = -h, x^{\prime}(0) =0. \tag{2,14}

\end{equation}

Then we have the solution

\begin{equation}

x(t) = \frac{g}{k}t + \frac{g(e^{-kt}- 1)}{k^2} - h. \tag{2,15}

\end{equation}

Then, for $k=0$, we obtain, immediately, by the division by zero

\begin{equation}

x(t) = \frac{1}{2}g t^2 -h. \tag{2,16}

\end{equation}

\begin{equation}

\left[\frac{z}{z-1}\right]_{z = 1} = \frac{1}{0} =0. \tag{2,17}

\end{equation}

However,

from the identity --

the Laurent expansion around $z=1$,

\begin{equation}

\frac{z}{z-1} = \frac{1}{z-1} + 1, \tag{2,18}

\end{equation}

we have

\begin{equation}

\left[\frac{z}{z-1}\right]_{z = 1} = 1. \tag{2,19}

\end{equation}

For analytic functions we can give uniquely determined values at isolated singular points by the values by means of the Laurent expansions as the division by zero calculus, however, the values by means of the Laurent expansions are not always reasonable. We will need to consider many interpretations for reasonable values. In many formulas in mathematics and physics, however, we can see that the division by zero calculus is reasonably valid. See [4,6]

3 Albert Einstein's biggest blunder

Note that the Big Bang also may be related to the division by zero like the blackholes.

4 Computer systems

By the division by zero calculus, we will be able to overcome troubles in Maple for specialization problems.

5 General ideas on the universe

We are standing on a new generation and in front of the new world, as in the discovery of the Americas. Should we push the research and education on the division by zero?

[2] J. A. Bergstra, Y. Hirshfeld and J. V. Tucker, Meadows and the equational specification of division $($arXiv:0901.0823v1[math.RA] 7 Jan 2009$)$.

[3] L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory 7 $($2013$)$, no. 4, 1049-1063.

[4] M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane, New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,

Int. J. Appl. Math. 27 $($2014$)$, no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.

[5] T. Matsuura and S. Saitoh, Matrices and division by zero $z/0=0$, Linear Algebra \& Matrix Theory,

[6] H. Michiwaki, S. Saitoh, and M.Yamada, Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. 6 $($2015$)$, 1--8. http://www.ijapm.org/show-63-504-1.html

[7] H. Michiwaki, H. Okumura, and S. Saitoh, Division by Zero $z/0 = 0$ in Euclidean Spaces. International Journal of Mathematics and Computation $($in press$)$.

[8] T. S. Reis and J.A.D.W. Anderson, Transdifferential and Transintegral Calculus, Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I WCECS 2014, 22-24 October, 2014, San Francisco, USA

[9] T. S. Reis and J.A.D.W. Anderson, Transreal Calculus, IAENG International J. of Applied Math., 45 $($2015$)$: IJAM 45 1 06.

[10] S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. 4 $($2014$)$, no. 2, 87--95. http://www.scirp.org/journal/ALAMT/

[11] S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, 38 $($2015$)$, no. 2, 369-380.

[12] Announcement 179 $($2014.8.30$)$: Division by zero is clear as $z/0=0$ and it is fundamental in mathematics.

[13] Announcement 185 $($2014.10.22$)$: The importance of the division by zero $z/0=0$.

[14] Announcement 237 $($2015.6.18$)$: A reality of the division by zero $z/0=0$ by geometrical optics.

[15] Announcement 246 $($2015.9.17$)$: An interpretation of the division by zero $1/0=0$ by the gradients of lines.

[16] Announcement 247 $($2015.9.22$)$: The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

[17] Announcement 250 $($2015.10.20$)$: What are numbers? - the Yamada field containing the division by zero $z/0=0$.

[18] Announcement 252 $($2015.11.1$)$: Circles and curvature - an interpretation by Mr. Hiroshi Michiwaki of the division by zero $r/0 = 0$.

[19] Announcement 281$($2016.2.1$)$: The importance of the division by zero $z/0=0$.

[20] Announcement 282$($2016.2.2$)$: The Division by Zero $z/0=0$ on the Second Birthday.

[21] Announcement 293$($2016.3.27$)$: Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.

[22] Announcement 300$($2016.05.25$)$: New challenges on the the division by zero 1/0=0.

Kawauchi-cho, 5-1648-16,

Kiryu 376-0041, Japan

E-mail: kbdmm360@yahoo.co.jp

May 22, 2016situation on the division by zero and propose basic new challenges.

By

\frac{b}{a} \tag{1.1}

\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{1.2}

\end{equation}incidentally in [10] 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 [4] for the case of real numbers.

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 [4] established a simple and decisive interpretation $($1.2$)$ by analyzing some full extensions of fractions and by showing the complete characterization for the property $($1.2$)$:

\begin{equation}

F (b, a)F (c, d)= F (bc, ad)

\end{equation}

$$

F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.

$$

\[

F (b, 0) = 0.

\]

Note that the complete proof of this proposition is simply given by 2 or 3 lines.

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

\begin{equation}

w = \frac{1}{z}, \tag{1.3}

\end{equation}the image of $z=0$ is $w=0$ $($

However, the division by zero $($1.2$)$ is now 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 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 [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,

and

D) by considering rotation of a right circular cone having some very interesting phenomenon from some practical and physical problem.

In [7], many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In [5], 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] 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 [8,9] 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:

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

For our ideas on the division by zero, see the survey style announcements 179, 185, 237, 246,247, 250 and 252 of the Institute of Reproducing Kernels [12, 13, 14, 15, 16, 17, 18, 21].

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.

3. Albert Einstein's biggest blunder

containing the division by zero. Now we should check the theory and the problems by the concept of the RIGHT and DEFINITE division by zero. Now is the best time since 100 years from Albert Einstein. It seems that the background knowledge is timely fruitful.

4. Computer systems

5. General ideas on the universe

We are standing on a new generation and in front of the new world, as in the discovery of the Americas.

References

[2] J. A. Bergstra, Y. Hirshfeld and J. V. Tucker, Meadows and the equational specification of division $($arXiv:0901.0823v1[math.RA] 7 Jan 2009$)$.

[3] L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory

[4] M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane, New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,

Int. J. Appl. Math.

[5] T. Matsuura and S. Saitoh, Matrices and division by zero $z/0=0$, Linear Algebra \& Matrix Theory $($ALAMT$)$ $($to appear$)$.

[6] H. Michiwaki, S. Saitoh, and M.Yamada, Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math.

[7] H. Michiwaki, H. Okumura, and S. Saitoh, Division by Zero $z/0 = 0$ in Euclidean Spaces. International Journal of Mathematics and Computation $($in press$)$.

[9] T. S. Reis and J.A.D.W. Anderson, Transreal Calculus, IAENG International J. of Applied Math.,

[10]

[11] S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics,

[12] Announcement 179 $($2014.8.30$)$: Division by zero is clear as $z/0=0$ and it is fundamental in mathematics.

[13] Announcement 185 $($2014.10.22$)$: The importance of the division by zero $z/0=0$.

[14] Announcement 237 $($2015.6.18$)$: A reality of the division by zero $z/0=0$ by geometrical optics.

[15] Announcement 246 $($2015.9.17$)$: An interpretation of the division by zero $1/0=0$ by the gradients of lines.

[16] Announcement 247 $($2015.9.22$)$: The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

[17] Announcement 250 $($2015.10.20$)$: What are numbers? - the Yamada field containing the division by zero $z/0=0$.

[18] Announcement 252 $($2015.11.1$)$: Circles and curvature - an interpretation by Mr. Hiroshi Michiwaki of the division by zero $r/0 = 0$.

[19] Announcement 281$($2016.2.1$)$: The importance of the division by zero $z/0=0$.

[20] Announcement 282$($2016.2.2$)$: The Division by Zero $z/0=0$ on the Second Birthday.

[21] Announcement 293$($2016.3.27$)$: Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.

Kawauchi-cho, 5-1648-16,

Kiryu 376-0041, Japan

E-mail: kbdmm360@yahoo.co.jp

March 27, 2016By

\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 [10] 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 result is a very special case for general fractional functions in [4].

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 $($[11]$)$ $($see also [5] $)$ established a simple and decisive interpretation $($1.2$)$ by analyzing some full extensions of fractions and by showing the complete characterization for the property $($1.2$)$:

\begin{equation}

F (b, a)F (c, d)= F (bc, ad)

\end{equation}

$$

F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.

$$

\[

F (b, 0) = 0.

\]

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

that is, for the mapping

\begin{equation}

w = \frac{1}{z},

\end{equation}

the image of $z=0$ is $w=0$ $($

However, the division by zero $($1.2$)$ is now 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 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 $($[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 $1/\overline{z}$ 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 [3] for the relationship between fields and the division by zero, and the importance of the division by zero for computer science.

Meanwhile, J. P. Barukcic and I. Barukcic $($[2]$)$ discussed recently the relation between the division $0/0$ and special relativity of Einstein. Furthermore, Reis and Anderson $($[8, 9]$)$ extends the system of the real numbers by defining division by zero. Meanwhile, we should refer to up-to-date information:

Kurt Arbenz https://www.researchgate.net/publication/272022137

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 divi-sion by zero is not valid; and Einstein admitted it as

For our results, see the survey style announcements 179, 185, 237, 246, 247, 250 and 252 of the Institute of Reproducing Kernels $($[13, 14, 15, 16, 17, 18, 19]$)$.

At this moment, the following theorem may be looked as the fundamental theorem of the division by zero:

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:

f(a) = \frac{1}{2\pi i} \int_{\gamma} f(z) \frac{dz}{z - a},

$$where the $\gamma$ is a rectifiable simple Jordan closed curve that surrounds one time the point $a$ on a regular region of the function $f(z)$.

In particular, note that the concept of parallel lines is very important in the Euclidean plane and non-Euclidean geometry. The essential results may be stated as known since the discovery of the division by zero $z/0=0$. However, for importance, we would like to state clearly the details.

\begin{equation}

\lim_{z \to \infty} z = \infty,\tag{2.1}

\end{equation}however,\begin{equation}

[z]_{z =\infty} =0,\tag{2.2}

\end{equation} by the division by zero. The difference of $($2.1$)$ and $($2.2$)$ is very important as we see clearly by 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 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,\tag{2.3}

\end{equation} however

\begin{equation}

[x]_{ +\infty } =0, \quad [x]_{ -\infty } =0.\tag{2.4}

\end{equation}

\begin{equation}

L_k: a_k x + b_k y + c_k = 0, k=1,2. \tag{3.1}

\end{equation} The common point is given by, if $a_1 b_2 - a_2 b_1 \ne 0$; that is, the lines are not parallel

\left(\frac{b_1 c_2 - b_2 c_1}{a_1 b_2 - a_2 b_1}, \frac{a_2 c_1 - a_1 c_2}{a_1 b_2 - a_2 b_1}\right).\tag{3.2}

\end{equation} By the division by zero, we can understand that if $a_1 b_2 - a_2 b_1 = 0$, then the commom point is always given by

\begin{equation}

(0,0),\tag{3.3}

\end{equation}even the two lines are the same. This fact shows that the image of the Euclidean space in Section 2 is right.

For a function

\begin{equation}

S(x,y) = a(x^2+y^2) + 2gx + 2fy + c,\tag{4.1}

\end{equation}the radius $R$ of the circle $S(x,y) = 0$ is given by

\begin{equation}

R = \sqrt{\frac{g^2 +f^2 -ac}{a^2}}.\tag{4.2}

\end{equation}If $a = 0$, then the area $\pi R^2$ of the circle is zero, by the division by zero; that is, the circle is line $($degenerate$)$.

Here, note that by the Theorem, $R^2$ is zero for $a = 0$, but for (4.2) itself

\begin{equation}

R = \frac{-c}{2} \frac{1}{\sqrt{g^2 + f^2}}\tag{4.3}

\end{equation}for $a=0$. However, this result will be nonsense, and so, in this case, we should consider $R$ as zero as $ 0^2 =0$. When we apply the division by zero to functions, we can consider, in general, many ways.

For example, for the function $z/(z-1)$, when we insert $z=1$ in numerator and denominator, we have\begin{equation}

\left[\frac{z}{z-1}\right]_{z = 1} = \frac{1}{0} =0.\tag{4.4}

\end{equation}However, in the sense of the Theorem,

from the identity\begin{equation}

\frac{z}{z-1} = \frac{1}{z-1} + 1,\tag{4.5}

\end{equation}we have\begin{equation}

\left[\frac{z}{z-1}\right]_{z = 1} = 1.\tag{4.6}

\end{equation} By the Theorem, for analytic functions we can give uniquely determined values at isolated singular points, however, the values by means of the Laurent expansion are not always reasonable. We will need to consider many interpretations for reasonable values.

In addition, the center of the circle (4.3) is given by\begin{equation}

\left( - \frac{g}{a},- \frac{f}{a}\right).\tag{4.7}

\end{equation}Therefore, the center of a general line

\begin{equation}

2gx + 2fy + c=0\tag{4.8}

\end{equation}may be considered as the origin $(0,0)$, by the division by zero.

We can see similarly the 3 dimensional versions. We consider the functions\begin{equation}

S_j(x,y) = a_j(x^2+y^2) + 2g_jx + 2f_jy + c_j.\tag{4.9}

\end{equation}The distance $d$ of the centers of the circles $S_1(x,y) =0$ and $S_2(x,y) =0$ is given by\begin{equation}

d^2= \frac{g_1^2 + f_1^2}{a_1^2} - 2 \frac{g_1 g_2 + f_1 f_2}{a_1 a_2} + \frac{g_2^2 + f_2^2}{a_2^2}.\tag{4.10}

\end{equation}If $a_1 =0$, then by the division by zero

\begin{equation}d^2= \frac{g_2^2 + f_2^2}{a_2^2}.\tag{4.11}

\end{equation}Then, $S_1(x,y) =0$ is a line and its center is the origin $(0,0)$.

ViXra.org $($Friday, June 5, 2015$)$ © Ilija Barukčić, Jever, Germany. All rights reserved. Friday, June 5, 2015 20:44:59.

[3] J. A. Bergstra, Y. Hirshfeld and J. V. Tucker, Meadows and the equational specification of division $($arXiv:0901.0823v1[math.RA] 7 Jan 2009$)$.

[4] L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory

[5] M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,

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

Int. J. Appl. Math.

[6] H. Michiwaki, S. Saitoh, and M.Yamada, Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. 6 $($2015$)$, 1--8. http://www.ijapm.org/show-63-504-1.html

[7] H. Michiwaki, S. Saitoh and M. Takagi, A new concept for the point at infinity and the division by zero z/0=0 $($manuscript$)$.

[8] T. S. Reis and James A.D.W. Anderson, Transdifferential and Transintegral Calculus,

Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I

WCECS 2014, 22-24 October, 2014, San Francisco, USA

[9] T. S. Reis and James A.D.W. Anderson, Transreal Calculus, IAENG International J. of Applied Math., 45: IJAM 45 10 6.

[10] S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra & Matrix Theory.

[11] S.-E. Takahasi, On the identities $100/0 = 0$ and $0/0 = 0$. $($note$)$

[12] S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics,

[13] Announcement 179 $($2014.8.30$)$: Division by zero is clear as z/0=0 and it is fundamental in mathematics.

[14] Announcement 185 $($2014.10.22$)$: The importance of the division by zero $z/0=0$.

[15] Announcement 237 $($2015.6.18$)$: A reality of the division by zero $z/0=0$ by geometrical optics.

[16] Announcement 246 $($2015.9.17$)$: An interpretation of the division by zero $1/0=0$ by the gradients of lines.

[17] Announcement 247 $($2015.9.22$)$: The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

[18] Announcement 250 $($2015.10.20$)$: What are numbers? - the Yamada field containing the division by zero $z/0=0$.

[19] Announcement 252 $($2015.11.1$)$: Circles and curvature - an interpretation by Mr. Hiroshi Michiwaki of the division by zero $r/0 = 0$.

[20] Announcement 281$($2016.2.1$)$, The importance of the division by zero $z/0=0$.

[21] Announcement 282$($2016.2.2$)$, The division by zero $z/0=0$ on the Second Birthday.

ゼロ除算は余りにも有名で，基本的，予想外の結果であったので，もちろん数学的に厳格に議論する必要があり，初歩的な数学的内容にも関わらず，広く国際的に意見を求め，繰り返して検討したのは当然である。新しいこと，新規性と数学的な論理の確定後は、実在感，実在性を示す方向で、検討している。最近はユークリッド幾何学や力学，解析幾何学に広範にゼロ除算が現れるとともに微積分学，微分方程式などについても新しい視点，解釈などが現れて，ゼロ除算は基本的な数学であることが確認されている状況であると言える。

詳しく歴史を調べている C.B. Boyer 氏の視点では、ゼロ除算を最初に考えたのはアリストテレスであると判断され，アリストテレスは ゼロ除算は不可能であると判断していたという。― 真空で比を考えること、ゼロで割ることはできない。アリストテレスの世界観は 2000年を超えて現代にも及び，我々の得たゼロ除算はアリストテレスの 物理的な世界は連続である に反しているので受け入れられないと 複数の数学者が言明されたり，情感でゼロ除算は受け入れられないという人は未だに結構多い。

数学界では，オイラーが積極的に $1/0$ は無限であるという論文を書き，その誤りを論じた論文がある。アーベルも記号として，それを無限と表し、リーマンもその流れで無限遠点の概念を持ち、リーマン球面を考えている。これらの思想は現代でも踏襲され，超古典アルフォースの複素解析の本にもしっかりと受け継がれている。現代数学の世界の常識である。これらが畏れ多い天才たちの足跡である。こうなると，ゼロ除算は数学的に確定し，何びとと雖も疑うことのできない、数学的真実であると考えるのは至極当然である。― ゼロ除算はそのような重い歴史で、数学界では見捨てられていた問題であると言える。

しかしながら，現在に至るも ゼロ除算は広い世界で話題になっている。まず，顕著な研究者たちの議論を紹介したい：

論理、計算機科学，代数的な体の構造の問題（J. A. Bergstra, Y. Hirshfeld and J. V. Tucker），特殊相対性の理論とゼロ除算の関係（J. P. Barukcic and I. Barukcic），計算器がゼロ除算に会うと実害が起きることから，ゼロ除算回避の視点から，ゼロ除算の研究（T. S. Reis and James A.D.W. Anderson）。またフランスでも、奇怪な抽象的な世界を建設している人たちがいるが、個人レベルでもいろいろ奇怪な議論をしている人があとを立たない。また、数学界の難問リーマン予想に関係しているという。

直接議論を行っているところであるが，ゼロ除算で大きな広い話題は 特殊相対性理論，一般相対性理論の関係である。実際，物理とゼロ除算の関係はアリストテレス以来，ニュートン，アインシュタインの中心的な課題で，それはアインシュタインの次の意味深長な言葉で表現される：

Albert Einstein:

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

ゼロ除算について長い論文を書いている Robert S. Miller 氏とも議論をしているが，上記3グループとも議論し，質問に答え，彼らの論点はおかしいと主張しているが，彼らは暫く沈黙していたが，最近議論が活発化して，我々の理論，立場について，了解が，理解が得られていると判断される。交流の輪もどんどん広がっている。ゼロ除算は 数学として完全な扱いができたばかりか，結果が世の普遍的な現象を表現していることが実証されたと考えている。それらは3篇の論文に公刊され，第4論文も出版が決まり，さらに４篇の論文原稿があり，討論されている。2つの招待された国際会議で報告され，日本数学会でも2件ずつ2回（ともに春の学会）発表された。また，ゼロ除算の解説（2015.1.14;14ページ）を1000部印刷配布，広く議論している。現在までに理に叶った批判は一切出ていない。そこで，ゼロ除算の位置づけを纏めるとともに，ゼロ除算の公認を求めている（再生核研究所声明 279（2016.01.28）ゼロ除算の意義; 再生核研究所声明280（2016.01.29）ゼロ除算の公認，認知を求める; Announcement 282: The Division by Zero $z/0=0$ on the Second Birthday）。

以上

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底円の半径がである直円錐を考える。それを半径の底円に平行な円で切る。2つの円板の間の距離を $d$とする。このとき、直円錐の頂点と底円板の間の直円錐の表面上での距離 $R$ はEM半径と呼ばれ、道脇愛羽（8歳）さんが計算され、

$$

R=\frac{r_2}{r_2-r_1}\sqrt{d^2+(r_2-r_1)^2}

$$となる。これは2つの円板で囲まれた部分の平面上での回転を考えたときに、底円が描く円の半径を計算されたものである。

半径Rの円の曲率は $K=K(R)＝1/R$ で定義される。いま、$r_1$ が $r_2$ に近づいた場合を考える。もちろん、$d$ を一定にしてである。まず、極限値を考えれば、$R$ は無限大に発散して、底円が描く円は直線に近づき、実際、$r_1=r_2$ の時は底円が描く円は直線になり、回転体は直線運動を行うことが分かる。

ところがゼロ除算は、$r_1=r_2$ のとき、愛羽さんの公式は意味を有し、$R$ がゼロであることを言っている。それは、一体何を意味するだろうか。ゼロ除算は $K=K(R)＝1/R$ が $R=0$ でゼロ と言っているから、その時の曲率がゼロ、すなわち、極限の場合と同様に、底円が描く円は直線になり、回転体は直線運動を行うことを述べている。

いまの場合、極限で考えた極限値とゼロ除算、すなわち、$R=0$ 自身の結果が同じことを述べている。

この現象は、ゼロ除算が現実の現象を良く表現しているものと考えられる。

同時に、半径ゼロの円（点）の曲率がゼロであることをよく、表している。

上記、回転体の運動の例は、ゼロ除算の強力な不連続性をよく捉えたものとして、大変面白いのではないだろうか。

$r_1$ が $r_2$ に近づいた場合と、一致した場合の発現の様は微妙で堪らなく楽しい。

以下次号

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ゼロ除算 $z/0=0$ は分数の自然な拡張として既に $1+1=2$ のように自明であり，しかもそれは，我々の数学そのものであり，自然現象もきちんと表している。しかしながら，永い間の偏見の世界史，それも千年を超える偏見であり，天才的な数学者たちの足跡を省みて，中々世の中で理解されない状況があるのは，世の関係サイトを見ても良く分かる。それらには，そもそもゼロに対する恐怖心とゼロ除算にからむ，不可思議で奇妙な論調を見ることができる。

ゼロ除算のこのような歴史は，やがて人類の愚かさの象徴であると世界史で記録されるだろう。

$1/0$ とは何だと，恐怖心を抱く者は尚世に多い状態と言える。公理論的に吟味したか，現代数学とは違う，変な世界の数学ではないかと特に優秀な人たちが述べてきたのは大変興味深い事実である。

最近，数学基礎論，公理論，計算機科学の専門家たちのゼロ除算に関する論文を発見した。

Meadows and the equational specification of division

J A Bergstra

Informatics Institute, University of Amsterdam,

Science Park 403, 1098 SJ Amsterdam, The Netherlands

Y Hirshfeld

Department of Mathematics, Tel Aviv University,

Tel Aviv 69978, Israel

J V Tucker

Department of Computer Science, Swansea University,

Singleton Park, Swansea, SA2 8PP, United Kingdom

が，結論ではとにかく，奇妙なことが書かれている：

5 Concluding remarks and further questions

We notice that a conference version of this paper, though with a quite different emphasis of presentation, has appeared as [2]. The theory of meadows depends upon the formal idea of a total inverse operator. We do not claim that division by zero is possible in numerical calculations involving the rationals or reals. But we do claim that zero totalized division is logically, algebraically and computationally useful: for some applications, allowing zero totalized division in formal calculations, based on equations and rewriting, is appropriate because it is conceptually and technically simpler than the conventional concept of partial division. Furthermore, one can make arrangements to track the use of the inverse operation in formal calculations and classify them as safe or unsafe dependent upon 0?1 is invoked: see [11]. We expect these areas to include elementary school algebra, specifying and understanding gadgets containing calculators, spreadsheets, and declarative programming. Of course, further research is necessary to test these expectations: at present, our theory of meadows is a theory of zero totalized division, constitutes a generalization of the theory of fields, and is known to be useful in specifying numerical data types using equations.

文献を見れば，彼らが相当な専門家であることが分かる。―上記は要するにゼロ除算を含むいろいろな公理系を建設できるが，幻のようであるが計算に役立つと言っているようである。きちんと書かれているのは，ゼロ除算が可能であるとは主張しないということである。

しかるに，我々はゼロ除算は可能であり，ゼロ除算は我々の数学そのものであると言っている。我々の本質的な原理は，ゼロ除算 $z/0$ は定義そのものであり，そのように定義し，導入することによって，数学は完全になり，新しい世界を拓くと言っている。いろいろな証拠を挙げて，解説してきた。

しかしながら，それでもなお，$1/0$ とは何者かという，思いが残っているかも知れない。それは数と言えるのだろうかなどの雑念が残っているかも知れない。

このような折り，2015.10.3. 山田正人氏が研究室を訪れ，上記の論文とともに氏の考えを夢中で討論した。そのときは，2人ともそんなには気にしなかったのであるが，山田氏は，ゼロ除算を含む体の構造を入れる方法を説明された。体とは，四則演算が自由にできる数学の述語で，言わば 数の資格もつ性質 を表している。こうなると，ゼロ除算 $z/0$ は代数的に堂々と数であると言明できることになる。

念を押したいのは，ゼロ除算 $z/0=0$ とは定義そのものであり，その定義で，全ての理論は現代数学の中で，新しい世界を展開できるということである。

実際，山田氏の上記の理論から，新しい結果は，何一つ得られない，数学の内容としては自明なものばかりである。

しかしながら，引用された上記論文や，体の概念の重要性から，山田氏の発見された体は極めて重要であり，数とは山田氏の発見された体の元，そのものであると言える。

山田氏の発見された，体の構造とは簡単であるが，新規な面白い概念を含んでいる。

極めて面白いのは，y軸の勾配はゼロであるという知見をゼロ除算の帰結として得ていたが，山田氏の上記の考えは，そのことの帰結を微妙な論理で同様に導いている事実である。複素数体では $z/0$ が定義できないが山田体

尚，上記研究者 J. A. Bergstra 教授とは直接メールによる交流に成功し，山田体の理論，その他，我々のゼロ除算の結果について，それらの 新規性と論理の正しさ について，認めていると質問内容などから，判断される。今後の交流を楽しみにしている。

以下次号

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半径1の原点に中心を持つ，円 $C$ を考える。いま，簡単のために，正の $x$ 軸方向の直線上の点 $(x,0)$ $(0<x<1)$ を考える。この点の円 $C$ に関する鏡像は $y=1/x$ である。この対応で $x$ がどんどん小さくなり，ゼロに近づけば，対応する鏡像 $y$ はどんどん大きくなって行くことが分かる。そこで，古典的な複素解析学では，$x=0$ に対応する鏡像として，極限の点が存在するものとして，無限遠点を考え，原点の鏡像として無限遠点を対応させている。この意味で $1/0 =\infty$ と表わされている。この極限で捉える方法は解析学における基本的な考え方で，アーベルやオイラーもそのように考え，そのような記号を用いていたという。

しかしながら，このような極限の考え方は，適切ではないのではないだろうか。正の無限，どこまで行っても切りはなく，無限遠点など実在しているとは言えないのではないだろうか。これは，原点に対応する鏡像は $x>1$ に存在しないことを示している。ところが，ゼロ除算は $1/0=0$ であるから，ゼロの鏡像はゼロであると述べていることになる。実際，鏡像として，原点の鏡像は原点で，我々の世界で，そのように考えるのが妥当であると考えられよう。これは，ゼロ除算の強力な不連続性を幾何学的に実証していると考えられる。

ゼロより大きな数の世界で，ゼロに対応する鏡像 $y=1/x$ は存在しないので，仕方なく，神はゼロにゼロを対応させたという，神の意思が感じられるが，それがこの世界における実態と合っているということを示しているのではないだろうか。

この説は，伝統ある複素解析学の考えから，鏡像と無限遠点の概念を変える歴史的な大きな意味 を有するものと考える。

無限の彼方に行くという考えは極限の考え方で，適切であるが，それは数値として無限ではなくて，数値ではゼロを意味するからである。ゼロ除算では無限遠点は存在するが，無限という数 の存在を否定している。そこで，複素解析の考え方，表現をいろいろ変える必要がある。

以下次号

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