It might seem odd to take a moment in our technology-driven lives to contemplate the humble and mysterious zero, the pivot upon which mathematics rotates.
At a time when the zero underpins practically everything we do, being an essential part of the binary code which saturates our world, its origins could prove interesting.
The University of Oxford, which recently published research findings that have pushed back the first recorded use of zero to a 3rd century CE Indian manuscript, clearly thinks so.
The findings, involving the carbon dating of an ancient mathematical treatise, the Bakhshila Manuscript, have opened up the door in terms of revealing the provenance of the elusive zero. Experts are clear that zero, as we know it, is an “Indian” invention — but how, why, and when it first emerged as a distinct mathematical concept is less a story of numbers, and more one of how different cultures looked at the world, how information was disseminated across countries, and how fundamental aspects of different models of philosophy and religion informed our consciousness and interests.
Origins
The concept of zero, or nothingness, did exist before any South Asian hand penned it down, though not in a usable form, and so before delving into the history of it, it’s important to make a distinction between a placeholder and numeral.
In mathematics, the main difference between the two is whether there is evidence that it was used in equations, and thus is a repeatable phenomenon.
Placeholder zeros have been present for thousands of years. According to Harvard math professor Robert Kaplan, they were first documented 5,000 years ago in Mesopotamia with the Sumerians using them.
Both the Chinese mathematical model of counting sticks and the Babylonians were clearly also aware of the concept of zero, but only as just that: a placeholder concept, something that could not be replicated with the same outcome each and every time a particular equation was used.
The concept then spread from Mesopotamia to places like China, Babylon, and India — but it was only the latter who made it more than just an idea.
India proved fertile ground for the evolution of zero from a placeholder to a numeral.
“Really the first place where zero was used as an equation was in India with the ancient Vedic mathematicians,” says Peter Gobets, secretary and leading member of ZerOrigIndia, or Project Zero, a collection of academics who are trying to determine the origin of zero.
“Indian mathematicians were able to conceive and fully invest in it because of their philosophical understandings of shunya — the nothingness that is the natural counterpart to something,” he adds.
It’s difficult to overstate the importance of this. Mathematics at the time was more an expression of philosophical ideas and reasoning, and directed towards astronomy and commerce.
This explains why zero could never have been conceptualised in the West. The Greeks, whose astronomical models and mathematical equations certainly did influence their Indian counterparts, abhorred the very idea of nothingness.
Pre-Socratic schools of thought contended that there could be no such thing as nothing. “Nothing cannot be something,” as Aristotle famously once reasoned.
“This deeply ingrained aversion to emptiness prevented any mathematical equation involving zero — the very idea was offensive to the ancient Greeks,” says Gobets.
India, on the other hand, proved fertile ground for the evolution of zero from a placeholder to a numeral.
“The concept of shunya is embraced completely, and is explored as being complementary to the concept of fullness,” Gobets explains. “It’s not just maths, or astronomy that it influences — you can see an evolution from the Vedas to being used in different cultural antecedents, in language, in dance, in music.”
“Indians actually utilised it in calculations — that’s what elevates zero from being a placeholder to a numeral. There’s no contention there. Where there isn’t a consensus is in when that first occurred.”
The first person to document zero as a number in its own right was the astronomer and mathematician Brahmagupta in 682 CE.
The Bakhshila Manuscript was unearthed by a farmer in 1881 in a field near what is now Mardan in modern day Pakistan. It swiftly came into the possession of the Raj’s premier Indologist at the time, AFR Hoernle, before making its way to the Bodleian Library in Oxford.
Its age has been in contention for the past century. The most authoritative academic study on the manuscript, conducted by Japanese scholar Dr Hayashi Takao, placed its age from between the 8th and 12th Centuries.
Oxford’s research appears to suggest that the manuscript was far earlier than that thanks to carbon dating — which would make it the first recorded use of zero as a numeral.
Gobets, however, is unconvinced that zero began with the manuscript, which his team hopes to independently study, but he says it is possible.
“Our biggest stumbling block is that there is very little evidence,” he says. “Most of the pertinent manuscripts were written on birch bark, which is pretty damn perishable, especially given the humid climate, and the ways they were stored.”
How zero was spread isn’t in contention, however.
The voyage of zero
Indian mathematical treatises began to be spread via trade and migration across the pre-Islamic empires of West Asia, particularly the Sasanian Empire in what is now Iran.
In 662, a Syrian Monophysite bishop named Severus Sebokht wrote approvingly about “the science of the Indians,” and “their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians, and their valuable methods of calculation which surpass description.”
He wasn’t the only fan. During the reign of Caliph al-Mansur, an Indian scholar bearing Sanskrit astronomical works was recorded as visiting Baghdad in the 770s. Those works were subsequently translated and adopted as Zij-al sindhind and Sindhind al-kabir — foundational mathematical treatises for Central Asian and Arabic scholars, based on Sanskrit models.
It’s important to understand just how momentous a change this system would bring in. Without zero, there would be no decimal system — and in turn no complicated equations suitable for commerce, for statistics, for figures.
The Arabs, in particular, began expanding and developing these ideas and concepts into what we now universally recognise as our numerals and our calculation system. The equations, and the methods that described them, were referred to as hisab al-hind — literally, “Indian calculation.”
In the 9th century, an Arabic scholar and trader named al-Khwarizmi would set the stage for Indian calculation to travel to the West — albeit without much explanation or usage of zero specifically. His kitab al-Jabr wa-al-muqabala became so influential via Latin translations in the late 10th Century that its Latinised name gave us a new word to define calculation systems: Algebra.
The shunya of ancient India became sifr in the Arabic works of al-Khwarizimi and his peers, which eventually became cipher, zero, and their respective cognates in European languages.
Without zero, and its many applications, everything from modern commerce, to banking, to statistics, to even coding would be hard to conceive of
It wasn’t immediately adopted, however. This was the time of the Crusades, and information flowing from the East was considered to be inherently suspect by the most significant authority in Western Europe — the Church.
It’s also important to realise just how abhorrent an idea a void is, philosophically and spiritually, in the eyes of the Church. While the Greeks despised the idea on grounds of rhetoric, the Church was shocked due to a different dogma. After all, this was a religion founded with the end goal of there being a Heaven and a Hell — not a void.
Nonetheless, al-Khwarizmi’s works caught the attention of a certain mathematician named Fibonacci, who immediately saw the practical applications of this strange calculation method from the East in terms of commerce and banking.
Commerce, trade, and industrialisation led to the birth of new scientific concepts and practices, which in turn underscored how sublime ancient India’s usage of zero truly was as Western scientific thought became influenced by it. When Blaise Pascal, for example, first being exploring the idea of there being a vacuum, he was elucidating a type of void.
Endgame
It’s curious to realise that, in a sense, nothing has come to underpin a great deal. Without zero, and its many applications, everything from modern commerce, to banking, to statistics, to even coding would be hard to conceive of, let alone put into practice.
Oxford’s research has already come under academic criticism. In a rejoinder written in the History of Science in South Asia journal, several of the most prominent mathematical historians on the subject have argued that the age of the document has still not been determined due to factors outside of carbon dating, and that the zeros utilised in the Bakshila Manuscript could potentially be more than a placeholder.
As the academic debate continues, the origin of zero continues to fascinate and confound us.
Its history is the history of ancient cultures, and of the intersection between philosophy, trade, war, and the rise and fall of said civilisations. Even the outliers are fascinating to consider — what of the Mayans, who, isolated from other cultures, also independently conceived of zero, only for that information to fade along with their empire into the jungles of the Yucatan?
More than just ancient, these are also decidedly non-Western histories. And the quest for zero’s origins provides an important opportunity to examine these cultures in a time of ever-increasing globalisation and uncertainty.
Something did come from nothing after all.
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…
https://sites.google.com/site/sandrapinelas/icddea-2017
2017.8.21.06:37
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
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