The History of Numbers - Part 1
Numbers and counting have become an integral part of our everyday life, especially when we take into account the modern computer. These words you are reading have been recorded on a computer using a code of ones and zeros. It is an interesting story how these digits have come to dominate our world.
Numbers Around the World
Presently, the earliest known archaeological evidence of any form of writing or counting are scratch marks on a bone from 150,000 years ago. But the first really solid evidence of counting, in the form of the number one, is from a mere twenty-thousand years ago. An ishango bone was found in the Congo with two identical markings of sixty scratches each and equally numbered groups on the back. These markings are a certain indication of counting and they mark a defining moment in western civilization. 1
[A-GRAPHIC OF Bone w markings]
Zoologists tell us that mammals other than humans are only able to count up to three or four, while our early ancestors were able to count further. They believed that the necessity for numbers became more apparent when humans started to build their own houses, as opposed to living in caves and the like.
Anthropologists tell us that in Suma, in about 4,000 BCE, Sumerians used tokens to represent numbers, an improvement over notches in a stick or bone. A very important development from using tokens to represent numbers was that in addition to adding tokens you can also take away, giving birth to arithmetic, an event of major significance. The Sumerian's tokens made possible the arithmetic required for them to assess wealth, calculate profit and loss and even more importantly, to collect taxes, as well as keep permanent records. The standard belief is that in this way numbers became the world's first writings and thus accounting was born.
More primitive societies, such as the Wiligree of Central Australia, never used numbers, nor felt the need for them. We may ask, why then did the Sumerians on the other side of the world feel the need for simple mathematics? The answer of course, was because they lived in cities which required organizing. For example, grain needed to be stored and determining how much each citizen received required arithmetic.
Egyptians loved all big things, such as big buildings, big statues and big armies. They developed numbers of drudgery for everyday labor and large numbers for aristocrats, such as a thousand, ten thousand and even a million. The Egyptians transformation of using "one" from counting things to measuring things was of great significance.
Their enthusiasm for building required accurate measurements so they defined their own version of "one." A cubit was defined as the length of a mans arm from elbow to finger tips plus the width of his palm. Using this standardized measure of "one" the Egyptians completed vast construction projects, such as their great pyramids, with astonishing accuracy.
[B-GRAPHIC OF PYRAMID]
Two and a half thousand years ago, in 520 BCE, Pythagorus founded his vegetarian school of math in Greece. Pythagorus was intrigued by whole numbers, noticing that pleasing harmonies are combinations of whole numbers. Convinced that the number one was the basis of the universe, he tried to make all three sides of a triangle an exact number of units, a feat which he was not able to accomplish. He was thus defeated by his own favorite geometrical shape, one for which he would be forever famous.
His Pythagorean theorem has been credited to him, even though ancient Indian texts, the Sulva Sutras (800 BCE) and the Shatapatha Brahmana (8th to 6th centuries BCE) prove that this theorem was known in India some two thousand years before his birth.
Later in the third century BCE, Archimedes, the renowned Greek scientist, who loved to play games with numbers, entered the realm of the unimaginable, trying to calculate such things as how many grains of sand would fill the entire universe. Some of these intellectual exercises proved to be useful, such as turning a sphere into a cylinder. His formula was later used to take a globe and turn it into a flat map.
Romans invading Greece were interested in power, not abstract mathematics. They killed Archimedes in 212 BCE and thereby impeded the development of mathematics. Their system of Roman numerals was too complicated for calculating, so actual counting had to be done on a counting board, an early form of the abacus.
Although the usage of the Roman numeral system spread all over Europe and remained the dominant numeral system for more than five hundred years, not a single Roman mathematician is celebrated today. The Romans were more interested in using numbers to record their conquests and count dead bodies.
Numbers in Early India
In India, emphasis was not on military organization but in finding enlightenment. Indians, as early as 500 BCE, devised a system of different symbols for every number from one to nine, a system that came to be called Arabic numerals, because they spread first to Islamic countries before reaching Europe centuries later.
What is historically known goes back to the days of the Harappan civilization (2,600-3,000 BCE). Since this Indian civilization delved into commerce and cultural activities, it was only natural that they devise systems of weights and measurements. For example a bronze rod marked in units of 0.367 inches was discovered and points to the degree of accuracy they demanded. Evidently, such accuracy was required for town planning and construction projects. Weights corresponding to units of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200 and 500 have been discovered and they obviously played important parts in the development of trade and commerce.
[C-GRAPHIC OF Weights from Harappa]
It seems clear from the early Sanskrit works on mathematics that the insistent demand of the times was there, for these books are full of problems of trade and social relationships involving complicated calculations. There are problems dealing with taxation, debt and interest, problems of partnership, barter and exchange, and the calculation of the fineness of gold. The complexities of society, government operations and extensive trade required simpler methods of calculation.
Earliest Indian Literary and Archaeological References
When we discuss the numerals of today's decimal number system we usually refer to them as "Arabian numbers." Their origin, however, is in India, where they were first published in the Lokavibhaga on the 28th of August 458 AD. This Jain astronomical work, Lokavibhaga or “Parts of the Universe,” is the earliest document clearly exhibiting familiarity with the decimal system. One section of this same work gives detailed astronomical observations that confirm to modern scholars that this was written on the date it claimed to be written: 25 August 458 CE (Julian calendar). As Ifrah2 points out, this information not only allows us to date the document with precision, but also proves its authenticity. Should anyone doubt this astronomical information, it should be pointed out that to falsify such data requires a much greater skill than it does to make the original calculations.
The origin of the modern decimal-based place value system is ascribed to the Indian mathematician Aryabhata I, 498 CE. Using Sanskrit numeral words for the digits, Aryabhata stated “Sthanam sthanam dasa gunam”or “place to place is ten times in value.” The oldest record of this value place assignment is in a document recorded in 594 CE, a donation charter of Dadda III of Sankheda in the Bharukachcha region.
The earliest recorded inscription of decimal digits to include the symbol for the digit zero, a small circle, was found at the Chaturbhuja Temple at Gwalior, India, dated 876 CE. This Sanskrit inscription states that a garden was planted to produce flowers for temple worship and calculations were needed to assure they had enough flowers. Fifty garlands are mentioned (line 20), here 50 and 270 are written with zero. It is accepted as the undisputed proof of the first use of zero.
[D-GRAPHIC - Chaturbhuja Inscription]
The usage of zero along with the other nine digits opened up a whole new world of science for the Indians. Indeed Indian astronomers were centuries ahead of the Christian world. The Indian scientists discovered that the earth spins on its axis and moves around the sun, a fact that Copernicus in Europe didn't understand until a thousand years later—a discovery that he would have been persecuted for, had he lived longer.
From these and other sources there can be no doubt that our modern system of arithmetic—differing only in variations on the symbols used for the digits and minor details of computational schemes—originated in India at least by 510 CE and quite possibly by 458 CE.
The first sign that the Indian numerals were moving west comes from a source which predates the rise of the Arab nations. In 662 AD Severus Sebokht, a Nestorian bishop who lived in Keneshra on the Euphrates river, wrote regarding the Indian system of calculation with decimal numerals:
" ... more ingenious than those of the Greeks and the Babylonians, and of their valuable methods of calculation which surpass description..." 3
This passage clearly indicates that knowledge of the Indian number system was known in lands soon to become part of the Arab world as early as the seventh century. The passage itself, of course, would certainly suggest that few people in that part of the world knew anything of the system. Severus Sebokht as a Christian bishop would have been interested in calculating the date of Easter (a problem to Christian churches for many hundreds of years). This may have encouraged him to find out about the astronomy works of the Indians and in these, of course, he would find the arithmetic of the nine symbols.
The Decimal Number System
The Indian numerals are elements of Sanskrit and existed in several variants well before their formal publication during the late Gupta Period (c. 320-540 CE). In contrast to all earlier number systems, the Indian numerals did not relate to fingers, pebbles, sticks or other physical objects.
The development of this system hinged on three key abstract (and certainly non-intuitive) principles: (a) The idea of attaching to each basic figure graphical signs which were removed from all intuitive associations, and did not visually evoke the units they represented; (b) The idea of adopting the principle according to which the basic figures have a value which depends on the position they occupy in the representation of a number; and (c) The idea of a fully operational zero, filling the empty spaces of missing units and at the same time having the meaning of a null number.4
The great intellectual achievement of the Indian number system can be appreciated when it is recognized what it means to abandon the representation of numbers through physical objects. It indicates that Indian priest-scientists thought of numbers as an intellectual concept, something abstract rather than concrete. This is a prerequisite for progress in mathematics and sciencein general, because the introduction of irrational numbers such as "pi," the number needed to calculate the area inside a circle, or the use of imaginary numbers is impossible unless the link between numbers and physical objects is broken.
[E-GRAPHIC of various forms of the Decimal Numbers evolving]
The Indian number system is exclusively a base 10 system, in contrast to theBabylonian (modern-day Iraq) system, which was base 60; for example, the calculation of time in seconds, minutes and hours. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system (based on 60, not 10). Despite the invention of zero as a placeholder, the Babylonians never quite discovered zero as a number.
The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. They added the "space" symbol for the zero in about 400 BC. However, this effort to save the first place-value number system did not overcome its other problems and the rise of Alexandria spelled the end of the Babylonian number system and its cuneiform (hieroglyphic like) numbers.
It is remarkable that the rise of a civilization as advanced as Alexandria also meant the end of a place-value number system in Europe for nearly 2,000 years. Neither Egypt nor Greece nor Rome had a place-value number system, and throughout medieval times Europe used the absolute value number system of Rome (Roman Numerals). This held back the development of mathematics in Europe and meant that before the period of Enlightenment of the 17th century, the great mathematical discoveries were made elsewhere in East Asia and in Central America.
The Mayans in Central America independently invented zero in the fourth century CE. Their priest-astronomers used a snail-shell-like symbol to fill gaps in the (almost) base-20 positional 'long-count' system they used to calculate their calendar. They were highly skilled mathematicians, astronomers, artists and architects. However, they failed to make other key discoveries and inventions that might have helped their culture survive. The Mayan culture collapsed mysteriously around 900 CE. Both the Babylonians and the Mayans found zero the symbol, yet missed zero the number.
Although China independently invented place value, they didn't make the leap to zero until it was introduced to them by a Buddhist astronomer from India in 718 CE.
Zero becomes a real number
The concept of zero as a number and not merely a symbol for separation is attributed to India where by the 9th century CE practical calculations were carried out using zero, which was treated like any other number, even in the case of division.
The story of zero is actually a story of two zeroes: zero as a symbol to represent nothing and zero as a number that can be used in calculations and has its own mathematical properties.
It has been commented that in India, the concept of nothing is important in its early religion and philosophy and so it was much more natural to have a symbol for it than for the Latin (Roman) and Greek systems. The rules for the use of zero were written down first by Brahmagupta, in his book “Brahmasphutha Siddhanta” (The Opening of the Universe) in the year 628 CE. Here Brahmagupta considers not only zero, but negative numbers, and the algebraic rules for the elementary operations of arithmetic with such numbers.
"The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol, but helpful power, is the characteristic of the Hindu race from whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power." - G. B. Halsted5
A very important distinction for the Indian symbol for zero, is that, unlike the Babylonian and Mayan zero, the Indian zero symbol came to be understood as meaning nothing.
As the Indian decimal zero and its new mathematics spread from the Arab world to Europe in the Middle Ages, words derived from ṣifr and zephyrus came to refer to calculation, as well as to privileged knowledge and secret codes. Records show that the ancient Greeks seemed unsure about the status of zero as a number. They asked themselves, "How can nothing be something?" This lead to philosophical and, by the Medieval period, religious arguments about the nature and existence of zero and the vacuum.
The word "zero" came via the French word zéro, and cipher came from the Arabic word safira which means "it was empty." Also sifr, meaning "zero" or "nothing," was the translation for the Sanskrit word śūnya, which means void or empty.
The number zero was especially regarded with suspicion in Europe, so much so that the word cipher for zero became a word for secret code in modern usage. It is very likely a linguistic memory of the time when using decimal arithmetic was deemed evidence of dabbling in the occult, which was potentially punishable by the all-powerful Catholic Church with death.6
END OF PART ONE
Part Two continues with the spread and acceptance of the decimal number system in the Arabic countries and later Europe.
This author has taken the liberty of directly quoting from some of the references given here, such as the quotations by some scholars or historians and, when necessary actual descriptions of the mathematical notations (rather than paraphrasing them). I am indebted to the original authors for their scholarly writings, without which justice could not have been done in narrating the contributions of Indians in Mathematics through history.
References for Part I
1-- BBC video presentation with Terry Jones, "The Story of 1," Impossible Pictures for the BBC, 2005.
2-- Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated from French by David Vellos, E. F. Harding, Sophie Wood and Ian Monk, John Wiley and Sons, New York, 2000.
3-- B. Datta and A. N. Singh, History of Hindu Mathematics, 1933.
4--Ifrah Ibid, pg. 346.
5-- G.B. Halsted, On the Foundation and Technique of Arithmetic, p. 20, Chicago, 1912. Quoted in Hogben’s Mathematics for the Million, London, 1942.
6--Ifrah Ibid, p. 588-589.