理論ならアインシュタイン
実験ならファラデー
総合ならニュートン
1.最初に火を点けた人
2.車輪を作った人
3.酒を作った人
ニュートンとか1000番目ぐらいだろ
マクスウェル
ディラック
ボルツマン
実験屋は俺もファラデーかなと思ったりする。
でもガリレオも凄いな。
ガリレオかもしれない。
ファラデーの時代はもう力学的世界観が普及してた頃だけど
ガリレオのはそんな時代じゃないから。
ガリレオによる慣性の思考実験は、発想が凄い。
今のような理論体系が整っていない時代に、数々の実験を行い
個々異なる実験でそれぞれ理論的考察も行うという、優れた人物。
何ていうか、実験と理論を両方できた偉大な物理学者という感じ。
やっぱ、ニュートンだと思うなぁ
ライプニッツとほぼ同時とはいえ微積分の発見と、
ニュートンの三法則、万有引力の法則の発見は
ガリレオ、ケプラー、フックその他の業績を統合し、最もシンプルな形
でまとめたのだから。。
アインシュタインも相対論をほぼ一人で全部作った点はすごいが、
総合的な意味ではやはりニュートンに劣るような気がする。
後、時代は離れるがアルキメデスもすごい。
あの時代にしては殆ど何でも自分で発見している。
0から組み立てるのは確かに凄いが（ニュートン）
あらかじめあるものを再構成するのはもっと凄いと思うんだ（アインシュタイン）
レゴブロックでお城を作るにしても最初に簡単な部品だけで作った奴と、
最初に作った奴のと全く同じ部品だけで色んな仕掛けが用意された新しい城を作ってしまったというか
あの辺の時代は色々と入り乱れてて頭の中で整理しきれないんだが
コペルニクス（地動説）
デカルト（唯物論）
ガリレイ（落体の法則・実験的手法の確立）
ニュートン（運動は単純な法則で記述できる）
の中で欠けたらアウトなのは誰なんだ？
個人的にはデカルトじゃないかと思うんだが…
ニュートンの運動の法則はそもそもデカルトやガリレイのをまとめたもの。
ガリレイが慣性の法則、デカルトが運動量保存則、
そしてニュートンが運動方程式の元たる第二法則の考案。
ガリレイの慣性の実験は、今でいう力学的エネルギー保存則の類。
球体を斜面のある高さから放して、Ｕ字の斜面をどう上るかを実験して
ほぼ同じ高さに上ることを実験を通して確認した。
ここからガリレイは、もし向かう先の斜面が水平だったらどうなるかと考えた。
そしてガリレイは上記の実験から、球体は水平面を進み続けるだろうと予想した。
そこから、そうならない現実に対して地面の摩擦という存在を仮定・解釈したんだな。
デカルトはガリレイのこの実験から、
以前から運動の法則として自分が考えていた運動量保存則に
慣性の法則を加えて彼の運動の２法則として定式化した。
デカルトは今の力学的世界観のイメージをつくったと思う。
これらを結構読んでいたらしいニュートンは、これを力学の概念でまとめた。
慣性の法則、運動量保存則の力学的解釈である作用反作用の法則、
そして自分が発案した力学の基本概念たる第二法則。
デカルトは運動量保存則というか、保存則の考えを導入したような気がする。
これに対し、ニュートンは変化の捉え方を考案導入した。
物体の運動の変化を次々に調べていけば、基本的な力を見出す。
これが自然を知るということだ、とニュートンは考えたんだな。
斬新過ぎて、他にこういう発想をした人物はいないと思うよ。
ガリレイ、デカルト、ニュートン、どれも偉大な物理学者だな。
最も偉大な物理学者は
究極理論の発見者
そこの君だよ！
http://uchu2ch.blog.fc2.com/blog-entry-155.html
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 293: Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0}
\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 declare that any parallel lines have the common point $(0,0) $ in the sense of the division by zero. From this fact we have to change our basic idea for the Euclidean plane and we will see a new world for not only mathematics, but also the universe.
\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 result is a very special case for general fractional functions in \cite{cs}.
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{taka}) (see also \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 } $
$$
F (b, 0) = 0.
$$
}
\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 (\cite{ahlfors}). Therefore, the division by zero will give great impacts 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 or by the Moore-Penrose generalized inverse,
\medskip
2) by the intuitive meaning of the fractions (division) by H. Michiwaki,
\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 $1/\overline{z}$ 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,
\medskip
and
\medskip
D) by considering rotation of a right circular cone having some very interesting
phenomenon from some practical and physical problem --- EM radius.
\medskip
See also \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 division $0/0$ and special relative theory of Einstein.
Furthermore, Reis and Anderson (\cite{ra,ra2}) extends the system of the real numbers by defining division by zero.
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.
For our results, see the survey style announcements 179,185,237,246, 247,250 and 252 of the Institute of Reproducing Kernels (\cite{ann179,ann185,ann237,ann246,ann247,ann250,ann252}).
At this moment, the following theorem may be looked as the fundamental theorem of the division by zero:
\bigskip
{\bf Theorem (\cite{mst}).} {\it Any analytic function takes a definite value at an isolated singular point }{\bf with a natural meaning.}
\bigskip
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:
\bigskip
{\bf Corollary.} {\it For an isolated singular point $a$ of an analytic function $f(z)$, we have the Cauchy integral formula
$$
f(a) = \frac{1}{2\pi i} \int_{\gamma} f(z) \frac{dz}{z - a},
$$
where the $\gamma$ is a rectifiable simple Jordan closed curve that surrounds one time the point $a$
on a regular region of the function $f(z)$.
}
\bigskip
The essential meaning of this theorem and corollary is given by that: the values of functions may be understood in the sense of the mean values of analytic functions.
\medskip
In this announcement, we will state the basic property of parallel lines by the division by zero on the Euclidean plane and we will be able to see that the division by zero introduces a new world and fundamental mathematics.
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.
\section{The point at infinity}
We will be able to see the whole Euclidean plane by the stereographic projection into the Riemann sphere --- {\it We think that in the Euclidean plane, there does not exist the point at infinity}.
However, we can consider it as a limit like $\infty$. Recall the definition of $z \to \infty$ by $\epsilon$-$\delta$ logic; that is, $\lim_{z \to \infty} z = \infty$ if and only if for any large $M>0$, there exists a number $L>0$ such that for any z satisfying $L <|z|$, $M<|z|$. In this definition, the infinity $\infty$ does not appear.
{\it The infinity is not a number, but it is an ideal space point.}
The behavior of the space around the point at infinity may be considered by that around the origin by the linear transform $W = 1/z$(\cite{ahlfors}). We thus see that
\begin{equation}
\lim_{z \to \infty} z = \infty,
\end{equation}
however,
\begin{equation}
[z]_{z =\infty} =0,
\end{equation}
by the division by zero. The difference of (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,
\end{equation}
however,
\begin{equation}
[x]_{ +\infty } =0, \quad [x]_{ -\infty } =0.
\end{equation}
\section{Interpretation by analytic geometry}
We write lines by
\begin{equation}
L_k: a_k x + b_k y + c_k = 0, k=1,2.
\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
\begin{equation}
\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).
\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),
\end{equation}
even the two lines are the same. This fact shows that the image of the Euclidean space in Section 2 is right.
\section{Remarks}
For a function
\begin{equation}
S(x,y) = a(x^2+y^2) + 2gx + 2fy + c,
\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}}.
\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}}
\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.
\end{equation}
However, in the sense of the Theorem,
from the identity
\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}
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).
\end{equation}
Therefore, the center of a general line
\begin{equation}
2gx + 2fy + c=0
\end{equation}
may be considered as the origin $(0,0)$, by the division by zero.
We can see similarly the 3 dimensional versions.
\medskip
We consider the functions
\begin{equation}
S_j(x,y) = a_j(x^2+y^2) + 2g_jx + 2f_jy + c_j.
\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}.
\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}.
\end{equation}
Then, $S_1(x,y) =0$ is a line and its center is the origin $(0,0)$.
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{bb}
J. P. Barukcic and I. Barukcic, Anti Aristotle - The Division Of Zero By Zero,
ViXra.org (Friday, June 5, 2015)
© Ilija Barukčić, Jever, Germany. All rights reserved. Friday, June 5, 2015 20:44:59.
\bibitem{bht}
J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,
Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).
\bibitem{cs}
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.
\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. 6(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{mst}
H. Michiwaki, S. Saitoh and M. Takagi,
A new concept for the point at infinity and the division by zero z/0=0
(manuscript).
\bibitem{ra}
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
\bibitem{ra2}
T. S. Reis and James A.D.W. Anderson,
Transreal Calculus,
IAENG International J. of Applied Math., 45: IJAM 45 1 06.
\bibitem{s}
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/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$.}
(note)
\bibitem{ttk}
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.
\end{thebibliography}
\end{document}
実験ならファラデー
総合ならニュートン
1.最初に火を点けた人
2.車輪を作った人
3.酒を作った人
ニュートンとか1000番目ぐらいだろ
マクスウェル
ディラック
ボルツマン
実験屋は俺もファラデーかなと思ったりする。
でもガリレオも凄いな。
ガリレオかもしれない。
ファラデーの時代はもう力学的世界観が普及してた頃だけど
ガリレオのはそんな時代じゃないから。
ガリレオによる慣性の思考実験は、発想が凄い。
今のような理論体系が整っていない時代に、数々の実験を行い
個々異なる実験でそれぞれ理論的考察も行うという、優れた人物。
何ていうか、実験と理論を両方できた偉大な物理学者という感じ。
やっぱ、ニュートンだと思うなぁ
ライプニッツとほぼ同時とはいえ微積分の発見と、
ニュートンの三法則、万有引力の法則の発見は
ガリレオ、ケプラー、フックその他の業績を統合し、最もシンプルな形
でまとめたのだから。。
アインシュタインも相対論をほぼ一人で全部作った点はすごいが、
総合的な意味ではやはりニュートンに劣るような気がする。
後、時代は離れるがアルキメデスもすごい。
あの時代にしては殆ど何でも自分で発見している。
0から組み立てるのは確かに凄いが（ニュートン）
あらかじめあるものを再構成するのはもっと凄いと思うんだ（アインシュタイン）
レゴブロックでお城を作るにしても最初に簡単な部品だけで作った奴と、
最初に作った奴のと全く同じ部品だけで色んな仕掛けが用意された新しい城を作ってしまったというか
あの辺の時代は色々と入り乱れてて頭の中で整理しきれないんだが
コペルニクス（地動説）
デカルト（唯物論）
ガリレイ（落体の法則・実験的手法の確立）
ニュートン（運動は単純な法則で記述できる）
の中で欠けたらアウトなのは誰なんだ？
個人的にはデカルトじゃないかと思うんだが…
ニュートンの運動の法則はそもそもデカルトやガリレイのをまとめたもの。
ガリレイが慣性の法則、デカルトが運動量保存則、
そしてニュートンが運動方程式の元たる第二法則の考案。
ガリレイの慣性の実験は、今でいう力学的エネルギー保存則の類。
球体を斜面のある高さから放して、Ｕ字の斜面をどう上るかを実験して
ほぼ同じ高さに上ることを実験を通して確認した。
ここからガリレイは、もし向かう先の斜面が水平だったらどうなるかと考えた。
そしてガリレイは上記の実験から、球体は水平面を進み続けるだろうと予想した。
そこから、そうならない現実に対して地面の摩擦という存在を仮定・解釈したんだな。
デカルトはガリレイのこの実験から、
以前から運動の法則として自分が考えていた運動量保存則に
慣性の法則を加えて彼の運動の２法則として定式化した。
デカルトは今の力学的世界観のイメージをつくったと思う。
これらを結構読んでいたらしいニュートンは、これを力学の概念でまとめた。
慣性の法則、運動量保存則の力学的解釈である作用反作用の法則、
そして自分が発案した力学の基本概念たる第二法則。
デカルトは運動量保存則というか、保存則の考えを導入したような気がする。
これに対し、ニュートンは変化の捉え方を考案導入した。
物体の運動の変化を次々に調べていけば、基本的な力を見出す。
これが自然を知るということだ、とニュートンは考えたんだな。
斬新過ぎて、他にこういう発想をした人物はいないと思うよ。
ガリレイ、デカルト、ニュートン、どれも偉大な物理学者だな。
最も偉大な物理学者は
究極理論の発見者
そこの君だよ！
http://uchu2ch.blog.fc2.com/blog-entry-155.html
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 293: Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0}
\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 declare that any parallel lines have the common point $(0,0) $ in the sense of the division by zero. From this fact we have to change our basic idea for the Euclidean plane and we will see a new world for not only mathematics, but also the universe.
\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 result is a very special case for general fractional functions in \cite{cs}.
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{taka}) (see also \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 } $
$$
F (b, 0) = 0.
$$
}
\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 (\cite{ahlfors}). Therefore, the division by zero will give great impacts 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 or by the Moore-Penrose generalized inverse,
\medskip
2) by the intuitive meaning of the fractions (division) by H. Michiwaki,
\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 $1/\overline{z}$ 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,
\medskip
and
\medskip
D) by considering rotation of a right circular cone having some very interesting
phenomenon from some practical and physical problem --- EM radius.
\medskip
See also \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 division $0/0$ and special relative theory of Einstein.
Furthermore, Reis and Anderson (\cite{ra,ra2}) extends the system of the real numbers by defining division by zero.
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.
For our results, see the survey style announcements 179,185,237,246, 247,250 and 252 of the Institute of Reproducing Kernels (\cite{ann179,ann185,ann237,ann246,ann247,ann250,ann252}).
At this moment, the following theorem may be looked as the fundamental theorem of the division by zero:
\bigskip
{\bf Theorem (\cite{mst}).} {\it Any analytic function takes a definite value at an isolated singular point }{\bf with a natural meaning.}
\bigskip
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:
\bigskip
{\bf Corollary.} {\it For an isolated singular point $a$ of an analytic function $f(z)$, we have the Cauchy integral formula
$$
f(a) = \frac{1}{2\pi i} \int_{\gamma} f(z) \frac{dz}{z - a},
$$
where the $\gamma$ is a rectifiable simple Jordan closed curve that surrounds one time the point $a$
on a regular region of the function $f(z)$.
}
\bigskip
The essential meaning of this theorem and corollary is given by that: the values of functions may be understood in the sense of the mean values of analytic functions.
\medskip
In this announcement, we will state the basic property of parallel lines by the division by zero on the Euclidean plane and we will be able to see that the division by zero introduces a new world and fundamental mathematics.
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.
\section{The point at infinity}
We will be able to see the whole Euclidean plane by the stereographic projection into the Riemann sphere --- {\it We think that in the Euclidean plane, there does not exist the point at infinity}.
However, we can consider it as a limit like $\infty$. Recall the definition of $z \to \infty$ by $\epsilon$-$\delta$ logic; that is, $\lim_{z \to \infty} z = \infty$ if and only if for any large $M>0$, there exists a number $L>0$ such that for any z satisfying $L <|z|$, $M<|z|$. In this definition, the infinity $\infty$ does not appear.
{\it The infinity is not a number, but it is an ideal space point.}
The behavior of the space around the point at infinity may be considered by that around the origin by the linear transform $W = 1/z$(\cite{ahlfors}). We thus see that
\begin{equation}
\lim_{z \to \infty} z = \infty,
\end{equation}
however,
\begin{equation}
[z]_{z =\infty} =0,
\end{equation}
by the division by zero. The difference of (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,
\end{equation}
however,
\begin{equation}
[x]_{ +\infty } =0, \quad [x]_{ -\infty } =0.
\end{equation}
\section{Interpretation by analytic geometry}
We write lines by
\begin{equation}
L_k: a_k x + b_k y + c_k = 0, k=1,2.
\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
\begin{equation}
\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).
\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),
\end{equation}
even the two lines are the same. This fact shows that the image of the Euclidean space in Section 2 is right.
\section{Remarks}
For a function
\begin{equation}
S(x,y) = a(x^2+y^2) + 2gx + 2fy + c,
\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}}.
\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}}
\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.
\end{equation}
However, in the sense of the Theorem,
from the identity
\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}
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).
\end{equation}
Therefore, the center of a general line
\begin{equation}
2gx + 2fy + c=0
\end{equation}
may be considered as the origin $(0,0)$, by the division by zero.
We can see similarly the 3 dimensional versions.
\medskip
We consider the functions
\begin{equation}
S_j(x,y) = a_j(x^2+y^2) + 2g_jx + 2f_jy + c_j.
\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}.
\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}.
\end{equation}
Then, $S_1(x,y) =0$ is a line and its center is the origin $(0,0)$.
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
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L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{bb}
J. P. Barukcic and I. Barukcic, Anti Aristotle - The Division Of Zero By Zero,
ViXra.org (Friday, June 5, 2015)
© Ilija Barukčić, Jever, Germany. All rights reserved. Friday, June 5, 2015 20:44:59.
\bibitem{bht}
J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,
Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).
\bibitem{cs}
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 $0/0=0$,
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\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. 6(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{mst}
H. Michiwaki, S. Saitoh and M. Takagi,
A new concept for the point at infinity and the division by zero z/0=0
(manuscript).
\bibitem{ra}
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
\bibitem{ra2}
T. S. Reis and James A.D.W. Anderson,
Transreal Calculus,
IAENG International J. of Applied Math., 45: IJAM 45 1 06.
\bibitem{s}
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/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$.}
(note)
\bibitem{ttk}
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.
\end{thebibliography}
\end{document}
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