The men who knew infinity: India’s lost history of mathematical genius

It was roughly a century ago that J.E. Littlewood, a renowned British mathematician, noted that every positive integer was a personal friend of Srinivasa Ramanujan. The Man Who Knew Infinity, a biopic on Ramanujan—a legendary mathematician born in 1887 in modern Tamil Nadu—released in India on Friday reminded his admirers of this dollop of history. A man with no formal training in mathematics, he would go on to secure several remarkable breakthroughs in his short life of 32 years.

Although a bright star, Ramanujan is just one among many distinguished Indian names who have made stellar contributions to the field of mathematics. The heritage of Indian mathematics is tremendously rich and diverse. The first comprehensive use of the place value system of arithmetic was found in Āryabhaṭīya (499CE), a famous work of Aryabhata. The trigonometric function “sine” traces its origin to jya-ardha series, a table of half-chords of a unit radius circle, compiled by him. Other prominent Indian names in mathematics include, chronologically, Varahamihira, Brahmagupta, Bhaskara I, Bhaskara II and Madhava.

Many of the discoveries which are attributed to European scholars had previously been worked out in India—in some cases centuries earlier. One of the most glaring examples is the Pythagorean theorem. There is no evidence to suggest that Pythagoras, the Greek mathematician, ever arrived at this theorem. The theorem, however, finds a place in Baudhayana’s Śulbasūtras, which dates back to about 800 BC—more than 200 years before Pythagoras was born. Pell’s equation, attributed by 18th century Swiss mathematician Leonhard Euler to 17th century English mathematician John Pell was originally solved by Bhaskara II, a 12th century Indian mathematician-astronomer.

Similarly, much of the work on calculus was done in India by the Kerala School of Mathematics—much before Isaac Newton and Gottfried Wilhelm Leibniz came into the picture—founded by Madhava in the 14th century CE. The entire list of wrong attributions is a much longer one. This is not an attempt to illegitimately usurp every work in the field of mathematics and claim it as Indian. The immense contributions of mathematicians from Europe, the Arab world, China and Africa cannot be wished away. Nor is this an effort to “Hinduize” or “saffronize” the achievements of India’s past. In fact, both the Jain and Buddhist traditions are inextricable parts of this heritage. For instance,Sūryaprajñapti, a Jain text had arrived to a close estimate of the value of π in the fourth century BC itself. A network of pearls described in the Buddhist text Avataṃsaka Sūtra as one where “in each pearl one can see the reflections of all the others, as well as the reflections within the reflections and so on” was worked upon by mathematicians in the US. The arrangement is exactly, they found, that of circles in what is known as Schottky groups. See the pictures here .

An important difference between the Indian tradition and the Greco-Western tradition of mathematics is the emphasis on proofs placed by the latter. This divergence is most distinctly observed in arguments between Ramanujan and his mentor G.H. Hardy at Trinity College, Cambridge. For Ramanujan, an equation had no meaning unless it expressed a thought of God. This fits in with the evolution of mathematics in India in a multi-disciplinarian framework. A regular osmotic process has sustained between Indian mathematics and other fields like astronomy, physics, linguistics, spiritualism and music.

Manjul Bhargava, a Princeton University mathematician of Indian origin and recipient of the prestigious Fields Medal, is one of the finest exponents of such cross-disciplinarian synergies. One of his favourites is how the number of rhythms in Sanskrit poetry consisting of long and short syllables—corresponding to beats on a musical instrument—can be calculated using the Hemachandra numbers, popularly known as the Fibonacci numbers after an Italian mathematician despite being first documented by the Indian polymath.

Barring the work of a few exceptions like Ramanujan, Indian advances in mathematics have seen an unprecedented decline in the last few centuries. Foreign conquests and colonization of the country seem to be the immediate factors contributing to the decline. Kerala remained untouched by the conflicts that had engulfed the northern parts of India, perhaps explaining why the mathematical tradition continued there longer than elsewhere else in the country.

Not much effort has been expended since independence on the revival of this great tradition. The cross-disciplinarian approach has been almost entirely done away with in the schools. A greater culture of commerce, trade and exchange of ideas that reinforced the intellectual quests of Indians in the past has also been lost; this is on the mend only in the last three decades. It is high time Indians took up this project.

A recovery of this great Indian tradition would involve a restoration of India’s glorious history. Mathematical concepts should be taught to students along with their histories. As Bhargava notes, “...knowing the correct history of mathematics was useful in my own research, because if you learn from the original source how an idea came about, that can give you great insight.” A beginning can be made by correcting the name and the origin of the so-called Pythagorean theorem in school textbooks.http://www.livemint.com/Opinion/d3ddePsFMHKZhHLmx1jAxL/The-men-who-knew-infinity-Indias-lost-history-of-mathemati.html

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\title{\bf  Announcement 300:  New challenges on the division by zero z/0=0\\

(2016.05.22)}

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

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

Kiryu 376-0041, Japan\\

%\date{\today}

\maketitle

{\bf Abstract: } In this announcement, for its importance we would like to state the

situation on the division by zero and propose basic new challenges.

\bigskip

\section{Introduction}

%\label{sect1}

By a {\bf natural extension} of the fractions

\begin{equation}

\frac{b}{a}

\end{equation}

for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$

\begin{equation}

\frac{b}{0}=0,

\end{equation}

incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the  case of real numbers.

 The division by zero has a long and mysterious story over the world (see, for example, Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628,  however,

  Sin-Ei Takahasi (\cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing the extensions of fractions and by showing the complete characterization for the property (1.2):

 \bigskip

 {\bf  Proposition 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.

\medskip

We thus should consider, for any complex number $b$, as  (1.2);

that is, for the mapping

\begin{equation}

w = \frac{1}{z},

\end{equation}

the image of $z=0$ is $w=0$ ({\bf should be defined}). This fact seems to be a curious one in connection with our well-established popular image for the  point at infinity on the Riemann sphere. Therefore, the division by zero will give great impacts to complex analysis and to our ideas for the space and universe.

However, the division by zero (1.2) is now clear, indeed, for the introduction of (1.2), we have several independent approaches as in:

\medskip

1) by the generalization of the fractions by the Tikhonov regularization or by the Moore-Penrose generalized inverse,

\medskip

2) by the intuitive meaning of the fractions (division) by H. Michiwaki,

\medskip

3) by the unique extension of the fractions by S. Takahasi,   as in the above,

\medskip

4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from $ {\bf C} \setminus \{0\} $ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z $ from  ${\bf C}$ onto ${\bf C}$,

\medskip

and

\medskip

5) by considering the values of functions with the mean values of functions.

\medskip

Furthermore, in (\cite{msy}) we gave the results in order to show the reality of the division by zero in our world:

\medskip

\medskip

A) a field structure  containing the division by zero --- the Yamada field ${\bf Y}$,

\medskip

B)  by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,

\medskip

C) by the reflection $W =1/\overline{z}$ of $W= z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero,

\medskip

and

\medskip

D) by considering rotation of a right circular cone having some very interesting

phenomenon  from some practical and physical problem.

\medskip

In (\cite{mos}),  many division by zero results in Euclidean spaces are given and  the basic idea at the point at infinity should be changed. In (\cite{ms}), we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.

\medskip

See  J. A. Bergstra, Y. Hirshfeld and J. V. Tucker \cite{bht} for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.

Meanwhile,  J. P.  Barukcic and I.  Barukcic (\cite{bb}) discussed recently the relation between the divisions $0/0$, $1/0$ and special relative theory of Einstein. However, their logic seems to be curious and their results contradict with ours.

 Furthermore,  T. S. Reis and J.A.D.W. Anderson (\cite{ra,ra2}) extend the system of the real numbers by introducing an ideal number for the division by zero $0/0$.

 Meanwhile, we should refer to up-to-date information:

{\it Riemann Hypothesis Addendum - Breakthrough

Kurt Arbenz

https://www.researchgate.net/publication/272022137 Riemann Hypothesis Addendum -   Breakthrough.}

\medskip

Here, we recall Albert Einstein's words on mathematics:

Blackholes are where God divided by zero.

I don't believe in mathematics.

George Gamow (1904-1968) Russian-born American nuclear physicist and cosmologist remarked that "it is well known to students of high school algebra" that division by zero is not valid; and Einstein admitted it as {\bf the biggest blunder of his life} [1]：

1. Gamow, G., My World Line (Viking, New York). p 44, 1970.

 For our ideas on the division by zero, see the survey style announcements 179,185,237,246,247,250 and 252 of the Institute of Reproducing Kernels (\cite{ann179,ann185,ann237,ann246,ann247,ann250,ann252,ann293}).

\section{On mathematics}

Apparently, the division by zero is a great missing in our mathematics and the result (1.2) is definitely determined as our basic mathematics, as we see from Proposition 1.  Note  its very general assumptions and  many fundamental evidences in our world in (\cite{kmsy,msy,mos}). The results will give great impacts  on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and  physical problems. See our announcements for the details.

The mysterious history of the division by zero over one thousand years is a great shame of  mathematicians and human race on the world history, like the Ptolemaic system (geocentric theory). The division by zero will become a typical  symbol of foolish human race with long and unceasing struggles. Future people will realize this fact as a definite common sense.

We should check and fill our mathematics, globally and beautifully, from the viewpoint of the division by zero. Our mathematics will be more perfect and beautiful,  and will give great impacts to our basic ideas on the universe.

\section{Albert Einstein's biggest blunder}

The division by zero is directly related to the Einstein's theory and various

physical problems

containing the division by zero.  Now we should check the theory and the problems by the concept of the RIGHT and DEFINITE division by zero. Now is the best time since 100 years from Albert Einstein. It seems that the background knowledge is timely fruitful.

\section{Computer systems}

The above Professors listed are wishing the contributions in order to avoid the zero division trouble in computers. Now,  we should arrange  new computer systems in order not to meet the division by zero trouble in computer systems.

\section{General  ideas on the universe}

The division by zero may be related to religion,  philosophy and the ideas on the universe, and it will creat a new world. Look the new world.

\bigskip

We are standing on a new  generation and in front of the new world, as in the discovery of the Americas.

 \bigskip

\bibliographystyle{plain}

\begin{thebibliography}{10}

\bibitem{bb}

J. P.  Barukcic and I.  Barukcic, Anti Aristotle—The Division of Zero by Zero. Journal of Applied Mathematics and Physics,  {\bf 4}(2016), 749-761.

doi: 10.4236/jamp.2016.44085.

\bibitem{bht}

J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,

Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).

\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.  {\bf 27} (2014), no 2, pp. 191-198,  DOI: 10.12732/ijam.v27i2.9.

\bibitem{ms}

T. Matsuura and S. Saitoh,

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

Linear Algebra \& Matrix Theory (ALAMT)(to appear).

\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

 (in press).

\bibitem{ra}

T. S. Reis and J.A.D.W. Anderson,

Transdifferential and Transintegral Calculus,

Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I

WCECS 2014, 22-24 October, 2014, San Francisco, USA

\bibitem{ra2}

T. S. Reis and J.A.D.W. Anderson,

Transreal Calculus,

IAENG  International J. of Applied Math., {\bf 45}(2015):  IJAM 45 1 06.

\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{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.

\end{thebibliography}

\end{document}