NOTATIONS AND NUMBERS Notations and Numbers We are so accustomed to today's number system, with its use of the ten decimal digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 (in Western countries), that it can come as a shock to realize that there are entirely different ways to write numbers. Even today, many cultures - Arabic, Chinese, Korean - use different symbols for the ten digits, although they all combine these symbols to form larger numbers using the same 'positional' method (hundreds, tens, units). But differences in notation can be more radical than that. There is nothing special about the number 10. It happens to be the number of human fingers and thumbs, which are ideal for counting, but if we had evolved seven fitigers, or twelve, very similar systems would have worked equally well, perhaps better in some cases. Roman numerals Mosl Westerners know of at least one alternative system, Roman numerals, in which - for example - the year 2012 is written MMXIl Musi of us are also aware, at least if reminded, that we ■ mplo) two distinct methods for writing numbers that are not w hole numbers - fractions like % and decimals such as 0.75. Yet .mother number notation, found on calculators, is the scientific 111 iiation for very large or very small numbers - such as 5 x 109 for live billion (often seen as 5E9 on calculator displays) or 5 x IO6 for live millionths. These symbolic systems developed over thousands of years, and n i,iiiy alternatives flourished in various cultures. We have already nieountered the Babylonian sexagesimal system (which would tome naturally to any creature that had 60 fingers), and the simpler tud more limited Egyptian number symbols, with their strange 11 raiment of fractions. Later, base-20 numbers were used in Central America by the Mayan civilization. Only recently did humanity setde • .i i (he current methods for writing numbers, and their use became l ublished through a mixture of tradition and convenience. ' In hematics is about concepts, not symbols - but a good choice of 111 hoi can be very helpful. Greek numerals Wc pick up the story of number symbols with the Greeks. Greek i niietry was a big improvement over Babylonian geometry, but Greek arithmetic - as far as we can tell from surviving sources - was not. The Greeks took a big step backwards; they did not use pi isitional notation. Instead, they used specific symbols for multiples ■ I 10 or 100, so that, for instance, the symbol for 50 bore no particular relationship to that for 5 or 500. The earliest evidence of Greek numerals is from about 1100 ec. (.1)0 bc the symbols had changed, and by 450 bc they had changed Igain, with the adoption of the Attic system, which resembles R< iinan numerals. The Attic system used I, II, III and llll for the numbers 1 ', J and 4. For 5 the Greek capital letter pi (fl) was employed, probably because it is the first letter of penta. Similarly, 10 was W ritten A, the first letter of deka; 100 was written H, the first letter Dl hckaton; 1000 was written E, the first letter of chilioi; and Ml TAMING THE INFINITE NOTATIONS ANN NIIMI1I -llli 10,000 was written M, the first letter of myrioi. Later 11 was changed to T. So the number 2178, for example, was written as HSHAAAAAAArlll Although the Pythagoreans made numbers the basis of their philosophy, it is not known how they wrote them.Their interest in square and triangular numbers suggests that they may have represented numbers by patterns of dots. By the classical period, 600-300bc, the Greek system had changed again, and the 27 different letters of their alphabet were used to denote numbers from 1 to 900, like this: 1 2 3 1 5 6 7 8 9 oc P Y O S 5 i >'i 9 10 20 40 50 60 70 80 90 \ K \ M V 4 O j>; P 100 200 300 400 500 600 700 800 900 p O T V X (0 T These are the lower-case Greek letters, augmented by three extra leiiers derived from the Phoenician alphabet: 5 (stigma), p (koppa), and T (sampi). I Jsing letters to stand for numbers might have caused ambiguity, ■.<>.i horizontal line was placed over the top of the number symbols, (■or numbers bigger than 999, the value of a symbol could be multiplied hy 1000 by placing a stroke in front of it. The various Greek systems were reasonable as a method for i ■ • ilin;,; the results of calculations, but not for performing the • ill illations themselves. (Imagine trying to multiply ou-Y by o>X.ft, |i ii instance.)The calculations themselves were probably carried oni Uling an abacus, perhaps represented by pebbles in the sand, lipecially early on. The Greeks wrote fractions in several ways. One was to write the numerator, followed by a prime ('), and then the denominator, I ol l< wed by a double prime ("). Often the denominator was written twice. So 21/+? would be written as Ka' |xE," where Ka is 21 and u5 is 47. They also used Egyptian-style fractions, and there was a special symbol for '/2. Some Greek lltronomers, notably Ptolemy, employed the Babylonian sexagesimal lystem for precision, but using Greek symbols for the component digits. It was all very different from what we use today. In fact, it was ,i mess. Indian number symbols rhe ten symbols currently used to denote decimal digits are often referred to as Hindu-Arabic numerals, because they originated in India and were taken up and developed by the Arabs. The earliest Indian numerals were more like the Egyptian system. II ii example, Khasrosthi numerals, used from 400 bc to ad 100, iqiresented the numbers from 1 to 8 as | || HI X IX ||X MIX XX nh a special symbol for 10. The first traces of what eventually became the modern symbolic system appeared around 300 bc in the Brahmi numerals. Buddhist inscriptions from the time include precursors of the later Hindu symbols for 1, 4 and 6. However, 11 ii ■ Brahmi system used different symbols for multiples of ten or multiples of 100, so it was similar to the Greek number TAMING THE infinite notation:; and niimiiers symbolism, except that it used special symbols rather than letters of the alphabet. The Brahmi system was not a positional system. By ad 100 there are records of the full Brahmi system. Inscriptions in caves and on coins show that it continued in use until the fourth century. Between the fourth and sixth centuries, the Gupta Empire gained control of a large part of India, and the Brahmi numerals developed into Gupta numerals. From there they developed into Nagari numerals. The idea was the same, but the symbols differed. The Indians may have developed positional notation by the first century, but the earliest datable documentary evidence for positional notation places it in 594. The evidence is a legal document which bears the date 346 in the Chedii calendar, but some scholars believe this date may be a forgery. Nevertheless, it is generally agreed that positional notation was in use in India from about 400 onwards. There is a problem with the use of only the symbols 1-9: the notation is ambiguous. What does 25 mean, for instance? It might (in our notation) mean 25, or 205, or 2005 or 250, etc. In positional notation, where the meaning of a symbol depends on its location, it is important to specify that location without ambiguity. Today we do that by using a tenth symbol, zero (0). But it took early civilizations a long time to recognize the problem and solve it in that manner. One reason was philosophical: how can 0 be a number when a number is a quantity of things? Is nothing a quantity? Another was practical: usually it was clear from the context whether 25 meant 25 or 250 or whatever. iiinu'i.ii'. i 'i Some time before 400bc - the exact date is unknown tin-l'..il lylonians introduced a special symbol to show a missing position m their number notation.This saved the scribes the effort of Iciving •i carefully judged space, and made it possible to work otit what .i number meant even if it was written sloppily. This invention w.ts forgotten, or not transmitted to other cultures, and eventually i' 'i liscovered by the Hindus.The Bakhshali manuscript, the date of \\ Inch is disputed but lies somewhere between ad200 and 11 Oil, uses a heavy dot •. The Jain text Lokavibhcwga of ad458 uses the 11 incept of zero, but not a symbol. A positional system that lacked the numeral zero was introduced by Aryabhata around ad 500. Later Indian mathematicians had names for zero, but did not use a lymboL The first undisputed use of zero in positional notation i k curs on a stone tablet in Gwalior dated to ad 876. Brahmagupta, Mahavira and Bhaskara i lie key Indian mathematicians were Aryabhata (born ad476), r.r.iIimagupta (born ad 598), Mahavira (9th century) and Bhaskara (born 1114). Actually they should be described as astronomers, because mathematics was then considered to be an astronomical technique. What mathematics existed was written down as i lupters in astronomy texts; it was not viewed as a subject in its i iwn right. Aryabhata tells us that his book Aryabhatiya was written when he as 23 years old. Brief though the mathematical section of his book 11, it contains a wealth of material: an alphabetic system of numerals, .ii iihmetical rules, solution methods for linear and quadratic < i|u,ttions, trigonometry (including the sine function and the 'wrsed sine' 1 - cos 0). There is also an excellent approximation, ; 1416, to n. Brahmagupta was the author of two books: Brahma Sputa Siddhanla ii n I Khanda Khadyaka.The first is the most important; it is an astronomy texi widi several sections on mathematics, with arithmetic and the TAMING the infinite notations and numbers What arithmetic did for them The oldest surviving Chinese mathematics text is the Chiu Chang, which dates from about ad100. A typical problem is: Two and a half piculs of rice are bought for 3/7 of a tael of silver. How many piculs can be bought for 9 taels? The proposed solution uses what medieval mathematicians called the 'rule of three'. In modern notation, let x be the required quantity. Then x _ % 9 sox = 52V2 piculs. A picul is about 65 kilograms. verbal equivalent of simple algebra. The second book includes a remarkable method for interpolating sine tables - that is, finding the sine of an angle from the sines of a larger angle and a smaller one. Mahavira was a Jain, and he included a lot of Jain mathematics in his Gcmita Sara Samgraha. This book included most of the contents of those of Aryabhata and Brahmagupta, but went a great deal further and was generally more sophisticated. It included fractions, permutations and combinations, the solution of quadratic equations, Pythagorean triangles and an attempt to fmd the area and perimeter of an ellipse. Bhaskara (known as 'the teacher') wrote three important works: l.iliiwiti, Bijaganita and Siddhanta Siromani. According to Fyzi, court poet of i ho Mogul emperor Akbar, Lilavati was the name of Bhaskara s < l.ilighter. Her father cast his daughter's horoscope, and determined the most auspicious time for her wedding.To dramatize his forecast, be pin a cup with a hole in it inside a bowl of water, constructed 10 thai ii would sink when the propitious moment arrived. But I il.wati leaned over the bowl and a pearl from her clothing fell into 'I.....i' .miI blocked the hole.The cup did not sink, which meant ih.ii 1.1 Lavali could never get married. To cheer her up, Bhaskara miitea mathematics textbook for her.The legend does not rvc< >rmoo o 2. \ 4 9 6 7 8 9 1 ■ill' hi dI ' • i' in number symbols [i ii «1 number notation as well as methods for doing calculations lulckly and accurately. \n influential figure was Leonardo of Pisa, also known as Fibonacci, whose book Liber Abbaci was published in 1202. (The ii ill.in word 'abbaco' usually means 'calculation', and need not iiii|>ly the use of the abacus, a Latin term.) In this book, Leonardo ......k luced Hindu—Arabic number symbols to Europe. The Liber Abbaci includes, and promoted, one further notational |l \ ii e that remains in use today: the horizontal bar in a fraction, ii« 11 as 4 for 'three-quarters'. The Hindus employed a similar ......iiion, but without the bar; the bar seems to have been introduced h i he Arabs. Fibonacci employed it widely, but his usage differed i1. ii 11 what we do today in some respects. For instance, he would use i in same bar as part of several different fractions. Because fractions are very important in our story, it may be • 'i ill adding a few comments on the notation. In a fraction like ■ I on the bottom tells us to divide the unit into four equal parts, mi I i lie 3 on top then tells us to select three of those pieces. More 11 lolly, 4 is the denominator and 3 is the numerator. For typographical .....venience, fractions are often written on a single line in the form I, or sometimes in the compromise form V4.The horizontal bar i In 11 mutates into a diagonal slash. < in the whole, however, we seldom use fractional notation in practical work. Mostly we use decimals — writing it as 3.14159, say, Inch is not exact, but close enough for most calculations, i Ir.iorically, we have to make a bit of a leap to get to decimals, but We are following chains of ideas, not chronology, and it will be much simpler to make the leap anyway. We therefore jump forward to I 585, when William the Silent chose the Dutchman Simon '.irvin as private tutor to his son Maurice of Nassau. Hi hiding on this recognition, Stevin made quite a career for himself, becoming Inspector of Dykes, Quartermaster-General of i he Army and eventually the Minister of Finance. He quickly realized TAMING THE INFINITE NOTATIONS AND NUMflEHS (Fibonacci) 1170-1250 Leonardo, Italian born, grew up in North Africa, where his i father Guilielmo was working as a diplomat on behalf of merchants trading at Bugia (in modern Algeria). He accompanied his father on his numerous travels, encountered the Arabic system for writing numbers and understood its importance. In his Liber Abbaci of 1202 he writes: 'When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else.' The book introduced the Hindu-Arabic notation to Europe, and formed a comprehensive arithmetic text, containing a wealth of material related to trade and currency conversion. Although it took several centuries for Hindu-Arabic notation to displace the traditional abacus, the advantages of a purely written system of calculation soon became apparent. Leonardo is often known by his nickname 'Fibonacci', which means 'son of Bonaccio', but this name is not recorded before the 18th century and was probably invented then by Guillaume Libri. the need for accurate accounting procedures, and he looked to the Kalian arithmeticians of the Renaissance period, and the 11111< 111 -Arabic notation transmitted to Europe by Leonardo of Pisa. Ih found fractional calculations cumbersome, and would havei |ni Irrrcii I he precision and tidiness of Babylonian sexagesimals, '......'ft lor die use of base-60. He tried to find a system that ■ in biued the best of both, and invented a base-10 analogue of the pbylonian system: decimals. I le published his new notational system, making it clear that h i 1.1 been tried, tested and found to be entirely practical by entirely |.i it lical men. In addition, he pointed out its efficacy as a business tool: "all computations that are met in business may be performed i.\ integers alone without the aid of fractions'. Negative numbers M.nl lematicians call the system of whole numbers the natural numbers. 11n biding negative numbers as well, we obtain the integers. The ......I numbers (or merely 'rationals') are the positive and negative 11 „ lions, the real numbers (or merely 'reals') are the positive and Hive decimals, going on forever if necessary. I low did negative numbers come into the story? Early in the first millennium, the Chinese employed a system of Minuting rods' instead of an abacus. They laid the rods out in i.....tiis to represent numbers. [he top row of the picture shows heng rods, which represented him is, hundreds, tens of thousands and so on, according to their 11 ni Chinese counting rods lit) TAMING the infinite notation:; and niimhers position in a row of such symbols. The bottom row shows tsung rods, which represented tens, thousands and so on. So the two type! alternated. Calculations were performed by manipulating the roc in systematic ways. When solving a system of linear equations, the Chinese calculators would arrange the rods in a table. They used red rods for terms that were supposed to be added and black rods for terms that were supposed to be subtracted. So to solve equations that wJ would write as 3x- 2y = 4 x + Sy = 7 they would set out the two equations as two columns of a table: on with the numbers 3 (red), 2 (black), 4 (red), and the other 1 (red), 5 (red), 7 (red). The red/black notation was not really about negative numbers, but the operation of subtraction. However, it set the stage for a concept of negative numbers, cheng fu shu. Now a negative number was represented by using the same arrangement of rods as for the corresponding positive number, by placing another rod diagonally over the top. To Diophantus, all numbers had to be positive, and he rejected negative solutions to equations. Hindu mathematicians found ''1 ■1111 vt1 nun ibers useful to represent i ■ivim.....il equations Chinese sly!;: SIi,iilull mil:; are red ii in financial calculations - owing someone a sum of money I worse, financially, than having no money, so a debt clearly nlil be less than zero. If you have 3 pounds and pay out 2, llini . | .ii ire left with 3 - 2 = 1. By the same token, if you owe a debt ■ 2 | >i Hinds and acquire 3, your net worth is —2 + 3 = 1. Bhaskara i. marks that a particular problem had two solutions, 50 and -5, but l' i named nervous about the second solution, saying that it was ..... in be taken; people do not approve of negative solutions'. 11. spite these misgivings, negative numbers gradually became ... i-| iied. Their interpretation, in a real calculation, needed care. ......limes they made no sense, sometimes they might be debts, .....ii limes they might mean a downwards motion instead of an B .nds one. But interpretation aside, their arithmetic worked |.. rfectly well, and they were so useful as a computational aid that n would have been silly not to use them. Arithmetic lives on ......nimber system is so familiar that we tend to assume that it is the . oil) possible one, or at least the only sensible one. In fact, it evolved, Laboriously and with lots of dead ends, over thousands of years, mere are many alternatives; some were used by earlier cultures, like iln Mayans. Different notations for the numerals 0-9 are in use today in .. line countries. And our computers represent numbers internally in liinary, not decimal; their programmers ensure that the numbers |rr uirned back into decimal before they appear on the screen or in i |'i int-out. nice computers are now ubiquitous, is there any point in leaching arithmetic any more?Yes, for several reasons. Someone has in he able to design and build calculators and computers, and in.ike them do the right job; this requires understanding arithmetic how and why it works, not just how to do it. And if your only .111111 uetical ability is reading what's on a calculator, you probably umi't notice if the supermarket gets your bill wrong. Without '. 1 63 TAMING THE INFINITE NOTATIONS AND NUMHFRS internalizing the basic operations of arithmetic, the whole of mathematics will be inaccessible to you. You might not worry ] about that, but modern civilization would quickly break down if we j stopped teaching arithmetic, because you can't spot the future engineers and scientists at the age of five. Or even the future bank managers and accountants. Of course, once you have a basic grasp of arithmetic by hand, using a calculator is a good way to save time and effort. But, just as] you won't learn to walk by always using a crutch, you won't learn to think sensibly about numbers by relying solely on a calculator. Mayan Numerals A remarkable number system, ■: 0 0 • • • • • • • which used base-20 notation I 2 3 4 5 instead of base-10, was developed by the Mayans, who lived in South America C 7 8 9 10 around 1000. In the base-20 (j • • • *« • t * • _ system, the symbols equivalent to our 347 11 12 13 14 15 would mean 3 x 400 + 4 X 20 + 7 x 1 0 a rj • a • • •»• 0 © (Since 20 x 20 = 400) 16 17 18 which is 1287 in our notation. 19 20 The actual symbols are • at t||| * shown here. © © © © © Early civilizations that use 40 60 80 100 120 base-10 probably did so because humans have ten fingers (including thumbs). It has been suggested that the Mayans counted on their toes as well, which is why they used base-20. What arithmetic does for us Wn use arithmetic throughout our daily lives, in commerce, and in ■,i;ience. Until the development of electronic calculators and computers, wii either did the calculations by hand, with pen and paper, or we used ,iids such as the abacus or a ready reckoner (a printed book of tables of multiples of amounts of money). Today most arithmetic goes on electronically behind the scenes - supermarket checkout tills now tell the operator how much change to give back, for instance, and banks total up what is in your account automatically, rather than getting their accountant in do it. The quantity of arithmetic 'consumed' by a typical person during Hie course of a single day is substantial. Computer arithmetic is not actually carried out in decimal format. Computers use base-2, or binary, rather than base-10. In place of units, lens, hundreds, thousands and so on, computers use 1,2,4, 8,16,32, Li, 128, 256, and so on-the powers of two, each twice its predecessor. (This is why the memory card for your digital camera comes in funny •;izes like 256 megabytes.) In a computer, the number 100 would be iimken down as 64+32+4 and stored in the form 1100100. 65 (r