世界最古の釣り針発見＝沖縄のサキタリ洞遺跡－県立博物館など

　

沖縄県南城市のサキタリ洞遺跡で、世界最古となる２万３０００年前の貝製の釣り針が発見された。沖縄の旧石器人がカニを捕獲するなど漁労を行っていたことも判明。沖縄県立博物館・美術館などの研究チームが発表し、論文は１９日（米東部時間）以降、米科学アカデミー紀要電子版に掲載される。
　釣り針は幅１．４センチで、巻き貝を割り、削って作られている。地表から約１メートルの場所で見つかり、周辺の炭の年代から２万３０００年前の物と判明。東ティモールで発見された１万６０００～２万３０００年前の貝製釣り針と並び、最古という。
　研究チームによると、日本国内で旧石器人の漁労の痕跡を示す道具が見つかるのは初めて。（2016/09/20-05:24）http://www.jiji.com/jc/article?k=2016092000060&g=soc

考古学にとても興味があります：

Announcement 213: An interpretation of the identity $ 0.999999...... =1$

カテゴリ：カテゴリ未分類

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\begin{document}

\title{\bf Announcement 213: An interpretation of the identity $ 0.999999...... =1$

}

\author{{\it Institute of Reproducing Kernels}\\

Kawauchi-cho, 5-1648-16,\\

Kiryu 376-0041, Japan\\

E-mail: kbdmm360@yahoo.co.jp\\

}

\date{}

\maketitle

{\bf Abstract: } In this announcement, we shall give a very simple interpretation for the identity: $ 0.999999......=1$.

\bigskip

\section{ Introduction}

On January 8, 2008, Yuusuke Maede, 8 years old boy, asked the question, at Gunma University, that (Announcement 9(2007/9/1): Education for genius boys and girls):

What does it mean by the identity:

$$

0.999999......=1?

$$

at the same time, he said: I am most interesting in the structure of large prime numbers. Then, a teacher answered for the question by the popular reason based on the convergence of the series: $0.9, 0.99, 0.999,... $. Its answer seems to be not suitable for the 8 years old boy with his parents (not mathematicians). Our answer seems to have a general interest, and after then, such our answer has not been heard from many mathematicians, indeed.

This is why writting this announcement.

\medskip

\bigskip

\section{An interpretation}

\medskip

In order to see the essence, we shall consider the simplist case:

\begin{equation}

\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + ... = 1.

\end{equation}

Imagine a tape of one meter length, we will give its half tape: that is,

\begin{equation}

\frac{1}{2}.

\end{equation}

Next, we will give its (the rest's half) half tape; that is, $\frac{1}{2}\cdot \frac{1}{2} = \frac{1}{2^2}$, then you have, altogether

\begin{equation}

\frac{1}{2} + \frac{1}{2^2} .

\end{equation}

Next, we will give the last one's half (the rest's half); that is, $\frac{1}{2}\cdot \frac{1}{2} \cdot \frac{1}{2}= \frac{1}{2^3}$,

then, you have, altogether

\begin{equation}

\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3}.

\end{equation}

By this procedure, you will be able to obtain the small tapes endressly. Imagine all the sum as in the left hand side of (2.1). However, we will see that this sum is just the division of the one meter tape. Therefore, we will be able to confim the identity (2.1), clearly.

The question proposed by Y. Maede is just the small change the ratio $\frac{1}{2}$ by $\frac{9}{10}$.

\bigskip

\section{ Conclusion}

Y. Maede asked the true sense of the limit in the series:

$$

0.999999.....

$$

that is, this series is approaching to 1; however, is it equal or not ? The above interpretation means that the infinite series equals to one and it is just the infinite division of one. By this inverse approarch, the question will make clear.

\medskip

\bigskip

\section{Remarks}

Y. Maede stated a conjecture that for any prime number $p$ $( p \geqq 7)$, for $1$ of $ - 1$

\begin{equation}

11111111111

\end{equation}

may be divided by $p$ (2011.2.6.12:00 at University of Aveiro, by skype)

\medskip

(No.81, May 2012(pdf 432kb)

www.jams.or.jp/kaiho/kaiho-81.pdf).

\medskip

This conjecture was proved by Professors L. Castro and Y. Sawano,

independently. Y. Maede gave later an interesting interpretation for his conjecture.

\medskip

(2015.2.26)

\end{document}

\title{\bf Announcement 214: Surprising mathematical feelings of a 7 years old girl

}

\author{{\it Institute of Reproducing Kernels}\\

Kawauchi-cho, 5-1648-16,\\

Kiryu 376-0041, Japan\\

E-mail: kbdmm360@yahoo.co.jp\\

}

\date{}

\maketitle

{\bf Abstract: } In this announcement, we shall give the two surprising mathematical feelings of 7 years old girl Eko Michiwaki who stated the division by 3 of any angle and the division by zero $100/0=0$ as clear and trivial ones. As well-known, these famous problems are historical, and her results will be quite original.

\bigskip

\section{ Introduction}

We had met, 7 years old girl, Eko Michiwaki on November 23, 2014 at Tokyo Institute of Technology and August 23, 2014 at Kusatu Seminor House, with our colleagues. She, surprisingly enough, stated there repeatedly the division by 3 of any angle and the division by zero $100/0=0$ as clear and trivial ones. As well-known, these famous problems are historical and her results will be quite original.

\section{The division of any angle by 3}

\medskip

Eko Michiwaki said:

divide a given angle with 4 equal angles; this is simly done. Next, we divide one divided angle

with 4 equal angles similarly and the three angles add to other 3 angles. By continuing this procedure, we will be able to obtain the division by 3 of any angle. Her idea may be stated mathematically as follows:

$$

\frac{1}{4} + \frac{1}{4^2} + \frac{1}{4^3} + ... ...= \frac{1}{3}.

$$

However, her idea seems to be more clear than the above mathematical formula. For this sentence, see \cite{ann3} for the sense of the limit.

\bigskip

\section{The division by zero $100/0=0$}

\medskip

As we stated in \cite{ann1}, she stated that division by zero $100/0=0$ is clear and trivial for our recent results \cite{cs,kmsy,s,ttk}. The basic important viewpoint is that division and product are different concepts and the division by zero $100/0=0$ is clear and trivial from the own sense of the division, independently of product \cite{ann1}. From the viewpoint, our colleagues stated as follows:

\medskip

On July 11, 2014, Seiichi Koshiba and Masami Yamane said at

Gunma University:

The idea for the division of Hiroshi Michiwaki and Eko Michiwaki (6 years

old daughter) is that division and product are different concepts and they

were calculated independently for long old years, by repeated addition and

subtraction, respectively. Mathematicians made the serious mistake for very

long years that the division by zero is impossible by considering that division

is the inverse operation of product. The division by zero was, however, clear

and trivial, as z/0=0, from the own nature of division.

\medskip

On February 21, 2015, Seiichi Koshiba and Masami Yamane visited our Institute and we confirmed this meaning of these sentences and the basic idea on the division by zero.

\medskip

(2015.2.27)

\bigskip

\bibliographystyle{plain}

\begin{thebibliography}{10}

\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. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.

\bibitem{s}

S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances inLinear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95.http://www.scirp.org/journal/ALAMT/

\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 (in press).

\bibitem{ann1}

Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics,

Institute of Reproducing Kernels, 2014.10.22.

\bibitem{ann2}

Announcement 185: The importance of the division by zero $z/0=0$, Institute of Reproducing Kernels, 2014.11.28.

\bibitem{ann3}

Announcement 213: An interpretation of the identity $ 0.999999...... =1$, Institute of Reproducing Kernels, 2015.2.26.

\end{thebibliography}

\end{document}