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Ubiquitous Indo Arabic Numerals

Ubiquitous Indo Arabic Numerals

The history of and is a fascinating one. 0 Read on to find out more. 1 Story 4 9 Cover 8 2 3 6

Number Sense It is not if numbers have been there all though the human history. Munduruku, ALK in to a supermarket in China, indigenous hunter-gatherers in the Brazilian Monkey even though the hoardings would Amazon, have no words for numbers Counting? Wbe unintelligible, the numbers on beyond five in their language while Piraha, the currency and coins and the telephone another tribal community, do not have any numbers would be easily readable. Take name for numbers beyond three. But, they a stroll in the marketplace at Barcelona, have a sense of numbers – they can the market babble wafting around you discriminate between a heap having more might be incomprehensible, but the price than the other even though they may not tags are easy to read. Walk into any count one by one. remote region in Africa and you find that the written numbers – numerals – make sense, even though we cannot make head or tail of their scripts and languages. The world over, counting styles are different, languages are varied, scripts are distinct, but symbols 0,1,2,3,4,5,6,7,8,9 are easily recognizable everywhere. The Indo- have hegemonized the world. It is as if the world speaks in various languages and writes in a of scripts, but works in only one kind of number symbols. Ubiquity of Indo-Arabic numerals is more than obvious.

People of the Munduruku tribe have no words for numbers beyond five

SCIENCE REPORTER, APRIL 2012 8 Cover Story .. VENKATESWARAN 5 UbiquitousUbiquitous 7 IndoIndo ArabicArabic NumeralsNumerals

Not just humans, it appears that thought he could deceive the crow, but ever since the case of “Clever Hans”, a animals also have, albeit primitive, a the crow did not fall into this trap and horse that is said to have responded to number sense. An eighteenth century carefully waited for the second man to questions requiring mathematical anecdote aptly summarizes the number come out before returning. Neither did calculations by tapping his hoof, but later sense observed in the animal world. A three, four, or five men fool the clever bird. revealed to be a case of subtle nobleman wanted to shoot down a crow Each time, the crow would wait until all manipulation by the handler. As Ray Hyman that had built its nest atop a tower on his the hunters had departed. Eventually, the puts it, “Hans was responding to a simple, domain. However, whenever he hunters came as a party of six. When five involuntary postural adjustment by the approached the tower, the bird flew out of them had left the tower, the bird, not so questioner, which was his cue to start of gun range, and waited until the man numerate after all, confidently came tapping, and an unconscious, almost departed. As soon as he left, it returned to back, and was shot down by the sixth imperceptible head movement, which its nest. The man decided to ask a hunter! was his cue to stop.” neighbour for help. The two hunters Scientists have been sceptical of Nevertheless, subsequent entered the tower together and later only claims of mathematical abilities in animals experiments have revealed an one of them came out. The nobleman

Scientists have been sceptical of claims of mathematical abilities in animals ever since the case Bone with of “Clever Hans”, a horse that is said to found in have responded to Ishango, a questions requiring village in Africa mathematical calculations by tapping his hoof.

9 SCIENCE REPORTER, APRIL 2012 Cover Story

LEFT HAND 1 2 3 4 5

RIGHT HAND 15 10 25 20 5

unexpected capacity for ‘quantity discrimination’ in animals as varied as bees, salamanders, rats, dolphins and primates, suggesting that mathematical abilities could be more fundamental in biology than previously thought. Trained Counting on fingers monkeys can, it seems, perform rudimentary maths – they can compare Babylonian counting two heaps of dots and identify which one long notch of double length separated the has more. first twenty-five marks from the rest. It was Number sense, though not strictly in clear that the carving and grouping were the sense of counting, could be important deliberate human action of counting for animals to survive in the wild. If a something; perhaps the number of spears chimpanzee is unable to look up a tree they had or the mammoths that they had and quantify the amount of ripe fruit most hunted in the season, rather than mere likely it would go hungry. If a lion wants to scribble. attack another pack of lions it has to make This is not an isolated case. Soon tally a judgment how many are in each side. If marks such as Lebombo bone, a piece of a grazer is unable to judge the relative baboon fibula, found in the caves of abundance of food in two patches, Lebombo Mountains in Swaziland, dated perhaps it would die of hunger. to be 35,000 years old, with 29 notches Evolutionary psychologists posit that such were found. A similar bone with 29 notches, evolutionary pressures, perhaps, have led and about 18000 years old, has been to evolution of sophisticated procedures discovered in Nicobar in India. for number sense in animals. Not Archaeologists interpret it as the counting surprisingly, primates and human toddlers of the cycle of waxing and waning of the too exhibit this primitive number sense. moon or perhaps menstrual period. At a fundamental level, these Birth of Counting archaeological artefacts attest that early When Karl Absolon, a Czech archaeologist, humans were counting rather than merely discovered a wolf-bone in the dust and making an estimate and the pattern debris of a 30,000-year-old Stone Age recognition had evolved into counting. settlement, it became an instant sensation, Further, whatever that they were counting The Mayan so much so that the London Illustrated was important enough for them to warrant Not all such tally sticks excavated so system of News, a weekly tabloid printed in London, the keeping of records. Every notch, far had just a few dreary lines carved on counting carried its picture in its 2 October 1937 called tally mark, precisely represented them. , a tiny 10 cm curved issue. This bone fragment discovered at something that they counted, implying a bone excavated at the Semliki River in Zaire Dolni Vestonice, a Palaeolithic site, is touted rudimentary version of an important dated to about 20,000 years old, is a to be one of the earliest evidences that mathematical cognition – the one-to-one significant find. The three rows of notched Stone Age mammoth hunters counted correspondence between elements of columns engraved on it are clearly something – said to be the dawn of two different sets of objects, or cardinality patterned and two of them add up to 60. . of numbers, in this case between the set Also one of these two rows is grouped in The bone had fifty-five little ‘tally’ of notches on the bone and the set of to 11,13,17,19 – all prime numbers notches carefully carved and they were whatever the prehistoric humans were between ten and twenty. The third row has arranged into groups of five. An additional counting. a particular pattern: 3 followed by its SCIENCE REPORTER, APRIL 2012 10 CoverCoverCoverCover Story Story

Counting tokens in stone

as “five ten seven”. On the other hand, and . Ten also has a unique symbol. In many European systems of number words like manner, fifty is and hundred is , five are irregular up to 100. For example in hundred is and thousand is . French, 92 is said as “four twenty twelve,” Of particular interest is the counting corresponding to 4*20 + 12. system of Siriona Indians of Bolivia and the But counting is not universal, Brazilian Yanoama. Their counting goes like nor counting upon ten fingers. “one,” “two,” “two and one,” “two and With the scientific Comparative studies have shown that two,” “two and two and one,” and so forth, revolution setting in counting of fingers, excluding the thumb, very similar to what mathematicians would is also an established practice among call binary counting. Similarly, in Australian , fetters fell and some North American aboriginal tribes and Aboriginal language, Kala Lagaw , the soon Indo-Arabic numerals thus numbers are grouped into fours numbers one through six are urapon, instead of five – that is, quaternary counting. ukasar, ukasar-urapon, ukasar-ukasar, became the only form to be Archaeologists contend that one of the ukasar-ukasar-urapon, ukasar-ukasar- used. ancient Indian numerals, Gandharan ukasar. But why two? We are yet to fathom. numerals, are partially Hundreds and thousands of quaternary – base 4 – counting. records have been excavated The Ndom language of Papua New from the ancient Sumer region, presently double, then 4, followed by its double, then Guinea is reported to be – that is in Iraq. In these written records preserved 10, followed by its half, and indicates more base 6. The Yuki Tribes in Americas count in clay, there are symbols only for 1, 10, 60 sophisticated arithmetical reasoning. the space between the fingers and have and 3600, suggesting that the ancient evolved an -base 8-number system. Sumerians used – base sixty – Variety of Counting Systems Consider the four fingers of your left hand: number system. But why 60, and not 10 The bunching of lines into fives by the Dolni ignoring the thumb. The joints divide each (fingers), or 20 (fingers and toes), or 5 Vestonice Stone Age people perhaps is finger into three parts; using the thumb as (fingers on one hand)? an indication of the primitive attempt at a pointer we can count up to 12 resulting Sixty seems like an odd choice. development of quinary counting system. in the dozanal counting. Perhaps, Sumerians chose sixty for it has so You count five fingers in one hand and Variations in terms of which parts of a many . It can be divided by 2, 3, 4, group them. Now six can be said to be hand people count with and to what other 5, 6, 10, 12, 15, 20 and 30. This makes it ‘one hand and one’, seven ‘one hand and body parts they extend counting are easier when one is doing and two’, thirteen becomes two hands and evident. An ethno-mathematical study of mental . Another theory three, and so on. Many tribal dialects have merchants from Maharashtra indicates speculates that Sumerians counted to 60 such quinary counting number names, that they use five fingers in one hand to using both hands like we do but with a including Niam Niam dialect of Central count from one to five and the fingers on difference: they used finger segments Africa. the other to keep tab of multiplies of five. instead of whole fingers. Take your left hand Tally is, in fact, the simplest numeral Body counting systems of Highland New and exclude the thumb finger. In the rest system, unary , in which Guinea such as the Oksapmin counting four fingers there are 12 segments. Now every is represented by a system make use of additional parts like instead of using the left thumb to point, we corresponding number of symbols. To write the wrist, elbow, shoulder, head and so on can use the five fingers of the right hand thirteen in this system you will have to carve counting up to 27. as pointers. Now, 12 x 5 = 60. lines thirteen times, which is indeed Mayans had a , that is base The Babylonians, who made great laborious. 20, number system, so were the Dzongkha, advances in mathematics and , Scholars speculate that using ten of Bhutan and Munda, a tribal community embraced the Sumerian sexagesimal fingers of both hands for counting resulted in India. Roman system is bi-quinary, that is base. Though Egyptians and later Greeks in base ten – that is, counting part base 5, part base 10. Roman number based their number system on ten, for system. In many languages, including Tamil symbols go like this: , II, III and IIII (in olden measuring time or computing angle they and Chinese, number names are clearly times they did not use IV for four). Then five continued to use the Babylonian base ten; 15 is spoken as “ten five” and 57 has a unique symbol V. Now it is VI, VII, VIII, IX sexagesimal method. This is why today we 11 SCIENCE REPORTER, APRIL 2012 Cover Story

Fibonacci

Al Khwarizmi’ Brahmagupta Pope Sylvester II

In 825 AD, Al Khwarizmi wrote a book “Calculations with Hindu Numerals” (Ketab Fi Istimal al adad al-hind). This book made the Indian system popular in the Arab world. The significance of this system was realized by , a European scholar, who learnt the Indian numeral system from the . Back in 967 CE, for example, the monk who became Pope Sylvester II figured out that the counting would be easier with a zero sign. Pope Silvester II was hounded for his advocacy of zero. Zero was a bizarre number. Brahmagupta found that zero multiplied by any number is zero; zero added or subtracted from any number makes no difference.

have 360 degrees of a circle and 60 heel bone representing 10, a coiled rope three rather than carve one hundred and seconds in a minute and 60 minutes in an representing 100, a lotus representing 1000 twenty three times tally marks. hour. In fact, it should be noted that in and so on. To write down a number, say Mayans had just three symbols: zero ancient India the time was not counted in 123, with this scheme, all an Egyptian (shell shape), one (a ) and five (a ). base 60 units. In ancient India, one day scribe had to do was write just six symbols For example, nineteen (19) is written as four was divided into 30 muhuratas. Each – coiled rope for hundred; two heal bones dots in a horizontal row above three muhurat was 2 nadikas or 30 kalas. for twenty and three vertical marks for unit horizontal lines stacked one upon the other. Surely two plus two is four; no one can say it is not, whatever base we use. Enthralment of European Mathematical truths cannot be influenced by culture or ideology. However, our traders and scholars at approach to mathematics is very much the simple and concise influenced by culture. Indo-Arabic numerals is captured elegantly in a Numerals – of Numbers woodcut printed in Scribbling lines for each count is laborious. Margarita Philosophica Soon people found that there was an easy published in 1503. way to write numbers. Archaeologists “Arithmetica”, the deity claim that even before writing became prolific, number systems had already been of mathematics, is shown well established. Transcribing the oral watching a competition number system into written form was a between an “abacist”, simple task: people just needed to figure out a coding method whereby scribes one who uses Roman could set the numbers down in a more numerals for permanent form. computation, and an In one of the earliest numerals, “algorist”, one who uses Egyptian, pictures stood for numbers; a single vertical mark representing a unit, a Indo-Arabic numerals.

SCIENCE REPORTER, APRIL 2012 12 Cover Story The world over, counting styles are different, languages are varied, ROMAN scripts are distinct, but numeral symbols 0,1,2,3,4,5,6,7,8,9 are NUMBERS easily recognizable everywhere. The Indo-Arabic numerals have hegemonized the world.

I = One in Mayan vigesimal it increases as 1 , 20, the areas of rectangles when they 400, 8000 and so on; in sexagesimal the multiplied two different values. So, numbers V = Five place value would be 1, 60, 360, 10800 were linked to geometry. Lengths, areas, and so on. In contrast, , and volumes resulting from geometrical X = Ten which had their lineage in the Egyptian constructions – all numbers had to be numerals, are not a place value system. positive; hence there was no place for L = Fifty C, for example, is 100, irrespective of the negative numbers. Obviously, lines with zero place it holds in the string of roman length, plot of land with zero area or a C = Hundred numerals; thus, CLXXX is 180, and cone with zero volume were unreal and MMDCCXXV is 2725. hence zero had no place in their D = Five hundred configuration of things. From Counting to Numbers Unlike the Greeks, the Indian system M = Thousand Discovered in the village of Bakhshali (now viewed numbers stripped of their in ), a catch of birch bark geometric significance and hence did not manuscript written in Gatha dialect, mixture worry whether the mathematical of and , is one of the oldest operations made any geometric sense. On the other hand the Babylonian Indian mathematical texts found so far. You cannot subtract four acres of land from sexagesimal system required fifty-nine Dated to be as old as 200 CE, the three acres of land; but you can do an distinct symbols. A vertical wedge manuscript provides evidence of the use algebraic sum three minus four and get – represented 1, and a fat horizontal one of just ten symbols for writing any number, 1 (negative number) as a legitimate represented 10. Suppose you wanted to use of zero and negative numbers in Indian mathematical answer. If your thoughts are write sixty-three. Then what do you do? mathematics. fettered by geometric notions, then Sixty-three is sixty + three. So you The Greeks, like the Romans, inherited negative numbers obviously make no combine the symbol for sixty and symbols their mathematics from Egyptians, and had sense. But to ancient Indian for three: Now seventy is actually 60+10. no place for either zero or negative mathematicians unencumbered by This is actually 60x1+10. So you take the numbers. For ancient Egyptians, tilling the geometric notions, negative numbers symbol for 1 and symbol for 10 and write land in the flood plains of the Nile was the made perfect sense. it as . Should we try writing 124 in the most profitable venture. Each household This was the birth of what we now Babylonian system? 124 is actually thus had a specific parcel of land assigned know as algebra. This mindset had a 2x60+4; so you take the symbol for ‘2’ to them and delineated by boundary significant import. Indians could embrace and the symbol for ‘4’ and write them as. markers. Come the next flood, the zero and negative numbers without any Can you try writing 2012 in the Babylonian boundary markers were washed away or hesitation. Indeed, it was in India (and in system? damaged. Therefore, pharaohs assigned China) that negative numbers first The Babylonian system is what surveyors to assess the land area and reset appeared. Brahmagupta, an Indian mathematicians would call as the place the boundary markers, after every flood; mathematician of the seventh century, value system. What does ‘2’ in ‘245’ in thus geometry was born. These surveyors alluded to positive and negative numbers modern number system denote? It is not understood that one could determine the as ‘profits’ and ‘losses’. He also gave rules just two. It is actually 2 x 100, that is, two area of a plot of land by dividing it into for doing arithmetic with negative hundred. Similarly, ‘4’ in 245 is actually 4 x rectangles and triangles. Further, Egyptian numbers. He said, “Positive divided by 10, that is, forty. The ‘2’ has a mathematicians also learnt to measure the positive, or negative by negative, is value depending upon the place it holds volumes of objects like pyramids. positive, Positive divided by negative is in the string of symbols. The notation ‘2’ in Length defined the number for negative, Negative divided by positive is this system could be just two, or twenty Greek mathematicians. When a surveyor negative” – arithmetic rules that we or two hundred depending on the placed the measuring rod on the ground, recognize today. ‘position’ it occupies. In like manner, the the ‘interval’ between the edge to the Zero as a place holder is a Babylonian number system was also a point marked ‘one’ represented one unit. requirement in any positional number place value system. The notation for Thus, all numbers were just multiples of system, unlike the non-positional Roman three in Babylonian system could mean the measuring rod. and numerals. The complex and sophisticated just three, or one hundred and eighty (3x division made sense when computing the sexagesimal (base-60) positional numeral 60) or 1080 (3x60x60) depending upon areas and volumes; you had a plot of system of the Babylonian mathematics the place it holds. land 3 feet wide and 4 feet long, then showed a gap in between the numerals Traditional Chinese numerals are also the area of the plot was 3 x 4 that is 12 to indicate absence of positional value a place value system. In decimal place square feet. (or zero); later a symbol value system, from right to left the value of Thus, Egyptians and later Greeks (indicated by two slant wedges) was used the symbol increases as 1, 10, 100 etc and imagined squares in square numbers and to fill the gap. 13 SCIENCE REPORTER, APRIL 2012 Cover Story What is two competitors, the abacist is a monk 0 ÷ 0 and and the algorist a worldly scholar. Arithmetica appears to favour the algorist; 1 ÷ 0? her own clothes are covered in Indian Today, we numerals, and she looks approvingly at the just banish algorist’ progress marking the triumph and this the final acceptance of Indo-Arabic question by numerals in Europe. Nevertheless, the spread of Indo- saying it is Arabic numerals in Europe was not smooth. ‘inde- Fibonacci’s book appeared during the terminate’. period of the Crusades against Islam, and the clergy was suspicious of anything with In Indo-Arabic numerals, in order to Arab connotations. Some, in fact, express the number 206, a symbol is considered the new arithmetic the Devil’s needed to show that there are no tens. work precisely because it was so ingenious The digit 0 serves this purpose. Mayans also – Indo-Arabic numerals were banned and had a symbol for zero, a seashell, much similar manner, all numbers are written by prohibited. Zero in particular was an before Indian numerals. But these are just0various combinations of these ten symbols. obstacle; even Fibonacci referred to the zero as place holder; a punctuation mark India and Arabia have had trade “nine [Indian numerals] and the sign 0”. This – not zero the concept or zero the number. and interaction from times immemorial. suspicion led to a series of actions against Just as three minus four was now a Arabian travellers and traders learnt the zero. number (negative number), three minus Indian system of numbers and found it Back in 967 CE, for example, the monk three could also be construed as a easier than Roman or . In who became Pope Sylvester II figured out number – zero. Since zero was equal to 825 AD, Al Khwarizmi wrote a book that the counting would be easier with a 3–3, then it had to be placed between “Calculations with Hindu Numerals” (Ketab zero sign. He was accused of hobnobbing positive one (2 – 1) and negative one (2 – Fi Istimal al adad al-hind). This book made with evil spirits and forced to abjure it. In 3). Nothing else made sense. No longer the Indian system popular in the Arab 1299, Florence city authorities banned the could zero sit next to nine, just as it does in world. use of Indo-Arabic numbers and in 1348 mobile telephone keyboards; zero had a The significance of this system was the ecclesiastical authorities of Padua distinct position in the number line. This was realized by Fibonacci, a European scholar, issued prohibition against the use of zero. a crucial breakthrough in mathematical who learnt the Indian numeral system from The fear of Indo-Arabic numerals is thinking. The famous mathematician the Arabs. He strongly advocated the use revealed through the etymology of some Alfred North Whitehead noted, “The point of Indo-Arabic numerals in his book Liber modern words. Sunya was translated as about zero is that we do not need to use it Abaci in 1202 and showed that complex sifr – meaning void in Arabic, which was in the operations of daily life. No one goes computations could be made using Indo- transliterated as zephirum in . The out to buy zero fish. It is in a way the most Arabic numerals without recourse to the Portuguese derivative word chifre, meant civilized of all the cardinals, and its use is ‘calculator’. ‘[Devil] horns’, and the English word cipher, only forced on us by the needs of cultivated Roman numerals were cumbersome meant ‘secret code’. modes of thought.” and unwieldy for making longhand This did not inhibit the practical minded However, zero was a bizarre number. calculations. Imagine trying to multiply the European merchants or progressive Brahmagupta found that zero multiplied following: MDCCLXVII (times) LVI or (1767 x scholars, who just went ahead using Indo- by any number is zero; zero added or 56). Do not even think about trying to Arabic numerals with Zero because it was subtracted from any number makes no perform Long Division using Roman so much easier. However, when Church difference. But what is 0 ÷ 0 and 1 ÷ 0? Numerals! With Indo-Arabic numerals one authorities unreasonably put restrictions the Brahmagupta and others were foxed. can do all the four basic arithmetical bankers simply created duplicate sets of Today, we just banish this question by operations just with pencil and paper. You books, one to show the church, one to do saying it is ‘indeterminate’. have a simple , however boring calculations in. it may have been when we learnt it in the With the scientific revolution setting in first place. Europe, fetters fell and soon Indo-Arabic Indian Numerals Go Places Enthralment of European traders and numerals became the only form to be Today, the world over we use ten with scholars at the simple and concise Indo- used. Through European colonialism Indo specific unique symbols for numbers: 0 1 Arabic numerals is captured elegantly in Arabic numerals went to different parts of 2 3 4 5 6 7 8 9. The shapes of the numbers a woodcut printed in Margarita the world and today the Roman numerals were unlike the modern 1, 2, etc, but this Philosophica published in 1503. are seen only on the faces, ingenious method for writing numbers just “Arithmetica”, the deity of mathematics, is numbering of prefaces in books and the by using ten symbols was invented in India shown watching a competition between like. thousands of years ago. After the number an “abacist”, one who uses Roman nine comes ten written as 10. Notice that numerals for computation, and an Dr T.V. Venkateswaran is a Scientist with Vigyan 10 is not a distinct symbol; it is a “algorist”, one who uses Indo-Arabic Prasar, A-50, Institutional Area, Sector 62, NOIDA- combination of two unique symbols. In numerals. Judging from the clothes of the 201307; Email: [email protected] SCIENCE REPORTER, APRIL 2012 14

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