【2011年11月14日 HubbleSite】
天文学における20世紀最大の発見のひとつ、「宇宙膨張」。一般的にはアメリカのハッブルが最初に発表したとされてきたが、一方で実はそれ以前にベルギーのルメートルが発表していたこともわかっている。この食い違いは、どうやらこの「先の発見者」自身の功名心のなさによるものだったようだ。Mario Livio氏が「ネイチャー」誌に発表した。
ルメートルとアインシュタイン
アルバート・アインシュタイン（右）と並ぶジョルジュ・ルメートル（左）（提供：Historical Image）
「宇宙は全方向に一様に膨張している」 ― この宇宙膨張の発見はエドウィン・ハッブルの功績として一般に知られ、「ハッブル定数」と呼ばれる宇宙の膨張率を求めた論文は1929年に発表されている。2011年のノーベル物理学賞は宇宙の加速膨張の発見に与えられたが、ハッブル自身はノーベル賞を受賞する前に亡くなった。だがハッブルの名はあの「ハッブル宇宙望遠鏡」にしっかりと残っている。
遠い銀河ほど地球から遠ざかる速度が速い、つまり宇宙は膨張しているというハッブルの発見は、銀河までの距離（ハッブル自身の測定）と、それらの銀河が遠ざかる速度（アメリカのヴェスト・スライファーによる測定）をもとになされた。この、銀河までの距離と後退（遠ざかる）速度の比率が、「ハッブル定数」として現在知られているものだ。
しかしこれに先立つこと2年、ベルギーの司祭であり天文学者であったジョルジュ・ルメートルが、ハッブルと同じような結果を既に発表していた。スライファーが測定した赤方偏移（後退速度の指標となる）と、1926年にハッブルが発表した銀河までの距離とを組み合わせた成果である。だがこの発表は、無名の仏語誌「ブリュッセル科学会年報（Annales de la Société Scientifique de Bruxelles）」でのものだったため、ほとんど注目されることはなかった。
しかし、話はここで終わらない。1931年に英国の王立天文学会月報（Monthly Notices of the Royal Astronomical Society）でルメートルの発表の英訳が発表されたが、なんとハッブル定数（と呼ばれるようになったもの）に関連した箇所が削られていたのだ。
この事実は1984年に判明して以来知る人ぞ知る話となり、誰が何の意図で削ったのかということがしばしば憶測の対象になった。月報の編集者が削ったのか、それともハッブルが何か手を加えたのか？
宇宙望遠鏡科学研究所（STScI）のMario Livio氏が英国王立天文学会の書簡記録を調査したところ、ルメートルから月報編集者に宛てた手紙にその答えはあった。なんとルメートル自身が英訳の際に削ったというのだ。この手紙で彼は、「面白くもない、視線速度に関する仮の議論を改めて載せる必要はないと考えた。代わりに過去の引用文献や新しい論文のテーマを載せたほうが良い」と綴っている。
「ルメートルは自分の発見をことさら強調し、自分の成果とするような気持ちは全く持っておらず、1929年にハッブルの結果が出ているので、1931年に自分が先に発見したと同じ話を繰り返す気はなかったのだろう」とLivio氏は結論付けている。
もし歴史がほんの少し違っていたら、私たちは「ルメートル宇宙望遠鏡」の素晴らしい画像を堪能することになっていたのかもしれない。http://www.astroarts.co.jp/news/2011/11/14lemaitre/index-j.shtml
Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics\\
}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
E-mail: kbdmm360@yahoo.co.jp\\
}
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall introduce the zero division $z/0=0$. The result is a definite one and it is fundamental in mathematics.
\bigskip
\section{Introduction}
%\label{sect1}
By a natural extension of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising 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 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 some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
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
\section{What are the fractions $ b/a$?}
For many mathematicians, the division $b/a$ will be considered as the inverse of product;
that is, the fraction
\begin{equation}
\frac{b}{a}
\end{equation}
is defined as the solution of the equation
\begin{equation}
a\cdot x= b.
\end{equation}
The idea and the equation (2.2) show that the division by zero is impossible, with a strong conclusion. Meanwhile, the problem has been a long and old question:
As a typical example of the division by zero, we shall recall the fundamental law by Newton:
\begin{equation}
F = G \frac{m_1 m_2}{r^2}
\end{equation}
for two masses $m_1, m_2$ with a distance $r$ and for a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty,
\end{equation}
however, in our fraction
\begin{equation}
F = G \frac{m_1 m_2}{0} = 0.
\end{equation}
\medskip
Now, we shall introduce an another approach. The division $b/a$ may be defined {\bf independently of the product}. Indeed, in Japan, the division $b/a$ ; $b$ {\bf raru} $a$ ({\bf jozan}) is defined as how many $a$ exists in $b$, this idea comes from subtraction $a$ repeatedly. (Meanwhile, product comes from addition).
In Japanese language for "division", there exists such a concept independently of product.
H. Michiwaki and his 6 years old girl said for the result $ 100/0=0$ that the result is clear, from the meaning of the fractions independently the concept of product and they said:
$100/0=0$ does not mean that $100= 0 \times 0$. Meanwhile, many mathematicians had a confusion for the result.
Her understanding is reasonable and may be acceptable:
$100/2=50 \quad$ will mean that we divide 100 by 2, then each will have 50.
$100/10=10 \quad$ will mean that we divide 100 by10, then each will have 10.
$100/0=0 \quad$ will mean that we do not divide 100, and then nobody will have at all and so 0.
Furthermore, she said then the rest is 100; that is, mathematically;
$$
100 = 0\cdot 0 + 100.
$$
Now, all the mathematicians may accept the division by zero $100/0=0$ with natural feelings as a trivial one?
\medskip
For simplicity, we shall consider the numbers on non-negative real numbers. We wish to define the division (or fraction) $b/a$ following the usual procedure for its calculation, however, we have to take care for the division by zero:
The first principle, for example, for $100/2 $ we shall consider it as follows:
$$
100-2-2-2-,...,-2.
$$
How may times can we subtract $2$? At this case, it is 50 times and so, the fraction is $50$.
The second case, for example, for $3/2$ we shall consider it as follows:
$$
3 - 2 = 1
$$
and the rest (remainder) is $1$, and for the rest $1$, we multiple $10$,
then we consider similarly as follows:
$$
10-2-2-2-2-2=0.
$$
Therefore $10/2=5$ and so we define as follows:
$$
\frac{3}{2} =1 + 0.5 = 1.5.
$$
By these procedures, for $a \ne 0$ we can define the fraction $b/a$, usually. Here we do not need the concept of product. Except the zero division, all the results for fractions are valid and accepted.
Now, we shall consider the zero division, for example, $100/0$. Since
$$
100 - 0 = 100,
$$
that is, by the subtraction $100 - 0$, 100 does not decrease, so we can not say we subtract any from $100$. Therefore, the subtract number should be understood as zero; that is,
$$
\frac{100}{0} = 0.
$$
We can understand this: the division by $0$ means that it does not divide $100$ and so, the result is $0$.
Similarly, we can see that
$$
\frac{0}{0} =0.
$$
As a conclusion, we should define the zero divison as, for any $b$
$$
\frac{b}{0} =0.
$$
See \cite{kmsy} for the details.
\medskip
\section{In complex analysis}
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 $w=0$. 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.
However, we shall recall the elementary function
\begin{equation}
W(z) = \exp \frac{1}{z}
\end{equation}
$$
= 1 + \frac{1}{1! z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdot \cdot \cdot .
$$
The function has an essential singularity around the origin. When we consider (1.2), meanwhile, surprisingly enough, we have:
\begin{equation}
W(0) = 1.
\end{equation}
{\bf The point at infinity is not a number} and so we will not be able to consider the function (3.2) at the zero point $z = 0$, meanwhile, we can consider the value $1$ as in (3.3) at the zero point $z = 0$. How do we consider these situations?
In the famous standard textbook on Complex Analysis, L. V. Ahlfors (\cite{ahlfors}) introduced the point at infinity as a number and the Riemann sphere model as well known, however, our interpretation will be suitable as a number. We will not be able to accept the point at infinity as a number.
As a typical result, we can derive the surprising result: {\it At an isolated singular point of an analytic function, it takes a definite value }{\bf with a natural meaning.} As the important applications for this result, the extension formula of functions with analytic parameters may be obtained and singular integrals may be interpretated with the division by zero, naturally (\cite{msty}).
\bigskip
\section{Conclusion}
The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.
The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.
The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.
Should we teach the beautiful fact, widely?:
For the elementary graph of the fundamental function
$$
y = f(x) = \frac{1}{x},
$$
$$
f(0) = 0.
$$
The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).
\medskip
If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.
\bigskip
section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\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}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\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. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 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 operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}
アインシュタインも解決できなかった「ゼロで割る」問題
http://matome.naver.jp/odai/2135710882669605901
私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 237: A reality of the division by zero $z/0=0$ by geometrical optics}
\author{{\it Institute of Reproducing Kernels}\\
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall state a reality of the division by zero $z/0=0$ by the reflection (geometrical optics) and from this fact we will be able to understand that the division by zero $z/0=0$ is natural in both mathematics and our physical world.
\bigskip
\section{Introduction}
%\label{sect1}
By {\bf a natural extension of the fractions}
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising 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 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 some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that {\bf our mathematics says} that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
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
Furthermore, note that Hiroshi Michiwaki with his 6 year old daughter gave the important interpretation of the division by zero $z/0=0$ by the intuitive meaning of the division, {\bf independently of the concept of the product }(see \cite{ann}) . See \cite{ann} for the basic meanings of the division by zero.
We shall state a reality of the division by zero $z/0=0$ by the concept of reflection (geometrical optics). It seems that the common interpretations for the reflections for the center of a circle and the point at infinity are not suitable.
\section{Reflection points}
For simplicity, we shall consider the unit circle ${|z| = 1}$ on the complex $z = x +iy$ plane.
Then, we have the reflection formula
\begin{equation}
z^* = \frac{1}{\overline{z}}
\end{equation}
for any point $z$, as well-known (\cite{ahlfors}). For the reflection point $z^*$, there is no problem for the points
$z \neq 0, \infty$. As the classical result, the reflection of zero is the point at infinity and conversely, for the point at infinity we have the zero point. The reflection is a one to one and onto mapping between the inside and the outside of the unit circle.
However, we wonder the following common facts:
Are these correspondences suitable?
Does there exist the point at $\infty$, really?
Is the point at infinity corresponding to the zero point? Is the point at $\infty$ reasonable from the practical point of view?
Indeed, where can we find the point at infinity? Of course, we know plesantly the point at infinity
on the Riemann sphere, however on the complex $z$-plane it seems that we can not find the corresponding point. When we approach to the origin on a radial line, it seems that the correspondence reflection points approach to {\it the point at infinity} with the direction (on the radial line).
\section{Interpretation by the division by zero $z/0=0$}
On the concept of the division by zero, there is no the point at infinity $\infty$ as the numbers. For any point $z$ such that $|z| >1$, there exists the unique point $z^*$ by (2.1). Meanwhile, for any point $z$ such that $|z| < 1$ except $z=0$, there exits the unique point $z^*$ by (2.1).
Here, note that for $z=0$, by the division by zero, $z^*=0$. Furthermore, we can see that
\begin{equation}
\lim_{z \to 0}z^* =\infty,
\end{equation}
however, for $z=0$ itself, by the division by zero, we have $z^*=0$. This will mean a strong discontinuity of the function
\begin{equation}
W = \frac{1}{z}
\end{equation}
at the origin $z=0$; that is a typical property of the division by zero. This strong discontinuity may be looked in the above reflection property, physically.
\section{Conclusion}
{\Large \bf Should we exclude the point at infinity, from the numbers?} We were able to look the strong discontinuity of the division by zero in the reflection with respect to circles, physically ( geometrical optics ).
The division by zero gives a one to one and onto mapping of the reflection (2.1) from the whole complex plane onto the whole complex plane.
{\Large \bf The infinity $\infty$ may be considered as in (3.1) as the usual sense of limits,} however, the infinity $\infty$ is not a definite number.
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\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}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\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. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\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
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 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 operators on the real and complex fields, Tokyo Journal of Mathematics (in press).
\bibitem{ann}
Announcement 185: Division by zero is clear as z/0=0 and it is fundamental in mathematics,
Institute of Reproducing Kernels, 2014.10.22.
\end{thebibliography}
\end{document}
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