インド人にビックリ!? 実はすごいことだった「ゼロ」の発見
世界史の教科書に出てくる「0（ゼロ）の発見」。日常で当たり前のように使っている「ゼロ」という概念ですが、教科書に載るほどの大発見なのでしょうか。今回の無料メルマガ『１日１粒！「幸せのタネ」』では著者の須田將昭さんが、思わず納得の理由をわかりやすく解説しています。
ゼロの発見
日本史や世界史の教科書を読んでいると、「ふぅ～ん、それがどうしたの？」と思うことはありませんか？ その中の一つが、「インドでのゼロの発見」。「ゼロって発見するものなの？」と思いませんでしたか？
今の私たちにはごくごく当たり前の概念ですが、「ゼロ」というものを「表す」というのは、実は実は結構大きなことなのです。
そもそも、数字は何のために必要となったのでしょう？
例えば、「今日、馬を一頭売った。もともと三頭持ってたから、今は、二頭あればいい」ということで
- III か 三
- I か 一
- II か 二
と、縦棒でも横棒でもいいのですが、印になるようなことを書いておく。そんなところから始まったと考えられます。
漢数字では3までは横棒で表しましたが、4以上は煩雑なので字が変わっていきます。ローマ数字も同じで、縦棒を順次増やしていきます。4は「5の一つ手前」ということで、5を表すVの前にIをつけてIVとしますが、それでも基本は順番に印を増やしていく方法です。
そして、それは「1対1」の対応です。馬一頭と縦棒か横棒が一本。馬が手元になければ表す必要がありません。だから「0」はないのです。
ローマ数字に「0」はないですよね？ メソポタミアのくさび形文字にも数字がありますが、これにも0はありません。だから10を超えて、100、1,000など「位」を表すためにそれぞれの数字が必要となります。
インドで「ここには何もない」ということで、「0」が発明されたことで、「1,025」というような「百の位に数字はない」ということが簡単に表されるようになり、それが計算を飛躍的に楽にしてくれました。どんな数でも0～9の10種類の数字で表せるのです。
これは本当に画期的なことだからこそ、歴史の教科書でも「ゼロの発見」というのが記述されるのです。
歴史の教科書に出てくるさりげない用語の一つにも、非常に大きな物語が隠れてるんですね。歴史の教科書にしても、授業にしても、紙幅と時間に限りがあるので、どうしても通り一遍で済ませざるをを得ません。
本当はすごく面白い、ワクワクするような話なのに、それに時間が割けない…、ということは、きっと歴史の先生も感じてらっしゃるんだろうなと思います。
そういうことをひっくり返して探してみると、いろんな発見がありそうです。気になっていた用語があったら、一度、それを深く調べてみませんか？
image by: Shutterstock.com
とても興味深く読みました：
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 380: What is the zero?\\
(2017.8.21)}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
}
\date{\today}
\maketitle
\section{What is the zero?}
The zero $0$ as the complex number or real number is given clearly by the axions by the complex number field and real number field.
For this fundamental idea, we should consider the {\bf Yamada field} containing the division by zero. The Yamada field and the division by zero calculus will arrange our mathematics, beautifully and completely; this will be our natural and complete mathematics.
\medskip
\section{ Double natures of the zero $z=0$}
The zero point $z=0$ represents the double natures; one is the origin at the starting point and another one is a representation of the point at infinity. One typical and simple example is given by $e^0 = 1,0$, two values. {\bf God loves two}.
\section{Standard value}
\medskip
The zero is a center and stand point (or bases, a standard value) of the coordinates - here we will consider our situation on the complex or real 2 dimensional spaces. By stereographic
projection mapping or the Yamada field, the point at infinity $1/0$ is represented by zero. The origin of the coordinates and the point at infinity correspond each other.
As the standard value, for the point $\omega_n = \exp \left(\frac{\pi}{n}i\right)$ on the unit circle $|z|=1$ on the complex $z$-plane is, for $n = 0$:
\begin{equation}
\omega_0 = \exp \left(\frac{\pi}{0}i\right)=1, \quad \frac{\pi}{0} =0.
\end{equation}
For the mean value
$$
M_n = \frac{x_1 + x_2 +... + x_n}{n},
$$
we have
$$
M_0 = 0 = \frac{0}{0}.
$$
\medskip
\section{ Fruitful world}
\medskip
For example, for very and very general partial differential equations, if the coefficients or terms are zero, then we have some simple differential equations and the extreme case is all the terms are zero; that is, we have trivial equations $0=0$; then its solution is zero. When we consider the converse, we see that the zero world is a fruitful one and it means some vanishing world. Recall Yamane phenomena (\cite{kmsy}), the vanishing result is very simple zero, however, it is the result from some fruitful world. Sometimes, zero means void or nothing world, however, it will show {\bf some changes} as in the Yamane phenomena.
\section{From $0$ to $0$; $0$ means all and all are $0$}
\medskip
As we see from our life figure (\cite{osm}), a story starts from the zero and ends with the zero. This will mean that $0$ means all and all are $0$. The zero is a {\bf mother} or an {\bf origin} of all.
\medskip
\section{ Impossibility}
\medskip
As the solution of the simplest equation
\begin{equation}
ax =b
\end{equation}
we have $x=0$ for $a=0, b\ne 0$ as the standard value, or the Moore-Penrose generalized inverse. This will mean in a sense, the solution does not exist; to solve the equation (6.1) is impossible.
We saw for different parallel lines or different parallel planes, their common points are the origin. Certainly they have the common points of the point at infinity and the point at infinity is represented by zero. However, we can understand also that they have no solutions, no common points, because the point at infinity is an ideal point.
Of course. we can consider the equation (6.1) even the case $a=b=0$ and then we have the solution $x=0$ as we stated.
We will consider the simple differential equation
\begin{equation}
m\frac{d^2x}{dt^2} =0, m\frac{d^2y}{dt^2} =-mg
\end{equation}
with the initial conditions, at $t =0$
\begin{equation}
\frac{dx}{dt} = v_0 \cos \alpha , \frac{d^2x}{dt^2} = \frac{d^2y}{dt^2}=0.
\end{equation}
Then, the highest high $h$, arriving time $t$, the distance $d$ from the starting point at the origin to the point $y(2t) =0$ are given by
\begin{equation}
h = \frac{v_0 \sin^2 \alpha}{2g}, d= \frac{v_0\sin \alpha}{g}
\end{equation}
and
\begin{equation}
t= \frac{v_0 \sin \alpha}{g}.
\end{equation}
For the case $g=0$, we have $h=d =t=0$. We considered the case that they are the infinity; however, our mathematics means zero, which shows impossibility.
These phenomena were looked many cases on the universe; it seems that {\bf God does not like the infinity}.
\bibliographystyle{plain}
\begin{thebibliography}{10}
\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. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{ms}
T. Matsuura and S. Saitoh,
Matrices and division by zero $z/0=0$, Advances in Linear Algebra
\& Matrix Theory, 6 (2016), 51-58. http://dx.doi.org/10.4236/alamt.2016.62007 http://www.scirp.org/journal/alamt
\bibitem{mos}
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, 2017), 1-16.
\bibitem{osm}
H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,
Journal of Technology and Social Science (JTSS), 1(2017), 70-77.
\bibitem{romig}
H. G. Romig, Discussions: Early History of Division by Zero,
American Mathematical Monthly, Vol. 31, No. 8. (Oct., 1924), pp. 387-389.
\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{s16}
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, {\bf 177}(2016), 151-182 (Springer).
\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.
\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.
\bibitem{ann326}
Announcement 326 (2016.10.17): The division by zero z/0=0 - its impact to human beings through education and research.
\bibitem{ann352}
Announcement 352(2017.2.2): On the third birthday of the division by zero z/0=0.
\bibitem{ann354}
Announcement 354(2017.2.8): What are $n = 2,1,0$ regular polygons inscribed in a disc? -- relations of $0$ and infinity.
\bibitem{362}
Announcement 362(2017.5.5): Discovery of the division by zero as
$0/0=1/0=z/0=0$.
\end{thebibliography}
\end{document}
The division by zero is uniquely and reasonably determined as 1/0=0/0=z/0=0 in the natural extensions of fractions. We have to change our basic ideas for our space and world
Division by Zero z/0 = 0 in Euclidean Spaces
Hiroshi Michiwaki, Hiroshi Okumura and Saburou Saitoh
International Journal of Mathematics and Computation Vol. 28(2017); Issue 1, 2017), 1
-16.
http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2016.62007
http://www.ijapm.org/show-63-504-1.html
http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf
http://okmr.yamatoblog.net/division%20by%20zero/announcement%20326-%20the%20divi
http://okmr.yamatoblog.net/
Relations of 0 and infinity
Hiroshi Okumura, Saburou Saitoh and Tsutomu Matsuura：
http://www.e-jikei.org/…/Camera%20ready%20manuscript_JTSS_A…
https://sites.google.com/site/sandrapinelas/icddea-2017
2017.8.21.06:37
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