History: Five millennia of Indian science

James Poskett applauds a show celebrating discovery on the subcontinent, from zero to the boson.

Illuminating India: 5000 Years of Science and Innovation

The Science Museum, London. Until 31 March 2018.

ISRO

An image of Mars captured by the Indian Space Research Organisation's Mangalyaan satellite.

If reason can be embodied, it stares us in the face at the entrance to the Illuminating India exhibition at London's Science Museum. A 1,000-year-old statue of the Buddha sits in the 'earth witness' pose (cross-legged, right hand touching the ground), symbolizing truth and rationality. A commitment to these concepts is woven through India's 5,000-year story of science, spanning religions and philosophies from Hinduism to Sikhism, and taking in mathematics, medieval astronomy, modern physics, the computer revolution and space exploration.

The star of the show is undoubtedly part of the Bakhshali manuscript, discovered in 1881 in what is now Pakistan. This is a piece of birch bark no bigger than my hand. But the Sanskrit text on it (last month carbon dated to AD 200–400) reveals one of the most important innovations in the history of mathematics: the first use of zero as a numeral.

For much of history, there was no straightforward way to represent the concept of nothing in maths. The ancient Greeks, such as the mathematician Ptolemy, weren't even sure if zero was a number. And the lack of zero in Roman numerals often made their arithmetic needlessly complex (try adding MDCCCLVIII to MCMXLVII). The number system we use today is ultimately derived from the ancient Hindu and Arabic mathematics found in the Bakhshali manuscript.

Bodleian Lib., Univ. Oxford

The Bakhshali manuscript contains the first known use of zero as numeral.

Other incredible objects abound. I find a spectroscope built by the Nobel-prizewinning physicist Chandrasekhara Venkata Raman. A sixteenth-century Mughal painting of the Emperor Akbar riding an elephant across the Ganges is effectively a work of early natural history. A prosthetic foot developed in the 1960s by craftsman Ram Chander Sharma and orthopaedic surgeon Pramod Karan Sethi points to India's impact on the world of medicine. This is the first time many of these items have been exhibited outside India.

Vibrant textile hangings divide the show into three themes. The first, 'Observation', spans six centuries, rich with intriguing juxtapositions. One moment I'm looking at a beautiful nineteenth-century astrological almanac. The next, I see a model of the camera sent to Mars in 2013 by the Indian Space Research Organisation. Part of the Mangalyaan satellite, it took detailed colour photographs of the red planet's largest canyon system, the Valles Marineris. (Incredibly, the whole mission cost under US$75 million, less than the budget for the 2013 science-fiction blockbuster Gravity.)

Then there is a bright orange auto-rickshaw (a hybrid of an electric scooter and cycle rickshaw), the centrepiece of the 'Innovation' gallery and a symbol of India's frugal innovation, or jugaad. Alongside other classic examples of Indian engineering — such as the country's railways, which carry more than 22 million passengers each day — there is a case on telecommunications. A small wooden and metal instrument looks like the sort of thing used to tap out Morse code, but is an early example of a coherer, a device for detecting radio signals. The technology was developed by Bengali polymath Jagadish Chandra Bose, who found time to study plant physiology and archaeology in between writing science fiction and pioneering the use of semiconductors in radios. Other Indian innovations on display include the USB connector, developed by a team led by Ajay Bhatt, and the Intel Pentium processor, developed by electrical engineer Vinod Dham.

The exhibition concludes with 'Calculation'. India has produced many great mathematicians. Works by some of the best are on display: the original notebooks of the mathematical genius Srinivasa Ramanujan (A. Robinson Nature 531, 576–577; 2016) appear alongside photographs of the incredible Jantar Mantar observatory, built in Jaipur in the early eighteenth century. From here, the Maharajah Jai Singh II produced detailed astronomical calculations, comparing his results to the latest findings from Paris.

A collection of letters reminds us that the boson has its origin in India. In the 1920s, the young Bengali physicist Satyendra Nath Bose struck up a correspondence with Albert Einstein. Together, the duo laid the foundations of quantum statistics and predicted the existence of undiscovered fundamental particles. Today, we call those particles bosons in honour of S. N. Bose. Coinciding with 70 years of Indian independence, Illuminating India is a celebration of South Asian contributions to science and technology.http://www.nature.com/nature/journal/v550/n7676/full/550332a.html

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…