3 matemáticos importantes: Diofanto de Alejandría, Brahmagupta y Al-Khwarizmi En esta oportunidad les contaré la historia de: Diofanto de Alejandría, Brahmagupta y Al-Khwarizmi.

3 matemáticos importantes: Diofanto de Alejandría, Brahmagupta y Al-Khwarizmi

En esta oportunidad les contaré la historia de: Diofanto de Alejandría, Brahmagupta y Al-Khwarizmi.

 

Diofanto de Alejandría
Matemático griego que floreció en Alejandría alrededor del año 275. Es sin duda el más grande algebrista griego: Nada se conoce de su vida, pero sí que han llegado a nuestras manos gran cantidad de trabajos. Resolvió problemas con ecuaciones algebraicas e inventó una fórmula particular. Su principal obra es la Arithmíteca, dedicada casi exclusivamente a la resolución exacta de ecuaciones determinadas e indeterminadas de forma que la rama de análisis que se dedica a esta tarea, se conoce hoy en día como análisis diofántico.

La solución en números enteros o racionales, de ecuaciones indeterminadas pertenece al análisis diofántico. Este nombre honra a Diofánto cuyo tratado de trece libros, de los cuales solamente sobreviven seis, fue el primero sobre el tema. La traducción latina (1621) de este fragmento sugestivo inspiró directamente a Fermat para que creara la moderna aritmética superior.

Brahmagupta (598 - 660)
Astrónomo y matemático indio. Es el mayor matemático de la antigua civilización india. Desarrolló su actividad en el noroeste de la India y resumió sus conocimientos astronómicos en un libro escrito en el año 628, en el que rechazaba la rotación de la tierra. El rasgo más importante de esta obra es la aplicación de métodos algebraicos a los problemas astronómicos. Los matemáticos indios rindieron un gran servicio al mundo, ya que alguno de ellos, posiblemente Brahmagupta ideó el concepto y el símbolo "cero".

Al-Khwarizmi (780 - 850)
A la Edad Media del mundo occidental corresponde la Edad de Oro del mundo musulmán que, en el año 700 al 1200, se extendió desde la India hasta España. Durante esa época, el árabe fue la lengua internacional de las matemáticas. Los matemáticos árabes conservaron el patrimonio matemático de los griegos, divulgaron los conocimientos matemáticos de la India, asimilaron ambas culturas e hicieron avanzar tanto el álgebra como la trigonometría.

El más recordado de los matemáticos árabes de esa época es Mohammed ibn Musa Al-Khwarizmi, también llamado el "Padre del Álgebra". Se sabe muy poco de su vida, solo que vivió en la primera mitad del siglo XIX y que trabajó en la biblioteca del califa Al-Mahmum, en Bagdad. Escribió varios libros de geografía, astronomía y matemática; dos de sus libros de matemáticas dejaron una huella imborrable en la historia de esta ciencia; uno de ellos viene la palabra "algoritmo" de otro la palabra "Álgebra".

En su obra Aritmética, explicó con detalle y claridad el funcionamiento del sistema decimal de numeración y del cero que usaban en la India (de ahí viene probablemente la creencia de que nuestro sistema de numeración es de origen árabe). La nueva notación se conocía en Europa como la de Al-Khwarizmi, pronunciado "algorismi", de donde después derivaron las palabras "guarismo" para indicar las cifras de un número y "algoritmo" para hablar de un proceso matemático que se repite o de una regla de cálculo.

En otro de sus libros, Al-jabr significa "restauración" del equilibrio mediante la trasposición de términos de una ecuación; "muqäbala" significa la simplificación de la expresión resultante mediante la cancelación de términos semejantes de cada lado de la ecuación.https://matessimples.blogspot.com/2017/09/matematicos-importantes-diofanto-de.html

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：

ゼロ除算の発見と重要性を指摘した：日本、再生核研究所：2014年2月2日

ゼロ除算関係論文・本

https://ameblo.jp/syoshinoris/entry-12370797278.html

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\title{\bf Announcement 433:\\ Puha's Horn Torus Model for the Riemann Sphere From the Viewpoint of  Division by Zero}

\author{

}

\date{2018.07.16}

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{\bf Abstract: }  In this announcement, we will introduce a beautiful horn torus model for the Riemann sphere in complex analysis from the viewpoint of the division by zero based on \cite{ps}.

\medskip

\section{Division by zero calculus and introduction}

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 $ax=b$, the division by zero was trivial and clear all as $a/0=0$ in the {\bf generalized fraction} that is defined by the generalized solution of the equation $ax=b$.

Division by zero is trivial and clear from the concept of repeated subtraction  - H. Michiwaki.

Recall the uniqueness theorem by S. Takahasi on the division by zero.

The simple field structure containing division by zero was established by M. Yamada.

Many applications of the division by zero to Wasan geometry were given by H. Okumura.

\medskip

The division by zero opens a new world  since Aristotelēs-Euclid.

See the references for recent related results.

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}

we should not 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}

\medskip

However, 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.  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 (1.5) 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 (1.6) for the function $f(z)$, we will call it {\bf the division by zero calculus}. By considering the formal derivatives in (1.5), 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 (1.6).}

 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

For the fundamental function $W =1/ z $ we did not consider any value at the origin $z = 0$, because we did not consider the division by zero

$1/ 0$ in a good way. 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 will 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}.

Now, with the division by zero we have to admit the strong discontinuity at the point at infinity.

On this situation, V. Puha discovered the mapping of the extended complex plane to a beautiful horn torus at (2018.6.4.7:22) and its inverse at (2018.6.18.22:18).

Incidentally, independently of the division by zero,  Wolfgang W. Daeumler has various special great ideas on horn torus as we see from his site:

\medskip

Horn Torus \& Physics ( https://www.horntorus.com/ ) 'Geometry Of Everything', intellectual game to reveal

engrams of dimensional thinking and proposal for a different approach to physical questions ...

\medskip

Indeed, Wolfgang Daeumler was presumably the first (1996) who came to the idea of the possibility of a mapping onto the horn torus. He expressed the idea of that on his private website (http://www.dorntorus.de). He was also, apparently, the first who to point out that zero and infinity are represented by one and the same point on the horn torus model of expanded complex plane.

\medskip

In this announcement, we will introduce simply the new horn torus model for the classical Riemann sphere from the viewpoint of the division by zero.

\section{Horn torus model}

 We will consider the three circles stated by

$$

\xi^2  + \left(\zeta-\frac{1}{2}\right)^2 = \left(\frac{1}{2}\right)^2,

$$

$$

\left(\xi-\frac{1}{4}\right)^2  + \left(\zeta-\frac{1}{2}\right)^2 = \left(\frac{1}{4}\right)^2,

$$

and

$$

\left(\xi+\frac{1}{4}\right)^2  + \left(\zeta-\frac{1}{2}\right)^2 = \left(\frac{1}{4}\right)^2.

$$

By rotation on the space $(\xi,\eta,\zeta)$ on the $(x,y)$ plane as in $\xi =x, \eta=y$ around $\zeta$ axis, we will consider the  sphere with $1/2$ radius as the Riemann sphere and the horn torus made in the sphere.

The stereographic projection mapping from $(x,y)$ plane to the Riemann sphere is given by

$$

\xi = \frac{x}{x^2 + y^2 + 1},

$$

$$

\eta = \frac{y}{x^2 + y^2 + 1},

$$

and

$$

\zeta = \frac{x^2 + y^2}{x^2 + y^2 + 1}.

$$

The mapping from $(x,y)$ plane to the horn torus by Puha is given by

$$

\xi = \frac{2x\sqrt{x^2 + y^2}}{(x^2 + y^2 + 1)^2},

$$

$$

\eta = \frac{2y\sqrt{x^2 + y^2}}{(x^2 + y^2 + 1)^2},

$$

and

$$

\zeta = \frac{(x^2 + y^2 -1)\sqrt{x^2 + y^2}}{(x^2 + y^2 + 1)^2} + \frac{1}{2}.

$$

The inversion is given by

$$

x = \xi \left(\xi^2 + \eta^2 + \left(\zeta - \frac{1}{2} \right)^2 -\zeta + \frac{1}{2} \right)^{(-1/2)}

$$

and

$$

y = \eta \left(\xi^2 + \eta^2 + \left(\zeta - \frac{1}{2} \right)^2 -\zeta + \frac{1}{2} \right)^{(-1/2)}.

$$

\section{Properties of horn torus model}

At first, the model shows the strong symmetry of the domains $\{|z|<1\}$ and  $\{|z|>1\}$ and they correspond to the lower part and the upper part of the horn torus, respectively. The unit circle $\{|z|=1\}$ corresponds to the circle

$$

\xi^2 + \eta^2 = \left(\frac{1}{2}\right)^2, \quad \zeta = \frac{1}{2}

$$

in one to one way. Of course, the origin and the point at infinity are the same point and correspond to $(0,0,1/2)$. Furthermore,

the inversion relation

$$

z \longleftrightarrow \frac{1}{\overline{z}}

$$

with respect to the unit circle $\{|z|=1\}$ corresponds to the relation

$$

(\xi,\eta,\zeta) \longleftrightarrow (\xi,\eta, 1-\zeta)

$$

and similarly,

$$

z \longleftrightarrow -z

$$

 corresponds to the relation

$$

(\xi,\eta,\zeta) \longleftrightarrow (- \xi,-\eta, \zeta)

$$

and

$$

z \longleftrightarrow - \frac{1}{\overline{z}}

$$

 corresponds to the relation

$$

(\xi,\eta,\zeta) \longleftrightarrow (-\xi,-\eta, 1-\zeta)

$$

(H.G.W. Begehr: 2018.6.18.19:20).

Furthermore, we can see directly the important properties that the mapping is isogonal (equiangular) and infinitely small circles correspond

 to infinitely small circles, as in analytic functions. However, of course, circles to circles mapping property is, in general, not valid as in the case of the stereographic projection mapping.

Horn torus, in contrast to the Riemann sphere, does not satisfy the definition of simply connected space because a closed nonzero path passing through the point $(0,0,1/2)$ can not be continuously shrinked to the point. In particular, note that a curve can pass the point $(0,0,1/2)$ on the horn torus.

We note  that only zero and numbers of the form $|a|=1$ have the property : $ |a|^b=|a|, b\ne 0.$

Here, note that we can also consider  $0^b =0$ (\cite{mms18}). The symmetry of the horn torus model agrees perfectly with this fact. Only zero and numbers of the form $|a|=1$ correspond to points  on the plane  described by equation $\zeta -1/2=0$.  Only zero and numbers of the form $|a| =1$ correspond to points whose tangent lines to the surface of the horn torus are parallel to the axis $\zeta$.

\section{Conclusion}

The division by zero shows the strong discontinuity at the point at infinity, however, the Riemann sphere model and stereographic projection mapping are fundamental and beautiful.

Many people feel  strange feelings for the strong discontinuity that is introduced by the division by zero to the Riemann sphere, however, the strong discontinuity appears in the universe naturally as we see from our new and many concrete results since Euclid.

However, the beautiful  horn torus model may be accepted with great pleasures as our space idea. In particular, note that the domains  $\{|z|<1\}$ and  $\{|z|>1\}$ are completely conformally equivalent and so the completely symmetric property of the corresponding domains on the horn torus is very fine and from this viewpoint, the Riemann sphere model will be curious, in particular, at the point at infinity and the point at infinity will be vague.

\section{Acknowledgements}

The Insitute of Reproducing Kernels wishes to express its deep thanks Professors and colleagues H.G.W. Begehr,  Wolfgang W. Daeumler, Hiroshi Okumura, Vyacheslav Puha and Tao Qian for their exciting communications.

\bibliographystyle{plain}

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L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.

\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{ms16}

T. Matsuura and S. Saitoh,

Matrices and division by zero $z/0=0$,

Advances in Linear Algebra \& Matrix Theory, {\bf 6}(2016), 51-58

Published Online June 2016 in SciRes.   http://www.scirp.org/journal/alamt

\\ http://dx.doi.org/10.4236/alamt.2016.62007.

\bibitem{mms18}

T. Matsuura, H. Michiwaki and S. Saitoh,

$\log 0= \log \infty =0$ and applications. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics. {\bf 230}  (2018), 293-305.

\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{mos}

H. Michiwaki, H. Okumura and S. Saitoh,

 Division by Zero $z/0 = 0$ in Euclidean Spaces,

 International Journal of Mathematics and Computation, {\bf 2}8(2017); Issue  1, 1-16.

\bibitem{osm}

H. Okumura, S. Saitoh and T. Matsuura, Relations of   $0$ and  $\infty$,

Journal of Technology and Social Science (JTSS), {\bf 1}(2017),  70-77.

\bibitem{os}

H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 (2017.11.14).

\bibitem{o}

H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947 International  Journal of Geometry.

\bibitem{os18april}

H.  Okumura and S. Saitoh,

Harmonic Mean and Division by Zero,

Dedicated to Professor Josip Peˇ cari´c on the occasion of his 70th birthday,　Forum Geometricorum,　{\bf 18} (2018), 155—159.

\bibitem{os18}

H. Okumura and S. Saitoh,

Remarks for The Twin Circles of Archimedes in a Skewed Arbelos by H. Okumura and M. Watanabe, Forum Geometricorum, {\bf 18}(2018), 97-100.

\bibitem{os18e}

H. Okumura and S. Saitoh,

Applications of the division by zero calculus to Wasan geometry.

GLOBAL JOURNAL OF ADVANCED RESEARCH ON CLASSICAL AND MODERN GEOMETRIES” (GJARCMG)(in press).

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S. Pinelas and S. Saitoh,

Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics. {\bf 230}  (2018), 399-418.

\bibitem{ps}

V. Puha and S. Saitoh,

  Horn Torus Model for the Riemann Sphere From the Viewpoint of  Division by Zero (manuscript).

\bibitem{s14}

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)

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\bibitem{s18}

S. Saitoh, Division by zero calculus (189 pages): http//okmr.yamatoblog.net/

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\end{thebibliography}

\end{document}

ダ・ヴィンチの名言 格言｜無こそ最も素晴らしい存在

https://systemincome.com/7521

                     

 

ゼロ除算の発見はどうでしょうか：
Black holes are where God divided by zero：

再生核研究所声明371（2017.6.27）ゼロ除算の講演―　国際会議　
https://ameblo.jp/syoshinoris/entry-12287338180.html

1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12276045402.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12263708422.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12272721615.html
Division By Zero（ゼロ除算）1/0=0、0/0=0、z/0=0

https://ameblo.jp/syoshinoris/entry-12392596876.html

ソクラテス・プラトン・アリストテレス　その他
https://ameblo.jp/syoshinoris/entry-12328488611.html

ドキュメンタリー 2017: 神の数式 第２回 宇宙はなぜ生まれたのか
https://www.youtube.com/watch?v=iQld9cnDli4
〔NHKスペシャル〕神の数式 完全版 第3回 宇宙はなぜ始まったのか
https://www.youtube.com/watch?v=DvyAB8yTSjs&t=3318s
〔NHKスペシャル〕神の数式 完全版 第1回 この世は何からできているのか
https://www.youtube.com/watch?v=KjvFdzhn7Dc
NHKスペシャル 神の数式 完全版 第4回 異次元宇宙は存在するか
https://www.youtube.com/watch?v=fWVv9puoTSs

再生核研究所声明 411(2018.02.02): 　ゼロ除算発見4周年を迎えて
https://ameblo.jp/syoshinoris/entry-12348847166.html

再生核研究所声明 416(2018.2.20): 　ゼロ除算をやってどういう意味が有りますか。何か意味が有りますか。何になるのですか　－　回答
再生核研究所声明 417(2018.2.23): 　ゼロ除算って何ですか　－　中学生、高校生向き　回答
再生核研究所声明 418(2018.2.24): 　割り算とは何ですか？　ゼロ除算って何ですか　－　小学生、中学生向き　回答
再生核研究所声明 420(2018.3.2): ゼロ除算は正しいですか，合っていますか、信用できますか　－　回答

2018.3.18．午前中　最後の講演：　日本数学会　東大駒場、函数方程式論分科会　講演書画カメラ用　原稿
The Japanese Mathematical Society, Annual Meeting at the University of Tokyo. 2018.3.18.
https://ameblo.jp/syoshinoris/entry-12361744016.html より

再生核研究所声明 424(2018.3.29): 　レオナルド・ダ・ヴィンチとゼロ除算

再生核研究所声明 427(2018.5.8): 神の数式、神の意志　そしてゼロ除算

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

Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.

https://notevenpast.org/dividing-nothing/

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。

1423793753.460.341866474681。

Einstein's Only Mistake: Division by Zero

http://refully.blogspot.jp/2012/05/einsteins-only-mistake-division-by-zero.html