과학자 가문 ‘금수저’라도 괜찮다
최성우의 데자뷔 사이언스(50)
요즈음 우리 사회에서 유행하는 용어 중의 하나가 이른바 ‘금수저’이다. 부모로부터 상당한 재력을 물려받은 이들뿐 아니라, 다른 여러 분야에서도 금수저 관련 논쟁이 자주 되풀이되는데, 부정적인 의미로 사용되는 경우가 대부분이다.
과학기술계에서도 대를 이어서 업적을 낸 이들이 적지 않다. 물론 부모인 저명 과학기술자의 후광을 입은 경우도 있겠지만, 재능을 물려받는 데에 그치지 않고 과학을 대하는 태도와 탐구정신 등을 어릴 적부터 익힌 결과라면 그리 나쁘게 볼 일은 아닐 듯하다.
베르누이 정리로 유명한 다니엘 베르누이. ⓒ Free Photo
스티븐슨 부자, 노벨 부자, 베르누이 가문
엔지니어로서 대를 이어서 큰 업적을 낸 인물로서, 증기기관차의 아버지 조지 스티븐슨(George Stephenson; 1781-1846)과 그의 아들을 들 수 있다. 그보다 앞서 증기기관차를 발명한 선구자들이 여럿 있었음에도 불구하고 스티븐슨이 유명해진 것은, 그가 성능 좋은 증기기관차를 발명하는 데에 그치지 않고 실용적으로 널리 보급시키는 데에 성공했기 때문이다.
특히 증기기관차가 기존의 마차를 대체하고 육상교통의 혁명을 불러 올 수 있었던 데에는, 조지 스티븐슨의 아들인 철도기술자 로버트 스티븐슨(Robert Stephenson; 1803-1859)의 공로가 컸다. 그는 아버지와 함께 철도 가설사업에 일생을 걸고 매진했는데, 특히 철도용 교량 건설에 탁월한 능력을 발휘했던 것으로 알려져 있다.
‘노벨상의 창시자’이자 다이너마이트와 무연화약의 발명으로 유럽 최고의 갑부가 되었던 알프레드 노벨(Alfred Bernhard Nobel;1833-1896) 역시 가문의 대를 이은 과학기술자이다. 그의 아버지 임마누엘 노벨(Immanuel Nobel; 1801-1872)은 기술자이자 발명가로서 일찍부터 화약제조사업에 종사하였고, 몇 차례 파산의 어려움을 겪기도 했지만 흑색화약으로 기뢰 등을 생산하였다.
알프레드 노벨은 어릴 적부터 아버지의 화약공장 일을 도왔고, 그가 다이너마이트를 발명하게 된 것도 아버지의 공장이 신종 액체화약인 니트로글리세린으로 인하여 몇 차례 폭발사고를 겪었기 때문이다.
천왕성을 처음으로 발견한 독일 태생의 영국 천문학자 윌리엄 허셜(William Herschel; 1738-1822)과 그의 누이동생 캐롤라인 허셜(Caroline Herschel; 1750-1848)은 남매가 눈물겹도록 협력하고 헌신해서 업적을 낸 것으로 잘 알려져 있다. ‘남매 명콤비’의 과학자에 그치지 않고, 윌리엄의 아들 존 프레더릭 허셜(John Frederick William Herschel; 1792-1871) 역시 탁월한 천문학자가 되었으니 대를 이어 가문의 명예를 떨친 셈이다.
존 프레더릭 허셜은 아버지의 유업을 계승하여 항성천문학을 더욱 발전시켰고, 광도계를 사용하여 1등성의 밝기가 6등성의 100배라는 사실을 밝혀냈으며, 5,000개 이상의 천체를 수록한 ‘성운, 성단 총목록’을 발표하는 등, 아버지 못지않은 업적을 남겼다.
과학자 가문으로 매우 유명한 사례로서, 스위스의 베르누이 가문을 들 수 있다. 이들 가문은 100년 이상에 걸쳐서 대대로 다수의 탁월한 수학자와 과학자들을 배출한 것으로 잘 알려져 있다. 유체역학에서 ‘베르누이 정리’는 비행기가 양력을 받아 하늘을 날 수 있는 것을 설명하는 매우 중요한 원리인데, 이를 발견한 것은 수학자이자 이론물리학자였던 다니엘 베르누이(Daniel Bernoulli; 1700-1782)이다.
그의 아버지 요한 베르누이(Johann Bernoulli; 1667-1748)는 해석학에서 여러 업적을 남겼고 역학에서의 가상변위의 원리를 정립하였으며, 탁월한 아들뿐 아니라 세기적 수학자였던 오일러(Leonhard Euler; 1707-1783)를 제자로 두었다. 요한 베르누이의 형 야콥 베르누이(Jakob Bernoulli; 1654-1705)는 동생과 함께 해석학을 연구하였고, 급수와 확률론 등에서도 업적을 남겼다.
알프레드 노벨의 생가. ⓒ Free Photo
대를 이은 노벨 과학상 수상자들
과학 분야에서 스승과 제자가 함께 또는 번갈아서 노벨상을 수상하는 경우는 너무 많아서 일일이 다 사례를 들기도 어려울 정도인데, 부자(父子)간에도 같은 일이 종종 벌어지곤 한다.
아버지와 아들이 다 노벨과학상을 수상한 첫 사례로는, 전자의 발견자 톰슨(Joseph John Thomson; 1856-1940)과 그의 아들을 들 수 있다. 톰슨은 전자를 발견하여 원자의 구조를 밝혀내는 데에 중요한 전기를 마련하였을 뿐 아니라, 네온의 동위원소를 발견한 공로 등으로 1906년도 노벨 물리학상을 수상하였다. 그의 아들인 조지 톰슨(George Paget Thomson; 1892-1975) 전자의 파동성, 즉 얇은 막에 의한 전자 빔의 회절현상을 발견하여 드브로이의 물질파 이론을 확증한 공로로 1937년도 노벨 물리학상을 수상하였다.
비교적 최근인 2006년도 노벨화학상 수상자 로저 콘버그(Roger David Kornberg;1947-) 역시 부자가 다 노벨상을 받은 경우이다. 그는 유전자 발현 경로의 첫 단계인 유전정보 전사를 분자 수준에서 규명하여 노벨화학상을 수상하였는데, 그의 아버지 아서 콘버그(Arthur Kornberg; 1918-2007)는 세포가 분열할 때 DNA가 복제되는 과정을 규명한 공로로 1959년도 노벨 생리의학상을 받은 바 있다.
노벨상을 아버지가 나중에, 아들이 훗날 받은 경우가 아니라, 아예 부자가 같은 해의 노벨과학상을 동시에 수상한 경우도 있다. X선에 의한 결정 구조의 해석으로 1915년도 노벨 물리학상을 받은 윌리엄 헨리 브래그(William Henry Bragg; 1862-1942)와 그의 아들 윌리엄 로렌스 브래그(William Lawrence Bragg; 1890-1971)이다.
브래그 부자는 X선 회절에 관한 이른바 브래그의 식을 함께 유도하고 X선 분광기를 고안한 공로로 노벨 물리학상을 공동으로 받았는데, 아들인 로렌스 브래그는 당시 나이 25세로 과학 분야 노벨상 수상자로는 역대 최연소 기록을 세웠다.
부자지간은 아니지만, 대를 이은 노벨상 수상자로 유명한 또 하나의 사례가 바로 퀴리 모녀이다. 퀴리부인이라 불리는 마리 퀴리(Marie Curie; 1867-1934)는 방사선에 대한 연구 및 폴로늄과 라듐의 발견 등으로 1903년도 노벨 물리학상을 스승인 베크렐(Antoine Henri Becquerel; 1852-1908), 남편인 피에르 퀴리(Pierre Curie; 1859-1906)과 함께 받았고, 남편이 죽은 후인 1911년도 노벨화학상을 단독으로 수상한 바 있다.
마리 퀴리의 큰 딸인 이렌 퀴리(Irène Joliot-Curie; 1897-1956)는 인공 방사성 원소의 존재를 확인한 공로로 1935년도 노벨화학상을 남편인 프레더릭 졸리오(Jean Frédéric Joliot-Curie; 1900-1958)와 공동으로 수상하였다. 퀴리 집안에서는 2대에 걸쳐서 세 차례의 과학 분야 노벨상을 받았으니, 역시 대단한 과학자 가문인 셈이다.http://www.sciencetimes.co.kr/?news=%EA%B3%BC%ED%95%99%EC%9E%90-%EA%B0%80%EB%AC%B8-%EA%B8%88%EC%88%98%EC%A0%80%EB%9D%BC%EB%8F%84-%EA%B4%9C%EC%B0%AE%EB%8B%A4
オイラー・ベルヌーイ
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\begin{document}
\title{\bf Announcement 326: The division by zero z/0=0 - its impact to human beings through education and research\\
(2016.10.17)}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
}
\date{\today}
\maketitle
{\bf 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.
\bigskip
\section{Introduction}
%\label{sect1}
By a {\bf natural extension} of the fractions
\begin{equation}
\frac{b}{a}
\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,
\end{equation}
incidentally in \cite{s} 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 \cite{kmsy} 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 the division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei Takahasi (\cite{kmsy}) 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):
\bigskip
{\bf Proposition 1. }{\it 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.
\medskip
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 $W=0$ ({\bf 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 impact to complex analysis and to our ideas for the space and universe.
However, the division by zero (1.2) is now clear, indeed, for the introduction of (1.2), we have several independent approaches as in:
\medskip
1) by the generalization of the fractions by the Tikhonov regularization and by the Moore-Penrose generalized inverse,
\medskip
2) by the intuitive meaning of the fractions (division) by H. Michiwaki - repeated subtraction method,
\medskip
3) by the unique extension of the fractions by S. Takahasi, as in the above,
\medskip
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 ${\bf C}$,
\medskip
and
\medskip
5) by considering the values of functions with the mean values of functions.
\medskip
Furthermore, in (\cite{msy}) we gave the results in order to show the reality of the division by zero in our world:
\medskip
\medskip
A) a field structure containing the division by zero --- the Yamada field ${\bf Y}$,
\medskip
B) by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,
\medskip
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.
\medskip
and
\medskip
D) by considering rotation of a right circular cone having some very interesting
phenomenon from some practical and physical problem.
\medskip
In (\cite{mos}), many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In (\cite{ms}), 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.
\medskip
See J. A. Bergstra, Y. Hirshfeld and J. V. Tucker \cite{bht} 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 (\cite{bb}) 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 (\cite{ra,ra2}) 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:
{\it Riemann Hypothesis Addendum - Breakthrough
Kurt Arbenz
https://www.researchgate.net/publication/272022137 Riemann Hypothesis Addendum - Breakthrough.}
\medskip
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 {\bf the biggest blunder of his life} [1]：
1. 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 (\cite{kmsy,msy,mos}). The results will give great impact 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 impact to our basic ideas on the universe.
For our ideas on the division by zero, see the survey style announcements.
\section{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 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 {\bf division by zero calculus}:
\medskip
For any formal Laurent expansion around $z=a$,
\begin{equation}
f(z) = \sum_{n=-\infty}^{\infty} C_n (z - a)^n,
\end{equation}
we obtain the identity, by the division by zero
\begin{equation}
f(a) = C_0.
\end{equation}
\medskip
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
\end{equation}
or
\begin{equation}
f^\prime(x) = -\infty,
\end{equation}
both cases, we can write them as follows:
\begin{equation}
f^\prime(x) = 0.
\end{equation}
\medskip
For the elementary ordinary differential equation
\begin{equation}
y^\prime = \frac{dy}{dx} =\frac{1}{x}, \quad x > 0,
\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,
\end{equation}
we see that, by the division by zero,
\begin{equation}
y^\prime (0)= \left[ \frac{1}{x}\right]_{x=0} = 0,
\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 (2.6) and (2.7). At $x = 0$, we see that we can not consider the limit in the sense (2.3). However, for $x >0$ we have (2.6) and
\begin{equation}
\lim_{x \to +0} \left(\log x \right)^\prime = +\infty.
\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
\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),
\end{equation}
we obtain, by the division by zero,
\begin{equation}
\int x(x^{2}+1)^{-1}dx=\frac{\log(x^{2}+1)}{2}.
\end{equation}
We will consider the fundamental ordinary differential equations
\begin{equation}
x^{\prime \prime}(t) =g -kx^{\prime}(t)
\end{equation}
with the initial conditions
\begin{equation}
x(0) = -h, x^{\prime}(0) =0.
\end{equation}
Then we have the solution
\begin{equation}
x(t) = \frac{g}{k}t + \frac{g(e^{-kt}- 1)}{k^2} - h.
\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.
\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'Hôpital'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.
\end{equation}
However,
from the identity --
the Laurent expansion around $z=1$,
\begin{equation}
\frac{z}{z-1} = \frac{1}{z-1} + 1,
\end{equation}
we have
\begin{equation}
\left[\frac{z}{z-1}\right]_{z = 1} = 1.
\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 \cite{kmsy,msy}.
\section{Albert Einstein's biggest blunder}
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.
\section{Computer systems}
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.
\section{General ideas on the universe}
The division by zero may be related to religion, philosophy and the ideas on the universe, and it will creat a new world. Look the new world introduced.
\bigskip
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?
\bigskip
\bibliographystyle{plain}
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J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,
Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).
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L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
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. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
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T. Matsuura and S. Saitoh,
Matrices and division by zero $z/0=0$, Advances in Linear Algebra
\& Matrix Theory, 6, 51-58. http://dx.doi.org/10.4236/alamt.2016.62007 http://www.scirp.org/journal/alamt
\bibitem{msy}
H. Michiwaki, S. Saitoh, and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{mos}
H. Michiwaki, H. Okumura, and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces.
International Journal of Mathematics and Computation
(in press).
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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
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T. S. Reis and J.A.D.W. Anderson,
Transreal Calculus,
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S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/
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S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.
\bibitem{ann179}
Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.
\bibitem{ann185}
Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.
\bibitem{ann237}
Announcement 237 (2015.6.18): A reality of the division by zero $z/0=0$ by geometrical optics.
\bibitem{ann246}
Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.
\bibitem{ann247}
Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.
\bibitem{ann250}
Announcement 250 (2015.10.20): What are numbers? - the Yamada field containing the division by zero $z/0=0$.
\bibitem{ann252}
Announcement 252 (2015.11.1): Circles and
curvature - an interpretation by Mr.
Hiroshi Michiwaki of the division by
zero $r/0 = 0$.
\bibitem{ann281}
Announcement 281 (2016.2.1): The importance of the division by zero $z/0=0$.
\bibitem{ann282}
Announcement 282 (2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.
\bibitem{ann293}
Announcement 293 (2016.3.27): Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.
\bibitem{ann300}
Announcement 300 (2016.05.22): New challenges on the division by zero z/0=0.
\end{thebibliography}
\end{document}
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