Zero and Infinity: invaluable contributions of ancient Indian Mathematics

In the system of numerals ‘zero’ is perhaps of least importance to any person unless it trails another numeral enhancing it ten times. Often used dismissively to describe someone’s competence in an area of study or performance it is an adjective or noun, depending on its use, which no one welcomes. That was perhaps the past, but in this age of computers and digitization, zero has assumed new meaning and significance; along with its complement ‘one’, the pair have metamorphosed our good old universe to a series of zeros and ones in its different manifestations as diverse as computing, literature, art, and technology, to name a few. Although used as complements, the origin of zero and one are as diverse as chalk and cheese and they occur at vastly different points of time in human history.

In one of his essays, the mathematician philosopher Bertrand Russel has speculated what could have given rise to a number system. He says that usually our introduction to the number system and later to arithmetic or mathematics begins with collection physical objects and their association with numbers, like one apple two books etc. and we move forward to more complex computing, rigorous logic and so on. But if we move in the other direction and enquire what ‘one’, ‘two’ etc. are, we enter into the realm of philosophy and intelligent speculation. The need for a number system, he speculated, must have arisen from the need to keep account of things man possessed once he moved from the stage of hunting to keeping domesticated animals (his wealth). In fact, numerals are symbols, a form of convenient representation of a certain character / property of collections of objects, but are not essential for such representation. Without numerals, a much later development, the ancient man did keep an account of his animals. He would keep one pebble against each animal, leave the collection of pebbles in his cottage and take animals for grazing. On return, he would again put one pebble against each animal; if there were unassigned pebbles, he would know as many animals as unassigned pebbles have strayed away and if pebbles were found short, he would infer that animals from other herds have joined his. Thus, without any number system he was able to keep an account of his animals in a very basic way. This is what is referred to in the language of mathematics as one-to-one correspondence. ?In fact, our number system has been developed over several stages, several years and across several places in the globe to reach the present stage, but essentially its character of presenting a one-to-one correspondence between a set of symbols and a set of collection of objects still holds.

If this is the origin of our numeral system then how does zero fit in in the scheme of things. The need to keep an account comes only if one possesses some animals, articles or any form of wealth. One does not need to keep an account of nothing. Therefore, zero as a numeral didn’t originate along with other numerals, because there was no need for it and conceptually it was way different than representing a collection of objects; it represents nothing. Thus, the composition of the series of natural numbers started from 1; zero was neither invented nor required then. David Berlinsky, in his book ‘A Tour of Calculus’ has this to say on two fantastic inventions:

“The first is number 0 (zero), the creation of some nameless but commanding Indian mathematician. When 5 is taken away from 5, the result is nothing whatsoever, the apples on the table vanishing from the table, leaving in their place a peculiar and somewhat perfumed absence…It required an act of profound intellectual audacity to assign a name and hence a symbol to all that nothingness. Nothing, Nada, zero,0. The negative numbers are the second of the great inventions” ???

A study of history of mathematics would reveal that early mathematics developed in the West never had any concept of zero as a number. The Egyptians are credited with invention of geometry, but never took mathematics /numbers seriously except to the extent of a tool for measuring land parcels and counting passage of days. Obviously zero has very little significance. The concepts of shapes, sizes, squares, rectangles etc. would lose all meaning if any one measure becomes zero / nothing. The Greeks on the other hand, took numbers and philosophy with extreme seriousness but zero never entered their consciousness as a numeric value. The great Greek mathematician-philosopher Pythagoras, famous for the theorem named after him: the square of the hypotenuse of a right-angled triangle equals the sum of squares of the other two sides, was remembered in ancient Greece more for the invention of musical scale. When a moving bridge was made to divide the string (a monochord) at a particular ratio, the sound generated was most resonant. The notes were different at different positions of the bridge, that is at different ratios of division of the string. For him dividing a string at different points is like the ratios of two numbers. Ratios and proportions not only controlled musical beauty, they also explained physical and mathematical beauty. Pythagoreans and later Greek mathematicians’ interest in numbers and ratios yielded the ‘golden ratio’, considered the mathematics behind the most beautiful physical structures of the world. If a line is divided into two parts so that the ratio of the small part to the larger one is the same as the ratio of the large part to the original line, what we get is the golden ratio, which at different values of original number converges to around 1.6.

?Later in 1200 AD, Italian mathematician Fibonacci gave the sequence where each term is the sum of two preceding terms (1, 1, 2, 3, 5, 8, 13, 21…and so on). The ratio of two consecutive terms also converged to the golden ration (1.625). As an aside, although this sequence of numbers is named after Fibonacci who introduced it to the western world, these numbers were first described in?Indian mathematics?as early as 200 BC, almost 1500 years earlier, in the works of?Pingala?on enumerating possible patterns of?Sanskrit?poetry formed from syllables of two lengths. During the same period Bharat Muni also used the knowledge of this sequence in his ‘Natya Satra’. Clearly this ratio is one that has been used extensively in art, sculptor, architecture, poetic patterns etc. since millennia. These days Fibonacci sequence is widely used in computer science in search algorithm, data structure etc.

Coming to the main topic of discussion, it is believed that the reason why the Greeks and the West in general could not conceptualize zero as a numeral and consequently the concept of infinity, has less to do with Mathematics than the dominant Philosophy of the time accepted in the West. Charls Seife in his book, ‘Zero- the Biography of a dangerous Idea’ has this to say:

?“Zero conflicted with the fundamental philosophical belief of the West, for contained within zero are two ideas that were poisonous to Western doctrine. Indeed, these concepts would eventually destroy Aristotelian philosophy after its long reign. These dangerous ideas are the void and the infinite.”

?It is beyond the scope of this article to discuss in some detail the philosophical conflicts the Greeks encountered. However, the following quotes from the author will show some clarity:

“It is not easy to reject both infinity and zero. Look back through time. Events have happened throughout history, but if there is no such thing as infinity, there cannot be infinite of events. Thus, there must be a first event: creation. But what existed before creation? Void? That was unacceptable to Aristotle. Conversely, if there was not a first event, then the universe must have always existed- and will aways exist in future. You got to have either infinity or zero; a universe without both of them makes no sense.”?

?While the West shunned the concept of void, both the concepts of void (shunya) and infinity (ananta) were integral to Hindu philosophy. Hinduism was replete with symbolism of duality; concepts of creation and destruction were intermingled in Hinduism. “Shiva was both the creator and destroyer of the world. He also represented nothingness. He was the ultimate void, the supreme nothing. (And) out of the void, the universe was born as was the infinity. … The Hindu cosmos was infinite in extent; beyond our universe were innumerable other universes.” Hindus believe that one attains liberation (Nirvana) when she shuns material desires and ‘embraces the silence and nothingness of the soul. The Atman escapes the web of human desire and join the collective consciousness-the infinite soul that suffuses the universe, at once everywhere and nowhere at the same time; It is infinite and it is nothing.’ It is but natural that India as a society which actively explored the concepts of nothingness and infinity as abstract thoughts should invent numerals representing them.

The earliest source of Hindu or Indian mathematical thoughts can be traced to Vedic literature, whose contents are primarily religious and ceremonial. The appendices to the Vedas, known as Vedangas contain important mathematical formulations. The Vedangas are classified into six fields: phonetics, grammar, etymology, verse, astronomy and rituals. The last two appendices of the Vedas, known as ‘Jyotisutra’ and ‘Kalpasutra’ provide insight into mathematics of the period. Kalpasutra deals with rules of rituals and also prescribes measurements and methods of constructing altars for rituals, known as ‘Sulbasutras’. The earliest Sulabsutras were written around 600-800 BC. The need for these rules (sutras) grew out of the need to conform precisely to the shapes, sizes and orientations of altars prescribed for different rituals in the main text (Vedas). The geometry was essentially expressed in three ways: explicitly stated theorems, procedures for construction and algorithms related to the earlier two. It was here that Pythagorean theorem for right-angled triangle was stated much earlier than it was ascribed to the Greek mathematician Pythagoras. ?

The use of zero or similar symbols as a placeholder was believed to have known thousands of years ago in Maya civilization (present day Mexico) and to Babylonians. It was more like the use of present day ‘comma’ in a number to help reading the right meaning. The seventh century Indian mathematician Brahmagupta, for the first time, treated zero as a number, not merely as a placeholder. Thinking of zero as a number rather than a place holder was a fundamental shift in thought and this reincarnation of zero gave it all its power.??

?The classical period of Indian Mathematics began in the middle of the first millennium when the imperial Guptas rules most part of India and who encouraged study of arts, science and Mathematics. Two most important mathematicians of the time were Aryabhata (476-550 CE) and Brahmagupta (598-670 CE). Aryabhata wrote ‘Aryabhatiya’, a compilation of 33 verses presenting algorithms for calculating squares, cubes, square roots, cube roots and also geometry, arithmetic and algebra. The tenth verse gives the value pf ‘pi’ as the ratio 62832:20000, equivalent to 3.1416, the most accurate value that would be obtained for nearly a thousand years. Even today, the value of ‘pi’ as the ratio of 22:7 largely conforms to this value. There is enough evidence in the works of Aryabhata that while he did not use the symbol of zero, “the knowledge of zero was implicit in Aryabhata’s place-value system as a place holder for powers of ten with null coefficients”.

Brahmagupta was the other important mathematician of the period.? He composed, at the age of 30, ‘Brahmasphutasiddhanta’, a comprehensive treatise on astronomy. The book consists of 24 chapters with 1008 verses. Besides astronomy, it has chapters on Mathematics including Algebra, Trigonometry, geometry and arithmetic. This is the earliest known text to treat zero as a number in its own right, rather than as a mere placeholder. He also set rules of arithmetic (addition, multiplication and subtraction) involving zero with both the positive and negative numbers. As a number, zero was placed at the origin of the number line and, as a logical corollary, those remaining to the left of the origin were Negative numbers. With the appearance of negative numbers, Indian mathematics treated numbers and their interplay as independent entities in their own right, stripped of their geometric significance (line, squares, cubes, area etc.). This approach made sure that mathematicians no longer had to worry about mathematical operations making geometric sense, unlike the Greeks, and thus expanded their ability to explore, which gave birth to what we now know as algebra.

Brahmagupta gave rules for dividing numbers by each other which included the negative and zero. His rules for addition, subtraction and multiplication and division are the following:

?“[The sum] of two positives is positives, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal, it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positives, [and that] of two zeros zero.”

?“A negative minus zero is negative, a positive [minus zero] is positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added.”

?“The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.”

?“A positive divided by a positive or a negative divided by a negative is positive; a zero divided by zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative.”

“A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square root.”

?While all these rules held, Brahmagupta’s rule for division by zero (1 ÷ 0 = 0 or 0 ÷ 0 = 0) ran into trouble. The 12th Century mathematician Bhaskara corrected this result to infinity. “This fraction of which denominator is a cipher, is termed an infinite quantity…. There is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God.”, wrote Bhaskara. “God was found in infinity – and in zero”.

?While on the works of Brahmagupta, it would not be out of place to touch upon some of his achievements in Mathematics besides Zero. Among his important works are the solution to the quadratic equation of the type ax^2 + bx = c.

He gave the sum of the squares of the first?n?natural numbers as? n(n?+ 1)(2n?+ 1)/6? and the sum of the cubes of the first n natural numbers as?(n*(n?+ 1)/2)2.

Brahmagupta gave an algorithm for the whole number solution of equation of the form ax^2 +/- c = y^2, which are equations for hyperbolas and also known as Pell’s equation. These methods were later refined by Bhaskara into a cyclic method known as ‘chakravala’. He gave solution to the famous problem

(61x^2 + 1 = y^2). In the 17th century, Pierre Format set this problem as a challenge, the solution to which was found by Joseph-Louis Lagrange a hundred years later. But it was found that ‘chakravala’ algorithm was far superior to Lagrange’s approach. The smallest solution Bhaskara gave was

(x= 226,153,980, y = 1,766,319, 049).

Among other notable mathematicians of the period was Mahavira who wrote the earliest Sanskrit textbook entirely devoted to Mathematics rather than having mathematics as an adjunct to astronomy. There were other texts also, but the most influential among them were two works by Bhaskar II (1114-1185 CE) of the 12th century, the Lilavati and the Bijaganita, on arithmetic and algebra respectively. He is also credited with some concepts that would much later be used in development of calculus. Between 14th and 16th century, in Vijayanagar empire, specifically in modern day Kerala, the mathematical school of Madhava (1340-1425) became established. During this period proofs of many results handed down from centuries earlier and infinite series, particularly those related to trigonometric functions, were written. “The Story of Mathematics”, a publication of Princeton University Press has this to say of Madhava’s works:

?“His (Madhava) works on infinite series have been lost, but are quoted extensively by later writers in sixteenth century. Many results which have been named after European mathematicians may need to be suffixed by the name of Madhava. These include infinite polynomial expansions of sines and cosines which have been credited to Newton as well as small angle approximation formulas, which are part of the general Taylor Series. These would have allowed trigonometry tables to be drawn up to any desired accuracy: Madhava’s table were accurate to eight decimal places.”

?With this brief detour, we return to the topics of zero and infinity.

?The concepts of void (zero) and infinity were not easily acceptable to the western mind, the mathematicians / philosophers of the West nurtured in the philosophical thoughts of Aristotle that rejected these concepts. By the seventh century the West had withered, but the East was flourishing. The Indian numeral system with ‘zero’ moved to the Arabic world mostly through travel and commerce and was readily accepted there for its obvious advantages. The nomenclature ‘zero’ came through a series of transformation from the Indian ‘sunya’ to the Arabic ‘sifr’, later substituted by western scholars to a Latin sounding ‘zephirus’, the root for the ‘zero’. Some scholars changed it to ‘cifra’ which later became ‘cipher’. Though Christianity initially rejected zero, the trade would soon demand it. The man who introduced zero to the West firmly was the son of an Italian trader, better known as Fibonacci, made famous for the sequence named after him and the ‘golden ratio’ discussed earlier. Fibonacci had learnt mathematics from the Islamic scholars and was well acquainted with Indo-Arabic numeral system with zero and its advantages. However, the transition was not smooth. In 1299, Florence banned these numerals but had to yield to the pressure from trade, commerce and banking to reintroduce it. From Italy it spread to the entire Europe. ‘Zero had arrived – as has the void’.

Amir D. Aczel in his book ‘Finding Zero’ has narrated in some detail his search for physical evidence of origin of zero in India and some South-East Asian countries, particularly Cambodia. In his quest for zero he visited different ancient temples / inscriptions, read available translations of ancient texts and had discussions with contemporary Indian scholars on the concept of zero. He was introduced to the works of the second century Buddhist philosopher and teacher Nagarjuna. It gave him the logic of the East:

Anything is either True or not True; or both True and not True; or neither True nor not True. This logic defied what in mathematics called the ‘Law of excluded middle’, that is, what is true cannot be untrue at the same time; these are mutually exclusive. This is also the basis of proving theorems in mathematics by contradiction, a very standard approach. This eastern idea of four logical possibilities, called ‘catuskoti or chatuskoti’ in Sanskrit, allowed for gradation of truth and falsity unlike the West’s strict ‘either or’ bias. This, together with the concept of ‘Sunyata’ or ‘Void’, the author was convinced, was the origin of concepts of zero and infinity, as a continuum. He had this to say, “My thesis was that the number system we use today developed in the East because of religious, spiritual, philosophical and mystical reasons – not for the practical concern of trade and industry as in the West. In particular, nothingness – the Buddhist concept of Shunyata – and the Jain concept of extremely large numbers and infinity played paramount role”. Although the author referred to the East - not India and the Buddhist-not Hindu philosophical thoughts as the source of concepts like zero and infinity in his narration, his search for zero started in India. The Buddhist thoughts also originated in India as a reformist branch from the Hinduism. Possibly his study of different texts and discussions with different scholars were not conclusive. Further, Shunyata (void) and Ananta (the limitless and infinite) are essentially Hindu concepts which were adopted by Buddhist and other eastern philosophies and thought processes.

In the course of his search, he came across a (4x4) magic square with Hindu numerals of the period and with the number zero engraved in the doorway of 'Parsvanatha Temple', an eastern group of temples in Khajuraho. It had numbers 1 to 16, in a 4x4 grid such that no number is repeated and the sum of each row, column and diagonal was 34. Another inscription showed that the temple was built in 934 CE. Even as early as middle of tenth century, the mathematics in India was enough advanced as to make such sophisticated numerical tables. But was this the earliest zero? The author’s search took him to Gwalior where the inscription in the walls of 'Chaturbhuja Temple' showed the year of construction as 933 of a calendar whose starting point was 57 BCE which made the year of built as 876 CE. The grant of land for the same temple was shown as 270 hastas (a measure of length). The author claimed that this was the earliest zero available in India today. “So, by 876 CE, the Indians had the crucially important use of a place-holding zero at their disposal in a number system that from our modern vantage point was perfect”, the author wrote.

But was this the oldest available zero? ?Finding a reference to discovery of zero in Cambodia in a published paper of a French archaeologist George Coedes, the author traced the evidence of zero to the ruins of the temples of 'Trapang Prei' in the basin of Mekong River. This stone inscriptions were discovered by another French Archeologist in 1891 and much later catalogued by George Coedes as K-127 and K-128. The inscription in K-127 in old Khmer language translated into English read, “The caca era has reached 605 on the fifth day of waning moon…”? Since the caca dynasty had started in 78 CE, the date clearly referred to 683 CE. But what was of interest was the use of zero and place-value in ‘605’. This was two hundred years earlier than Gwalior ‘zero’ and claimed to be the evidence of earliest zero available. Those stones K-127 and K-128 are presently kept in the national museum of Cambodia in Phnom Penh.

Acceptance of zero as a number in its own right and the concept of infinity led to several further explorations and outcomes in Mathematics. A few important developments are briefly discussed below.

?Rene Descartes, a 17th century French mathematician philosopher adopted zero as the center of the number line, the concept of negative numbers as those lying to the left of the center as postulated in 6/7th century by Brahmagupta and gave one of the fundamental inventions of mathematics, the ‘Cartesian Coordinates’. The center of the Cartesian coordinates was zero (0,0) and any point in a plane (two dimensional) or space (three dimension) could be represented by its position with reference to zero, like say (3,4) or (1, 3, -5) etc. Originally Descartes did not extend the coordinates to negative numbers which was done by later scholars as a follow up. This was the birth of analytical geometry as a discipline in mathematics which essentially meant that any geometrical object – squares, triangles, circles, ellipse etc. – could be represented by a mathematical relationship (equation). Interestingly at a philosophical level, Descartes argued that nothing, not even knowledge, can be created out of nothing. To him all ideas – philosophies, all notions, concepts – already existed in people’s brains when they were born and the process of learning only uncovered that previously coded knowledge. Since we had a concept of infinite and perfect being in our minds, that infinite and perfect being – God - must exist. All other beings were some where between God and naught, a combination of zero and infinity.

?Although the rules for processing zero and infinity had been given and accepted by mathematics community, there were situations where these terms yielded abnormal, indeterminate outcomes. The process which encountered (0 ÷ 0) or adding of infinite numbers of decreasing finite terms (like 1 + ? + ?^2 + ?^3 + …+1/2^n + …) had inconsistent outcome. Although such situations were inconsistent to pure mathematical thinking, adding infinite things to each other often times yielded accurate finite results in physical world. This process was used by Johannes Kepler to estimate the volume of wine barrels. He mentally sliced the barrel to infinitely thin (Δx) slices of near zero width (Δx) and added the area of each tiny slice to assess the volume ∑f(x)Δx. This was the idea of differentiation and integration refined later. In the second half of 17th century, the British scientist Newton and a German lawyer and philosopher Leibniz gave to the world of mathematics a set of tools, combinedly known a Calculus, comprising the processes of differentiation, integration and solving differential equations. They worked independently for the same outcome, but what survived as calculus today is the methods propounded by Leibniz. Calculus offered the tools and concepts with which most real-world physical phenomena could be represented in the form of equations and analyzed. However, the conceptual issues of division by zero and adding infinite number of near zero (Δx) terms remained. This problem was finally addressed by the French scientist D’Alembert who introduced the concept of ‘limit’. This was formally adopted to the world of mathematics by three prominent mathematicians of the period, Augustine Cauchy, Bernhard Bolzano and Karl Weierstrass. With the concept of limit, one can carry on mathematical operations with finite quantities and one variable and arrive at the final outcome by making the variable approach either infinity or zero as the problem demands.

The other important development concerning understanding zero and infinity, considered as twins or two sides of the same coin, is the world of the ‘imaginary’, “a bizarre world where circles are lines, lines are circles and infinity and zero sit on opposite poles”. The solution to a quadratic equation of the form (ax^2 + bx + c = 0) has been known from centuries since the days of Brahmagupta in 7th century. The roots this equation is given by

(x=(-b±√(b^2-4ac))/2a). Clearly one root of this equation will be negative. Also, what happens if the second part of the numerator is negative? There could also be quadratic equations where it is impossible to find the value of x, such as (x^2 + 1 = 0). Here x can take two values; x = -√(-1,) and x=√(-1) . Square root of negative number was inconsistent with mathematical thinking of the time since squares of both positive numbers and negative numbers are positive. The 12th century mathematician Bhaskara wrote, ‘there is no square root of a negative number, for a negative number is not a square’. To overcome the above intractable problem, the square root of (-1) was defined as an imaginary number (i). In the hands of mathematicians, particularly in algebra discipline, i soon became a potent tool to solve polynomials of any degree, n. Although there were those who were skeptical of imaginary numbers, numbers with imaginary components (x + i*y), called Complex numbers were in use as early as the 16th century.? Carl Friedrich Gauss, famous German mathematician, proved that a polynomial of degree 'n' would have 'n' roots only if one accepts both the real and imaginary numbers. In 1830, Gauss constructed the complex plain using the Cartesian grid where the real part was represented by x-axis (horizontal axis) and the imaginary part by the y-axis (vertical axis). The association of zero and infinity to the complex plain was constructed by the German mathematician Riemann, a student of Gauss, by merging projective geometry with the complex numbers and complex plain. The outcome was that lines and circles in the real plane became circles and lines, respectively in projection on to the complex plain and vice versa and most importantly, zero and infinity could be seen as opposite poles in the Riemann’s complex sphere. They could switch place and have equal and opposite powers and properties. ‘The infinity was no longer mystical; it became an ordinary number. But in the deepest infinity – nestled within the vast continuum of numbers- zero kept appearing. Most appalling of all, infinity itself can be zero’.

To conclude, although, as of now, the physical evidence of earliest zero has been traced to Cambodia, it is accepted by the mathematics community that the origin and use of the concept of zero (and infinity) and the place-value system can be traced to the seminal works of Aryabhata and Brahmagupta as early as in 5/6th century. Given that religion and culture of the kingdom of Cambodia was highly influenced by India (Cambodia had Hindu rulers for centuries) and there were regular exchange of ideas, people and trade between two regions, there is possibility that these ideas of mathematics travelled from India to the entire eastern region. However, with the presence of large volume of original works dealing with zero, infinity and other areas of mathematics, India is acknowledged as the original source which gave to the world of mathematics the modern numerals with zero, place-value system and infinity.

References:

1.????? A History of Mathematics; Victor J Katz, published by Pearson

2.????? The story of mathematics: Richard Mankiewicz, Princeton University Press

3.????? Zero The Biography of a Dangerous Idea: Charles Seife, Penguin Books

4.????? Finding Zero: Amir D. Aczel, Macmillan

5.????? A Tour of Calculus: David Berlinski, Vintage Books