なぜ西暦には0年がないのか/笑う数学
西暦には0年がない。紀元前1年の次の年は、紀元後1年だ。なぜ西暦には0年がないのか？
答えは至ってシンプル。
西暦が使われ始めたのは紀元後500年代。0が西洋に伝わったのは紀元後800年代。
つまり、西暦が使われ始めたときには、まだ0は西洋に存在しなかったのだ！
ちなみに、天文学の世界では、西暦X年における星の位置をYとし、XとYの関係を式で表す。その際に西暦0年がないのは都合が悪いため、紀元後2年→西暦2年、紀元後1年→西暦1年、紀元前1年→西暦0年、紀元前2年→西暦-1年、紀元前3年→西暦-3年としている。
ちなみに、ちなみに、日本の建物は0階がない。地下1階のひとつ上の階は地上1階だ。でも海外では、日本の地上1階の部分を0階と呼んでいるらしい。
え～と、つまり私が何を言いたいかというと、
「まぎらわしいから統一しろ！ 」
［テストに出る度］ 0
［盛り上がり度］ ★★
［社会で役に立つ度］ ★★
（つづく）
タカタ先生（日本お笑い数学協会）
【著者紹介】日本お笑い数学協会
協会メンバー、タカタ先生、横山明日希、さんきゅう倉田、秋田崇宏、平井基之、小林裕人、鯵坂もっちょ。理系企画プロデューサー、現役高校教師兼お笑い芸人、受験戦略家、予備校講師、元国税局職員のお笑い芸人などからなる、数学ユニット。お笑い×数学のイベントや、出張授業を多数行い、数学のおもしろさ&すばらしさを伝えている。「日本お笑い数学協会（JOMA）」Twitterはこちら
【書き手】タカタ先生
日本お笑い数学協会会長。日本お笑い数学協会（通称「JOMA」）を立ち上げ、同日開催された第1回JOMAじゃんけん大会で勝利し会長となる。高校の数学教師、よしもと芸人、数学系YouTuberの4つの顔を持ち、大人から子供まで幅広い層に数学の楽しさを広める活動を行っている。Twitterアカウントは@takatasennsei
【書籍紹介】
『笑う数学』（KADOKAWA）
数々の数学イベントで活躍中の日本お笑い数学協会のメンバーが書き下ろした、とっておきの数学の話100。バカらしいことをマジメに数学的に考察したもの、美しい数字の世界、あんなに苦労して覚えた定理を使わずに解を導き出す方法など、多種多様な数学の話を1冊にまとめました。https://news.walkerplus.com/article/159723/
協会メンバー、タカタ先生、横山明日希、さんきゅう倉田、秋田崇宏、平井基之、小林裕人、鯵坂もっちょ。理系企画プロデューサー、現役高校教師兼お笑い芸人、受験戦略家、予備校講師、元国税局職員のお笑い芸人などからなる、数学ユニット。お笑い×数学のイベントや、出張授業を多数行い、数学のおもしろさ&すばらしさを伝えている。「日本お笑い数学協会（JOMA）」Twitterはこちら
【書き手】タカタ先生
日本お笑い数学協会会長。日本お笑い数学協会（通称「JOMA」）を立ち上げ、同日開催された第1回JOMAじゃんけん大会で勝利し会長となる。高校の数学教師、よしもと芸人、数学系YouTuberの4つの顔を持ち、大人から子供まで幅広い層に数学の楽しさを広める活動を行っている。Twitterアカウントは@takatasennsei
【書籍紹介】
『笑う数学』（KADOKAWA）
数々の数学イベントで活躍中の日本お笑い数学協会のメンバーが書き下ろした、とっておきの数学の話100。バカらしいことをマジメに数学的に考察したもの、美しい数字の世界、あんなに苦労して覚えた定理を使わずに解を導き出す方法など、多種多様な数学の話を1冊にまとめました。https://news.walkerplus.com/article/159723/
ゼロ除算の発見は日本です：
∞？？？
∞は定まった数ではない・
人工知能はゼロ除算ができるでしょうか：
とても興味深く読みました：2014年2月2日 4周年を超えました：
ゼロ除算の発見と重要性を指摘した：日本、再生核研究所
ゼロ除算関係論文・本
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\begin{document}
\title{\bf Announcement 448:\\ Division by Zero;\\
Funny History and New World}
\author{再生核研究所}
\date{2018.08.20}
\maketitle
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{\bf Abstract: } Our division by zero research group wonder why our elementary results may still not be accepted by some wide world and very recently in our Announcements: 434 (2018.7.28),
437 (2018.7.30),
438(2018.8.6), \\
441(2018.8.9),
442(2018.8.10),
443(2018.8.11),
444(2018.8.14),
in Japanese, we stated their reasons and the importance of our elementary results. Here, we would like to state their essences. As some essential reasons, we found fundamental misunderstandings on the division by zero and so we would like to state the essences and the importance of our new results to human beings over mathematics.
We hope that:
close the mysterious and long history of division by zero that may be considered as a symbol of the stupidity of the human race and open the new world since Aristotle-Eulcid.
From the funny history of the division by zero, we will be able to realize that
human beings are full of prejudice and prejudice, and are narrow-minded, essentially.
\medskip
\section{Division by zero}
The division by zero with mysterious and long history was indeed trivial and clear as in the followings:
\medskip
By the concept of the Moore-Penrose generalized solution of the fundamental equation $az=b$, the division by zero was trivial and clear as $b/0=0$ in the {\bf generalized fraction} that is defined by the generalized solution of the equation $az=b$.
Note, in particular, that there exists a uniquely determined solution for any case of the equation $az=b$ containing the case $a=0$.
People, of course, consider as the division $b/a$ that it is the solution of the equation $ az =b$ and if $a=0$ then $0 \cdot z =0$ and so, for $b\ne0$ we can not consider the fraction $a/b$. We have been considered that the division by zero $b/0$ is impossible for mysteriously long years, since the document of zero in India in AD 628. In particular, note that Brahmagupta (598 -668 ?) established 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 right and suitable. However, he did not give its reason and did not consider the importance case $1/0$ and the general fractions $b/0$. The division by zero was a symbol for {\bf impossibility} or to consider the division by zero was {\bf not permitted}. For this simple and clear conclusion, we did not definitely consider more on the division by zero. However, we see many and many formulas appearing the zero in denominators, one simple and typical example is in the function $w=1/z$ for $z=0$.
We did not consider the function at the origin $z=0$.
In this case, however, the serious interest happens in many physical problems and also in computer sciences, as we know.
When we can not find the solution of the fundamental equation $az=b$, it is fairly clear to consider the Moore-Penrose generalized solution in mathematics. Its basic idea and beautiful mathematics will be definite.
Therefore, we should consider the generalized fractions following the Moore-Penrose generalized inverse. Therefore, with its meaning and definition we should consider that $b/0=0$.
It will be very curious that we know very well the Moore-Penrose generalized inverse as a very fundamental and important concept, however, we did not consider the simplest case $ az =b$.
Its reason may be considered as follows: We will consider or imagine that the fraction $1/0$ may be like infinity or ideal one.
For the fundamental function $W =1/ z $ we did not consider any value at the origin $z = 0$. Many and many people consider its value by the limiting like $+\infty $ and $- \infty$ or the
point at infinity as $\infty$. However, their basic idea comes from {\bf continuity} with the common sense or
based on the basic idea of Aristotle. --
For the related Greece philosophy, see \cite{a,b,c}. However, as the division by zero we have to consider its value of
the function $W =1 /z$ as zero at $z = 0$. We will see that this new definition is valid widely in
mathematics and mathematical sciences, see (\cite{mos,osm}) for example. Therefore, the division by zero will give great impacts to calculus, Euclidian geometry, analytic geometry, complex analysis and the theory of differential equations in an undergraduate level and furthermore to our basic ideas for the space and universe.
For the extended complex plane, we consider its stereographic projection mapping as the Riemann sphere and the point at infinity is realized as the north pole in the Alexsandroff's one point compactification.
The Riemann sphere model gives a beautiful and complete realization of the extended complex plane through the stereographic projection mapping and the mapping has beautiful properties like isogonal (equiangular) and circle to circle correspondence (circle transformation). Therefore, the Riemann sphere is a very classical concept \cite{ahlfors}.
\medskip
Now, with the division by zero we have to admit the strong discontinuity at the point at infinity. To accept this strong discontinuity seems to be very difficult, and therefore we showed many and many examples for giving the evidences over $800$ items.
\medskip
We back to our general fractions $1/0=0/0=z/0=0$ for its importances.
\medskip
H. Michiwaki and his 6 years old daughter Eko Michiwaki stated that in about three weeks after the discovery of the division by zero that
division by zero is trivial and clear from the concept of repeated subtraction and they showed the detailed interpretation of the general fractions. Their method is a basic one and it will give a good introduction of division and their calculation method of divisions.
We can say that division by zero, say $100/0$ means that we do not divide $100$ and so the number of the divided ones is zero.
\medskip
Furthermore,
recall the uniqueness theorem by S. Takahasi on the division by zero:
\medskip
{\bf Proposition 1.1 }{\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.
In the long mysterious history of the division by zero, this proposition seems to be decisive.
Indeed, Takahasi's assumption for the product property should be accepted for any generalization of fraction (division). Without the product property, we will not be able to consider any reasonable fraction (division).
Following Proposition 1.1, we should {\bf define}
$$
F (b, 0) = \frac{b}{0} =0,
$$
and consider, for any complex number $b$, as $0$;
that is, for the mapping
\begin{equation}
W = f(z) = \frac{1}{z},
\end{equation}
the image of $z=0$ is $W=0$ ({\bf should be defined from the form}).
\medskip
Furthermore,
the simple field structure containing division by zero was established by M. Yamada.
\medskip
In addition, for the fundamental function $f(z) = 1/z$, note that
the function is odd function
$$
f(z) = - f(-z)
$$
and if the function may be extended as an odd function at the origin $z=0$, then the identity $f(0) = 1/0 =0$ has to be satisfied. Further, if the equation
$$
\frac{1}{z} =0
$$
has a solution, then the solution has to be $z=0$.
\medskip
\section{Division by zero calculus}
As the number system containing the division by zero, the Yamada field structure is complete.
However, for applications of the division by zero to {\bf functions}, we need the concept of the division by zero calculus for the sake of uniquely determinations of the results and for other reasons.
For example, for the typical linear mapping
\begin{equation}
W = \frac{z - i}{z + i},
\end{equation}
it gives a conformal mapping on $\{{\bf C} \setminus \{-i\}\}$ onto $\{{\bf C} \setminus \{1\}\}$ in one to one and from \begin{equation}
W = 1 + \frac{-2i}{ z - (-i)},
\end{equation}
we see that $-i$ corresponds to $1$ and so the function maps the whole $\{{\bf C} \}$ onto $\{{\bf C} \}$ in one to one.
Meanwhile, note that for
\begin{equation}
W = (z - i) \cdot \frac{1}{z + i},
\end{equation}
if we enter $z= -i$ in the way
\begin{equation}
[(z - i)]_{z =-i} \cdot \left[ \frac{1}{z + i}\right]_{z =-i} = (-2i) \cdot 0= 0,
\end{equation}
we have another value.
\medskip
In many cases, the above two results will have practical meanings and so, we will need to consider many ways for the application of the division by zero and we will need to check the results obtained, in some practical viewpoints. We referred to this delicate problem with many examples.
Therefore, we will introduce the division by zero calculus that give important values for functions. For any Laurent expansion around $z=a$,
\begin{equation}
f(z) = \sum_{n=-\infty}^{-1} C_n (z - a)^n + C_0 + \sum_{n=1}^{\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}
Note that here, there is no problem on any convergence of the expansion (2.5) at the point $z = a$, because all the terms $(z - a)^n$ are zero at $z=a$ for $n \ne 0$.
\medskip
For the correspondence (2.6) for the function $f(z)$, we will call it {\bf the division by zero calculus}. By considering the formal derivatives in (2.5), we {\bf can define any order derivatives of the function} $f$ at the singular point $a$; that is,
$$
f^{(n)}(a) = n! C_n.
$$
\medskip
{\bf Apart from the motivation, we define the division by zero calculus by (2.6).}
With this assumption, we can obtain many new results and new ideas. However, for this assumption we have to check the results obtained whether they are reasonable or not. By this idea, we can avoid any logical problems. -- In this point, the division by zero calculus may be considered as an axiom.
\medskip
This paragraph is very important. Our division by zero is just definition and the division by zero is an assumption. Only with the assumption and definition of the division by zero calculus, we can create and enjoy our new mathematics. Therefore, the division by zero calculus may be considered as a new axiom.
Of course, its strong motivations were given. We did not consider any value {\bf at the singular point} $a$ for the Laurent expansion (2.5). Therefore, our division by zero is a new mathematics entirely and isolated singular points are a new world for our mathematics.
We had been considered properties of analytic functions {\bf around their isolated singular points.}
The typical example of the division zero calculus is $\tan (\pi/2) = 0$ and the result gives great impacts to analysis and geometry.
See the references for the materials.
\medskip
For an identity, when we multiply zero, we obtain the zero identity that is a trivial.
We will consider the division by zero to an equation.
For example, for the simple example for the line equation on the $x, y$ plane
$$
ax + by + c=0
$$
we have, formally
$$
x + \frac{by + c}{a} =0,
$$
and so, by the division by zero, we have, for $a=0$, the reasonable result
$$
x = 0.
$$
However, from
$$
\frac{ax + by}{c} + 1 =0,
$$
for $c=0$, we have the contradiction, by the division by zero
$$
1 =0.
$$
For this case, we can consider that
$$
\frac{ax + by}{c} + \frac{c}{c} =0,
$$
that is always valid. {\bf In this sense, we can divide an equation by zero.}
\section{Conclusion}
Apparently, the common sense on the division by zero with a long and mysterious history is wrong and our basic idea on the space around the point at infinity is also wrong since Euclid. On the gradient or on derivatives we have a great missing since $\tan (\pi/2) = 0$. Our mathematics is also wrong in elementary mathematics on the division by zero.
We have to arrange globally our modern mathematics with our division by zero in our undergraduate level.
We have to change our basic ideas for our space and world.
We have to change globally our textbooks and scientific books on the division by zero.
From the mysterious history of the division by zero, we will be able to study what are human beings and about our narrow-minded.
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T. Matsuura and S. Saitoh,
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$\log 0= \log \infty =0$ and applications. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics. {\bf 230} (2018), 293-305.
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H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 (2017.11.14).
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Harmonic Mean and Division by Zero,
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２０１８．９．17. 展示書籍などを拝見させて頂きました。大変賑わっていて関心の大きさが感じられました。時間の関係で じっくり、詳しくとは行きませんでしたが、全体の案内（知の連鎖ゾーン）で、初期、初めにアリストテレスとユークリッドが 在って、中間くらいにニュートン、最後がアインシュタインで 世界史を総攬する想いがしました。 数学では 非ユークリッド幾何学の扱いにおけるガウスの記述、資料の欠落と算術の発見、ゼロの発見の Brahmagupta (598 -668 ?) の欠落は 残念に思われました。書籍など無くても大事な事実と思いますので、 大きく取り上げて欲しかった。 この世界史年表で凄いことに気づいて興奮して後にしました。
ゼロ除算がこれらで基本的な関与があるからです。
まず、ゼロ除算は、ユークリッド幾何学の変更を求め、連続性のアリストテレスの世界観に反して、強力な不連続性の世界を示しているからです。 アインシュタインは ゼロ除算が人生の最大の関心事で、今でもなお、ゼロ除算とアインシュタインの相対性理論との関係が議論され、ブラックホールは 神がゼロで割ったところに存在するなどと 神秘的な問題を提供しているからです。
もちろん、Brahmaguptaは ゼロ除算を議論していて、その後、１３００年に亘って、世界史で議論されてきて、 ニユートン力学でも基本的な問題を提起している。 当然、非ユークリッド幾何学とも関係していて、それらの空間とも違う新しい幾何学を提案している。 このように考えると、検討中の Division by Zero Calculus の著書（出版契約済み）は 世界史上で大きな扱いになるだろうと発想して、大変興奮して、展示会を後にしました。
広く世界に意見を求め、この著書の出版計画を進めたい。 途中経過も公表して行きたい。
最後に素晴らしい展示会を企画され、そのために努力された人たちに 感謝の気持ちを表明したい。
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