「0(ゼロ)」をインド人が発明した時期が定説よりも前だと放射性炭素年代測定で判明

「数学史上、最大のブレークスルー」だと言われ、その意義が高く評価される「0(ゼロ)」の発明に関して、インド人が0(ゼロ)を発明した年代が定説よりも数百年も早かったことが明らかになりました。

History of zero pushed back 500 years by ancient Indian text | New Scientist
https://www.newscientist.com/article/2147450-history-of-zero-pushed-back-500-years-by-ancient-indian-text/

Carbon dating reveals earliest origins of zero symbol - BBC News
http://www.bbc.com/news/uk-england-oxfordshire-41265057

何もない状態を表すだけでなく、空位を表す記号としても極めて重要な「0(ゼロ)」の価値については、以下の記事を見ればよくわかります。

「無」を形ある物で表現するという「0(ゼロ)」の概念はどのように誕生し、人類の発展に大きく貢献したのか - GIGAZINE

これまで、インドで0(ゼロ)の概念が確立されたのは早くても5世紀ごろだと考えられてきました。最初に0(ゼロ)の表記が登場するのは、数学者ブラーマグプタによる著書「ブラーマ・スプタ・シッダーンタ」だと言われています。当時は「0」ではなく「●」として表記されていましたが、その後、●の中に空白の円が描かれて現在の数字の「0」という表記になりました。

今回、0(ゼロ)の起源が定説よりもさかのぼることになったのは、「Bakhshali」という古代インドの仏教の修道士向けトレーニングマニュアルが原因です。このマニュアルは1881年にパキスタンの農家で発見され、1902年以来、オックスフォード大学のボドリアン図書館に収蔵されてきました。

Bakhshaliの中にある以下のページには0(ゼロ)を意味する「●」が記述されています。

これまでの研究ではBakhshaliは8世紀から12世紀に作られたと考えられてきましたが、書物が70種類もの異なる樹皮のページから構成されているため正確な年代を特定するのが困難だったとのこと。そこで、放射性炭素年代測定によってBakhshaliが誕生した年代を正確に調べたところ、なんと最も古い樹皮は224年から383年の間のものと判明。これまでの通説よりも約500年も早く0(ゼロ)が発明されていたことが明らかになりました。

なお、放射性炭素年代測定の結果、Bakhshaliは最も古いページと最も新しいページとの間で500年の隔たりがあることも判明しており、すべてのページがどのようにして集められ編さんされたのかについて新たに謎が残されたそうです。http://gigazine.net/news/20170919-zero-symbol-origin/

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…