Zero

A Landmark Discovery, the Dreadful Void, and the Ultimate Mind

Syamal K. Sen,Ravi P. Agarwal

• Éditeur: Academic Press
• Année: 2015
• ISBN:
  9780128046241
• Sujet: Matematica
• Sous-sujet: Storia e filosofia della matematica

Zero indicates the absence of a quantity or a magnitude. It is so deeply rooted in our psyche today that nobody will possibly ask "What is zero?" From the beginning of the very creation of life, the feeling of lack of something or the vision of emptiness/void has been embedded by the creator in all living beings. While recognizing different things as well as the absence of one of these things are easy, it is not so easy to fathom the complete nothingness viz. the universal void. Although we have a very good understanding of nothingness or, equivalently, a zero today, our forefathers had devoted countless hours and arrived at the representation and integration of zero and its compatibility not only with all non-zero numbers but also with all conceivable environments only after many painstaking centuries. Zero can be viewed/perceived in two distinct forms: (i) as a number in our mundane affairs and (ii) as the horrific void or Absolute Reality in the spiritual plane/the ultimate state of mind.

Presented are the reasons why zero is a landmark discovery and why it has the potential to conjure up in an intense thinker the dreadful nothingness unlike those of other numbers such as 1, 2, and 3. Described are the representation of zero and its history including its deeper understanding via calculus, its occurrences and various roles in different countries as well as in sciences/engineering along with a stress on the Indian zero that is accepted as the time-invariant unique absolute zero. This is followed by the significant distinction between mathematics and computational mathematics and the concerned differences between the unique absolute zero and non-unique relative numerical zeros and their impact and importance in computations on a digital computer.

• Introduces the history of the value of zero and why it was a landmark discovery
• Discusses how zero is used in science and engineering and its use in different countries
• Explains how zero affects different mathematics and calculus

1

Introduction

Abstract

Zero is introduced stressing on its practical usage and spiritual significance. Big Bang and Steady State theories with their pros and cons are highlighted. The implication of division by zero is specifically pointed out.

Keywords

Bakhshali manuscript; circle for the zero; decimal-based place value notation; division by exact zero; five properties of zero; natural mathematics; numerical zero; seven-segment display; slashed zero; steady state theory; universal nothingness; zero density; zero hour

One of the remarkable things about the behaviour of the world is how it seems to be grounded in mathematics to a quite extraordinary degree of accuracy. The more we understand about the physical world, and the deeper we probe into the laws of nature, the more it seems as though the physical world almost evaporates and we are left only with mathematics.

—Roger Penrose (born 1931).

…With his eyes open, he (Swami Vivekananda, 1863–1902) saw the walls and everything in the room, nay, the whole universe and himself within it, whirling and vanishing into an all-encompassing void. He was frightened as he thought he might be on the verge of death, and cried out: “What are you (Sri Ramakrishna, 1836–1886) doing to me? I have my parents at home.”

—Mahendra Nath Dutta (younger brother of Swami Vivekananda). The incidence occurred in November, 1881 in Kolkata.

Zero indicates the absence of a quantity or a magnitude. It is so deeply rooted in our psyche today that nobody will possibly ask “What is zero?” From the beginning of the very creation of life, the feeling of the lack of something or the vision of emptiness/void has been embedded by the creator in all living beings. While recognizing different things as well as the absence of one of these things are easy, it is not so easy to fathom the complete nothingness, viz. the universal void. Although we have a very good understanding of nothingness or, equivalently, a zero today, our forefathers had devoted countless hours and arrived at the representation and integration of zero and its compatibility not only with all nonzero numbers but also with all conceivable environments only after many painstaking centuries. Zero can be viewed/perceived in two distinct forms: (i) as a number in our mundane affairs and (ii) as the horrific void or Absolute Reality in the spiritual plane/the ultimate state of mind. Presented are the reasons why zero is a landmark discovery and why it has the potential to conjure up in an intense thinker the dreadful nothingness unlike those of other numbers such as 1, 2, and 3. Described are the representation of zero and its history including its deeper understanding via calculus, its occurrences and various roles in different countries as well as in sciences/engineering along with a stress on the Indian zero that is accepted as the time-invariant unique absolute zero. This is followed by the significant distinction between mathematics and computational mathematics and the concerned differences between the unique absolute zero and nonunique relative numerical zeros, and their impact and importance in computations on a digital computer.

1.1 Matter versus nonmatter

While dealing with zero meaning “nothing” or void, its significance in the realm of both matter and nonmatter, its birth and properties, abstract (symbolic) presentation and various names in different contexts, occurrences and uses in science and engineering as well as in different countries are discussed, along with the reasons why it can be portrayed as the most fearful void, the highest state of mind, and also considered as one of the greatest innovations of mankind.

With our current conditioned mind, it appears to us easy to conceive the physical significance of just a zero. It is not difficult to imagine “nothing” in the background of something. It is, on the other hand, very difficult or even dreadful to think “nothing” in the background of “nothing” (achieved by eliminating everything including even the background). Just attempt to think/imagine about something that exists and remove that thing. Continuing the successive removal of one thing after the other and reaching the state in which everything including all relations, the surroundings, and even one’s own body from the conscious state of one’s mind has vanished, could lead one to a dreadful experience! Was there any universal void—a situation when nothing existed in the Universe? Physics has been sticking until today, and possibly will continue to stick to the point for an indefinite period of time, that something cannot be created from nothing. In other words, nothing can be created out of nothing. This implies there is always something eternally. That is, infinite years ago there was something (matter including energy, assuming that matter is convertible to energy and vice versa), this exists today, and will continue to exist infinite years hence (its form, however, may be changing with time).

In any science including physics, something cannot be created out of nothing. There is no evidence that a thing has been created from complete void. The valiant effort of Fred Hoyle (1915–2001 AD) and Jayant Vishnu Narlikar (born 1938 AD) during the early 1960s to propound the Steady State Theory in Cosmology (an alternative to the Big Bang Theory of the universe’s origin), which says that new matter is continuously created as the universe expands, thus adhering to the cosmological principle, did not succeed and the theory is now obsolete. This is true for both matter and nonmatter. The mind of any one individual contains all the knowledge (nonmatter). There exists no knowledge outside the mind. It is the specific knowledge-mining that a scientist does in the ocean of knowledge residing in his/her own mind.

According to today’s physics, the void, that is, the universal void, was never there, is not there, and will never be there. Two aspects are important to be considered here. One aspect is that of matter while the other aspect is that of nonmatter or, may be termed, Spirit or Nature or God (encompassing all knowledge) or Consciousness that is omnipotent (having unlimited power), omnipresent (present everywhere), and omniscient (knowing everything). One’s realization/experience is the proof of the existence of spirit, which is the best proof (better than even a mathematical proof). This Nature (or God if you wish to call it) pervades all matter, all spaces containing matter of varying density including numerically zero (not exactly zero) density. Matter with exactly zero density, that is, completely/absolutely empty space, does not seem to be fathomable by a physicist or possibly by anybody within the realm of science that we are taught conventionally and traditionally. Is there a sharp boundary (maybe static or dynamic) just beyond which matter has absolutely zero density and just within (maybe closest to the boundary) which it has nonzero density? Is there a discontinuity of density (in a mathematical term)? Interestingly, when we attempt to create vacuum in a container, we successively reduce the density of the gas (say, air) but we will never be able to make the density exactly zero by any process that we know of in physics.

1.2 Zero in universal nothingness

Under these circumstances, the zero—the way we understand it today—is distinctly different from other numbers such as 1, 2, 3, and 4 (denoting one, two, three, and four physical objects), which can be very easily comprehended from the physical world which we live in. In this context, we may consider zero, that is, nothingness, in the well-known environment/surrounding of many things which we live with. This zero is well within our understanding but has been playing hide-and-seek over centuries in terms of unambiguous unique representation as well as unambiguous integration with other nonzero numbers (mainly for arithmetic operations) and complete compatibility with everything under all circumstances. But the zero in the environment of complete vacuum state or absolute nothingness is not well within our grasp. We therefore stick to the former zero in most of our following discussion.

1.3 Birth and five properties of zero

The exact date of birth of zero is not known although the very feeling of nothingness or of absence (of something) did exist in the minds of living beings since time immemorial. This nothingness is conceived against the visible world around us. The question of uniquely representing this nothingness and its function in relation to other numbers (representing nonnothingness), such as 1, 2, 3, and 4, under all circumstances and in all sciences without any noncompatibility, which has no inner contradiction or clash and which solves all our arithmetic and algebraic problems without any ambiguity, continued to remain elusive to mathematicians for centuries. Today we are so accustomed/conditioned with using zero (0) along with other numbers that we, with our existing mental set-up, will not ask the aforementioned question in the realm of not only arithmetic and algebra but also in the whole of mathematics. For instance, when one subtracts the number 825 from 825, the result is nothing and so an accountant in a business transaction used to keep the result-space blank indicating “nothing.” Among a large number of computations, leaving the result-space empty could mean either (i) the accountant has forgotten (a nontrivial possibility) to write the result of the arithmetic expression involving several numbers or (ii) the result of the expression is “nothing” or zero. With our present day conditioned mind it might appear to us that this is not a serious issue as we would readily fill the result-space by one or more zeros. This is a role of zero as a number. Determining (or finding) a symbol for zero different from all other existing symbols was also an issue that might appear trivial to us today, but it was not so during the third or earlier millennium BC. Since zero is the bottom of all positive numbers, it should act as a direction separator to accommodate negative numbers which are unavoidable almost everywhere in science and engineering. In addition, to denote the magnitude of a quantity, a number is used. If the magnitude happens to be nil (that might occur quite often in our physical world, for instance no money or no cow), then the same zero should represent that magnitude. In the Indo-Arabic number system, zero should also act as the place holder. For example, 1 in the unit position and 1 in the tens position are completely different. Adding a zero on the right side of 1 would uniquely decide the value. These five problems did not exist with other nonzero numbers occurring in any arithmetic/mathematical computation that does not encounter zero or “nothing.” Thus we should define and represent a zero which have all the foregoing five properties. Such a zero has been found to be (would then be) usable everywhere without any context dependence and any ambiguity. There appears to be no other distinct property (besides the foregoing five) that must be satisfied for absolute compatibility with numbers and nonnumbers in any context.

Since the exact date of birth of zero, rather the physical meaning of zero, is unknown and will never be known, one could imagine that zero existed eternally, that is, before the universe (if it is assumed born out of a birthless (visible or nonvisible, perceivable or nonperceivable) seed) came into existence and will remain after the universe is gone, like the number Pi (ratio of the circumference and the diameter of any circle or, in other words, the area of the circle with unit radius), but with much more pervasiveness. A primitive/prehistoric man can easily comprehend the absence of something in the background of things around. Thus the concept of zero has been in-built in any primitive man and possibly in any living being from the very beginning of creation of life in the universe. The exact date of birth of the very first primitive man is not known, we can only attempt, based on some controversial logic/reasoning, the approximate large period of time that might contain the exact date of birth of the first primitive man. However, imagining the existence of nothing in the backdrop of (Universal) Nothing (analogously, finding a black snake in a dark environment) or allowing the mind to remove everything including even one’s own body—one thing after the other by the process of successive exclusions (or, simply allowing things to vanish all at a time)—could be much tougher for most of us, the human beings—primitive, historic, and modern. This needs an extraordinary sense of detachment (meaning giving up the notion of “I” and “mine” referring not so much to the renunciation of possession but renouncing the idea of possessor) and spirituality.

1.4 Zero is the very life of all sciences and engineering

Zero is very much more extensively known than the famous constants such as Pi, e (exponential function of argument 1), and Phi (Golden ratio). Everything in any science, any engineering, and any technology will simply collapse and die readily if zero is taken out (unlike the numbers Pi, e, and Phi). Even an irrational number (having infinity of digits), such as Pi, e, and Phi, which contain zeros in their numerical values, will become nonrepresentable (as a number) if zeros are dropped. Not only in the conventional decimal number system, but also in any other number system of any radix, the symbol of zero along with its unique physical meaning is preserved. This is not so true with any other symbol implying a nonzero number, say, 11 (in octal, i.e., in base-8 (i.e., radix-8) number system, its physical meaning is 9 and in binary, i.e., in base-2 number system, its physical meaning is different and it is 3), while 00 in any number system of any positive integral radix (e.g., 8, 2, 16, 20, 60) has its physical meaning preserved, that is, it is always 0.

1.5 Nomenclature, symbols, and terms concerning zero and place–value system

The word zero came from Venetian zero via French zero, which (together with cipher or, equivalently, cypher) came, via Italian zefiro from Arabic safira meaning “it was empty” or, equivalently, sifr (the Persian mathematician Mohammed ibn-Musa al-Khwarizmi (around 780–850 AD) called zero “sifr,” from which our cipher is derived.) denoting “zero” or “nothing.” This was a translation of the Sanskrit word śūnya (shoonya meaning “empty”). Brahmagupta (born 30 BC), a renowned Indian mathematician and astronomer and author of many important works on mathematics and astronomy, used dots or, equivalently, points (a dot is called bindu in Sanskrit and many other Indian languages such as the Bengali language) underneath numbers to indicate a zero. These dots were alternately referred to as “sunya.” which means empty, or “kha,” which means place. Much earlie...

Table des matières

• Cover image
• Title page
• Table of Contents
• Copyright
• Preface
• 1. Introduction
• 2. Zero a landmark discovery, the dreadful void, and the ultimate mind: Why
• 3. History of zero including its representation and role
• 4. Zero in sciences, engineering, its uses in various countries, and opposition faced
• 5. Conclusions
• Bibliography
• Index