LURE OF THE UNKNOWN Lure of the Unknown X marks the spot The use of symbols in mathematics goes well beyond thei appearance in notation for numbers, as a casual glance at any mathematics text will make clear. The first important step towards symbolic reasoning - as opposed to mere symbolic representation - occurred in the context of problem-solving. Numerous ancient texts, right back to the Old Babylonian period, present their readers with information about some unknown quantity, and then ask for its value. A standard formula (in the literary sense) in Babylonian tablets goes 'ill found a stone but did not weigh it'. After some additional information - 'when I had added a second stone of half th«| weight, the total weight was 15 gin' - the student is required toy calculate the weight of the original stone. Algebra IVoblems of this kind eventually gave rise to what we now call] llgebra, in which numbers are represented by letters. The unknown i|u.uiiily is traditionally denoted by the letter x, the conditions thai ri'ly "> a are stated as various mathematical formulas, and thd 11ii li'in is taught standard methods for extracting die value of x from i lormulas. For instance, the Babylonian problem above would ritten as x + l/1x~ 15, and we would learn how to deduce that 10. i school level, algebra is a branch of mathematics in which unknown numbers are represented by letters, the operations of ■' 111 metic are represented by symbols and the main task is in ilcduce the values of unknown quantities from equations. A typical problem in school algebra is to find an unknown number x, given the equation x2 + 2x = 120.This quadratic equation iii «Hie positive solution, x = 10. Here x2 + 2x = 102 + 2 X10 = i iki i 20 = 120. It also has one negative solution, x = -12. Now 2x= (-12)2 + 2x(-12) = 144-24= 120.The ancients would ■ accepted the positive solution, but not the negative one.Today ■ ,n 11nit both, because in many problems negative numbers have usible meaning and correspond to physically feasible answers, hi- I because the mathematics actually becomes simpler if negative lumbers are permitted. In advanced mathematics, the use of letters to represent (lumbers is only one tiny aspect of the subject, the context in Willi Ii it got started. Algebra is about the properties of symbolic ncssions in their own right; it is about structure and form, not i i number. This more general view of algebra developed when mathematicians started asking general questions about school-i. i I .ilgebra. Instead of trying to solve specific equations, they ■I'd at the deeper structure of the solution process itself. 11. »w did algebra arise? What came first were the problems and ■m. 11m ids for solving them. Only later was the symbolic notation -I in Ii we now consider to be the essence of the topic - invented, I line were many notational systems, but eventually one eliminated «11 dI its competitors.The name 'algebra' appeared in the middle of process, and it is of Arabic origin. (The initial 'al\ Arabic for i In', indicates its origin.) TAMING THE INFINITE LURE OF THE UNKNOWN Equations What we now call die solution of equations, in which an unknown quantity must be found from suitable information, is almost as olm as arithmetic. There is indirect evidence that the Babylonians werd solving quite complicated equations as early as 2000 bc, and direct! evidence for solutions of simpler problems, in the form ofl cuneiform tablets, dating from around 1700bc. The surviving portion of Tablet YBC 4652, from the Old Babylonian period (1800-1600 bc) , contains eleven problems for solution; the text on the tablet indicates that originally there were 22 of them. A typical question is: 'I found a stone, but did not weigh it. After I weighed out six times its weight, added 2 gin and added one third of one seventh [of this new weight] multiplied by 24,1 weighed it. The result was 1 ma-na. What was the original weight of the stone?' A weight of 1 ma-na is 60 gin. In modern notation, we would let x be the required weight ii gin. Then the question tells us that (6x + 2) + y x y x 24(6x + 2) = 60 and standard algebraic methods lead to the answer x = 41 /3 grn.The tablri states this answer but gives no clear indication of how it is obtained. We can be confident that it would not have been found using symbol™ methods like the ones we now use, because later tablets prescribe solution methods in terms of typical examples - 'halve this number, add H the product of these two, take the square root ...' and so on. This problem, along with the odiers on YBC 4652, is what we novfl call a linear equation, which indicates that the unknown x enters only to the first power. All such equations can be rewritten in the form ax + b = 0 u nli solution x = —b/a. But in ancient times, with no concept of .....111 u■ 111111 ilkts and no symbolic manipulation, finding a solution ll not so straightforward. Even today, many students would Itrnggle with the problem from YBC 4652. More interesting are quadratic equations, in which the unknown > in also appear raised to the second power - squared.The modern I"i.....lation takes the form ax2 + bx + c = 0 ■ i |i 1111 ere is a standard formula to find the value of x.The Babylonian Ipproach is exemplified by aproblem on Tablet BM 13901: 'I have added up seven times the side of my square and eleven ■Ties the area, [getting] 6; 15.' (Here 6; 15 is the simplified form of pbylonian sexagesimal notation, and means 6 plus 15/60, or 6V4 in modern notation.) rhe stated solution runs: .....write down 7 and 1 l.You multiply 6; 15 by 11, [getting] I. K. lr>. You break off half of 7, [getting] 3;30 and 3;30.You iln|>ly, [getting] 12; 15.You add [this] to 1,8;45 [getting] result i II. This is the square of 9. You subtract 3;30, which you .....In I died, from 9. Result 5;30.The reciprocal of 11 cannot be .....d. By what must I multiply 11 to obtain 5;30? [The answer is] 10, i he side of the square is 0;30.' Notice that the tablet tells its reader what to do, but not why. It I* ,i recipe. Someone must have understood why it worked, in ■ ii t In to write it down in the first place, but once discovered it could tin ii l>o used by anyone who was appropriately trained. We don't know whether Babylonian schools merely taught the recipe, or i'»|d.lined why it worked. Hie recipe as stated looks very obscure, but it is easier to interpret ih' ii ripe than we might expect.The complicated numbers actually In l|> i hey make it clearer which rules are being used. To find < In iii. we just have to be systematic. In modern notation, write a = n,b = 7,c = 6;15 = 6V4 i,m 69 TAMING THE INFINITE LURE OF THE UNKNOWN Then the equation takes the form ax2 + bx — c with those particular values for a, b, c. We have to deduce x. The | Babylonian solution tells us to: (1) Multiply c by a, which gives uc. (2) Divide b by 2, which is b/2. (3) Square b/2 to get b2/4. (4) Add this to cic, which is ac + b2/4. (5) Take its square root Vac+bV4. (6) Subtract b/2, which makes /oc+bV4 - b/2. (7) Divide tins by a, and the answer is x = hc+b2/4- b/2 This is equivalent to the formula _ - b + Jb2 - 4ac 2a that is taught today because we put the term c on the left hand side, where it becomes -c. It is quite clear that the Babylonians knew that their procedure was a general one. The quoted example is too complex for dieII solution to be a special one, designed to suit that problem alone. How did the Babylonians think of their method, and how did they think about it? There had to be some relatively simple idea lying behind such a complicated process. It seems plausible, though there is no direct evidence, that they had a geometric idea, completing the square. An algebraic version of this is taught today, too. We can represent the question, which for clarity we choose to write in the form x2 + ax = b, as a picture: □ ♦ D - □ xz + ax = o Min - ilie square and the first rectangle have height x; their f Idths are, respectively, x and a. The smaller rectangle has area b. Ilir Babylonian recipe effectively splits the first rectangle into Iwn pieces, + r - □ x2 + 2(a/2xx) = b We can then rearrange the two new pieces and stick them on the dge of the square: - □ x2 +2(72xx) = b The left-hand diagram now cries out to be completed to a larger m.u'e, by adding the shaded square; - rj Hi keep the equation valid, the same extra shaded square is 4i li led to the other diagram too. But now we recognize the left-hand yj3.m as the square of side (x + a/2), and the geometric picture i|iiivalent to the algebraic statement x2 + 2(°/2xx) + (a/2y = b + (a/2y i, i 70 71 TAMING THE INFINITE LURE OF THE UNKNOWN Since the left-hand side is a square, we can rewrite this as (x + a/2y = b + (a/2y and then it is natural to take a square root x + a/2 = Vb+(a/2)2 and finally rearrange to deduce that x = jb+(y2y - °/2 J which is exacdy how the Babylonian recipe proceeds. There is no evidence on any tablet to support the view that thi geometric picture led the Babylonians to their recipe. However, this suggestion is plausible, and is supported indirectly by varioul diagrams that do appear on clay tablets. Al-jabr The word algebra comes from the Arabic al-jabr, a term employee by Muhammad ibn Musa al-Khwarizmi, who flourished around 820. His work The Compendious Book on Calculation by al-jabr w'al-muqabalo explained general methods for solving equations by manipulating unknown quantities. Al-Khwarizmi used words, not symbols, but his methods arc recognizably similar to those taught today. Al-jabr means 'adding equal amounts to both sides of an equation', which is what we do when we start from x- 3 = 5 and deduce that In effect, we make this deduction by adding 3 to both sides. M-muqabala has two meanings.There is a special meaning: 'subtracting equal amounts from bodi sides of an equation', which we do to pass from lii Mil- answer x + 3 = 5 x = 2 mi also has a general meaning: 'comparison'. Al-Khwarizmi gives general rules for solving six kinds of ••i|ii.ition, which between them can be used to solve all linear and ludratic equations. In his work, then, we find the ideas of Hm ternary algebra, but not the use of symbols. Cubic equations I In' Babylonians could solve quadratic equations, and their method is essentially the same one that is taught today. Algebraically, it involves nothing more complicated than a square root, beyond the i.indard operations of arithmetic (add, subtract, multiply, divide). I In- obvious next step is cubic equations, involving the cube of the unknown. We write such equations as (ix + bx2 + cx + d = 0 72 ■> here x is the unknown and the coefficients a, b, c, d are known iinnibers. But until the development of negative numbers, mathematicians classified cubic equations into many distinct types ■ > that, for example, x3 + 3x = 7 and x3 - 3x = 7 were considered i" he completely different, and required different methods for [heir solution. The Greeks discovered how to use conic sections to solve some I nliic equations. Modern algebra shows that if a conic intersects nidi her conic, the points of intersection are determined by an tquation of third or fourth degree (depending on the conies).The i ; reeks did not know this as a general fact, but they exploited its I ■ >iisequences in specific instances, using the conies as a new kind i»1 'geometrical instrument. This line of attack was completed and codified by the Persian i >iiiar Khayyam, best known for his poem the Rubaiyat. Around 1075 73 TAMING THE INFINITE he classified cubic equations into 14 kinds, and showed how tQ solve each kind using conies, in his work On the Proofs of the Problems of Algebra and Muqabala.The treatise was a geometric tour de force, and it polished off the geometric problem almost completely. A modem mathematician would raise a few quibbles - some of Omar's cases are not completely solved because he assumes that certain geometrically constructed points exist when sometimes they do not That is, he assumes his conies meet when they may fail to do so. Bin these are minor blemishes. Geometric solutions of cubic equations were all very well, but could there exist algebraic solutions, involving such things as cube roots, but nothing more complicated? The mathematicians of Renaissance Italy made one of the biggest breakthroughs in algebra when they discovered that the answer is 'yes'. In those days, mathematicians made their reputation by taking part in public contests. Each contestant would set his opponent I problems, and whoever solved the most was adjudged the winner* Members of the audience could place bets on who would win. The I contestants often wagered large sums of money - in one recorded I instance, the loser had to buy the winner (and his friends) thirty banquets. Additionally, the winner's ability to attract paying students, mostly from the nobility, was likely to be enhanced. So, public j mathematical combat was serious stuff. In 1535 there was just such a contest, between Antonio Fior and Niccolo Fontana, nicknamed Tartaglia, 'the stammerer'. Tartaglia wiped the floor with Fior, and word of his success spread, coming to the ears of Girolamo Cardano. And Cardano's ears pricked up. He was in the middle of writing a comprehensive algebra text, and the 111 km ions that Fior and Tartaglia had posed each other were — cubic equations. At that time, cubic equations were classified into three distil id types, again because negative numbers were not recognized, i i"i I new liow to solve just one type. Initially, Tartaglia knew how in ■ I. <■ jiist one different type. In modern symbols, his solution of LURE OF THE UNKNOWN i cubic equation of the type xi + ax = b is a3 + /— + 2 V27 4r + 3 £l + _bi 27 4 11 i hurst of inspired desperation, a week or so before the contest, I,ii i.iglia figured out how to solve the other types too. He then set ii ii only the types that he knew Fior could not solve. < '.irdano, hearing of the contest, realized that the two combatants i id devised methods for solving cubic equations. Wanting to add hi mi to his book, he buttonholedTartaglia and asked him to reveal in a sentence or so of mention. I lowever, Cardano had an excuse, quite a good one. And he also i.id ,i strong reason to bend his promise toTartaglia.The reason was li.it Cardano's student Lodovico Ferrari had found a method for "Iviug quartic equations, those involving the fourth power of the unknown.This was completely new, and of huge importance. So of 75 T."ií','iI(!G THE INFINITE Fibonacci Sequence The third section of the Liber Abbaci contains a problem that seems to have originated with Leonardo: 'A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if in every month, each pair begets a new pair, which from the second month onwards becomes productive?' This rather quirky problem leads to a curious, and famous, sequence of numbers: 1 2 3 5 8 13 21 34 55 and so on. Each number is the sum of the two preceding numbers. This is known as the Fibonacci Sequence, and it turns up repeatedly in mathematics and in the natural world. In particular, many flowers have a Fibonacci number of petals, This is not coincidence, but a consequence of the growth pattern of the plant and the geometry of the 'primordia' -tiny clumps of cells at the tip of the growing shoot that give rise to important structures, including petals. Although Fibonacci's growth rule for rabbit populations is unrealistic, more general rules of a similar kind (called Leslie models) are used today for certain problems in population dynamics, the study of how animal populations change in size as the animals breed and die. course Cardano wanted quartic equations in his book, too. Since it was his student who had made the discovery, this would have been legitimate. But Ferrari's method reduced the solution of any quartic to that of an associated cubic, so it relied on Tartaglia's solution of cubic equations. Cardano could not publish Ferrari's work without also publishing Tartaglia's. Then news reached him that offered a way out. Fior, who had lost to Tartaglia in public combat, was a student of Scipio del Ferro. Cardano heard that del Ferro had solved all three types of cubic, not LIIHI (II III! UNKNOWN What algebra did for them Several chapters of the Liber Abbaci contain algebraic problems relevant to the needs of merchants. One, not terribly practical, goes like this: 'A man buys 30 birds - partridges, doves and sparrows. A partridge costs 3 silver coins, a dove 2, and a sparrow % He pays 30 silver coins. How many birds of each type does he buy?' In modern notation, if we let x be the number of partridges, y the number of doves, and i the number of sparrows, then we must solve two equations x + y + z = 30 3x + 2y + %z = 30 In real or rational numbers, these equations would have infinitely many solutions, but there is an extra condition implied by the question: the numbers x, y, z are integers. It turns out that only one solution exists: 3 partridges, 5 doves and 22 sparrows. Leonardo also mentions a series of problems about buying a horse. One man says to another, 'If you give me one-third of your money, I can buy the horse'. The other says, 'if you give me one-quarter of your money, I can buy the I lorse'. What is the price of the horse? This time there are many solutions; the smallest one in whole numbers sets the price of the horse at 11 silver coins. 111'.i i he one that he had passed on to Fior. And a certain Annibale del N.ive was rumoured to possess del Ferro's unpublished papers. So 1 irdano and Ferrari went to Bologna in 1543 to consult del Nave, \n wed the papers - and there, as plain as the nose on your face, were lohitions of all three types of cubic. So Cardano could honestly say ill.ii he was not publishing Tartaglia's method, but del Ferro's. Fartaglia didn't see things that way. But he had no real answer to 1 .i u Lino's point that the solution was not Tartaglia's discovery at all, I nil del Ferro's. Tartaglia published a long, bitter diatribe about the 76 77 TAMING THE INFINITE (aka Hieronymus Cardanus, Jerome Cardan) 1501-1576 Girolamo Cardano was the illegitimate son of the Milanese lawyer Fazio Cardano and a young widow named Chiara Micheria who was trying to bring up three children. The children died of the plague in Milan while Chiara was giving birth to Girolamo in nearby Pavia. Fazio was an able mathematician, and he passed on his passion for the subject to Girolamo. Against his father's wishes, Girolamo studied medicine at Pavia University; Fazio had wanted him to study law. While still a student, Cardano was elected rector of the University of Padua, to which he had moved, by a single vote. Having spent a small legacy from his recently deceased father, Cardano turned to gambling to augment his finances: cards, dice and chess. He always carried a knife and once slashed the face of an opponent whom he believed he had caught cheating. In 1525 Cardano gained his medical degree, but his application to join the College of Physicians in Milan was rejected, probably because he had a reputation for being difficult. He practised medicine in the village of Sacca, and married Lucia Bandarini, a militia captain's daughter. The practice did not prosper, and in 1533 Girolamo again turned to gambling, but now he lost heavily, and had to pawn his wife's jewellery and some of the family furniture. Cardano struck lucky, and was offered his father's old position as lecturer in mathematics at the Piatti Foundation. He continued practising medicine on the side, and some miraculous cures enhanced his reputation as a doctor. By 1539, after several attempts, he was finally admitted to the College of Physicians. He began to publish scholarly texts on a variety of topics, including mathematics. I Uli! ill III; UNKNOWN Cardano wrote a remarkable autobiography, The Book ot My Life, a miscellany of chapters on numerous topics. His fame was at its peak, and he visited Edinburgh to treat the Archbishop of St Andrews, John Hamilton. Hamilton suffered Irom severe asthma. Under Cardano's care, his health improved dramatically, and Cardano left Scotland 2000 gold crowns the richer. He became professor at Pavia University, and things were yoing swimmingly until his eldest son Giambatista secretly married Brandonia di Seroni, 'a worthless, shameless woman' In Cardano's estimation. She and her family publicly humiliated and taunted Giambatista, who poisoned her. Despite Cardano's best efforts, Giambatista was executed. In 1570 Cardano was tried for heresy, having cast the horoscope of Jesus. He was imprisoned, then released, but banned from university employment. He went to Rome, where the Pope unexpectedly gave him a pension and he was admitted to the College of Physicians. He forecast the date of his own death, and allegedly made sure he was right by committing suicide. Despite many tribulations, he remained an optimist to the end. iil.iir, and was challenged to a public debate by Ferrari, defending lir. master. Ferrari won hands down, and Tartaglia never really 11 ■covered from the setback. Algebraic symbolism I lie mathematicians of Renaissance Italy had developed many klgebraic methods, but their notation was still rudimentary. It look hundreds of years for today's algebraic symbolism to develop. < )ne of the first to use symbols in place of unknown numbers i> IMophantus of Alexandria. His ^rithmetica, written around 250, ■i 11 ally consisted of 13 books, of which six have survived as later ipies. Its focus is the solution of algebraic equations, either in TAMING THE INFINITE LURE OF Tllf UNKNOWN whole numbers or in rational numbers - fractions P/q where p and q are whole numbers. Diophantus's notation differs considerably from what we use today. Although the Arithmetica is the onlyj surviving document on this topic, there is fragmentary evidenco that Diophantus was part of a wider tradition, and not just ad isolated figure. Diophantus's notation is not very well suited to calculations, but it does summarize them in a compact form. The Arabic mathematicians of the Medieval period developed sophisticated methods for solving equations, but expressed them in words, not symbols. Diophantus's Notation and Ours Meaning Modern symbol Diophantus's symbol The unknown X Y Its square X2 Ay Its cube :;• Ky Its fourth power X4 AyA Its fifth power Xs AKy Its sixth power X6 KyK Addition + Juxtapose terms (use AB for A+B) Subtraction - A Equality .CT I I he move to symbolic notation gained momentum in the i»< uaissance period. The first of the great algebraists to start using . > i ■ 11 iols was Francois Vieta, who stated many of his results in niliolic form, but his notation differed considerably from the Uiodern one. He did, however, use letters of the alphabet to ii i iicsent known quantities, as well as unknown ones.To distinguish ihrse, he adopted the convention that consonants B, C, D, F, G... ■ 11 iresented known quantities, whereas vowels A, E, Í,... represented unknowns. In the 15th century, a few rudimentary symbols made their |p| learance, notably the letters p and m for addition and subtraction: I'lir. and minus.These were abbreviations rather than true symbols. I In' symbols + and - also appeared around this time. They arose in commerce, where they were used by German merchants to distinguish overweight and underweight items. Mathematicians quickly began to employ them too, the first written examples ipi learing in 1481. William Oughtred introduced the symbol x for .....Implication, and was roundly (and rightly) criticized by Leibniz mi the grounds that this was too easily confused with the letter x. I n 15 5 7, in his The Whetstone of Witte, the English mathematician Robert Recorde invented the symbol = for equality, in use ever nice. He wrote that he could think of no two things that were more alike than two parallel lines of the same length. However, he Used much longer lines than we do today, more like =. Vieta initially wrote the word 'aequalis' for equality, but later replaced it by the symbol ~. René Descartes used a different lymbol, cc. The current symbols > and < for 'greater than' and 'less than' ire due to Thomas Harriot. Round brackets () show up in 1544, nul square [ ] and curly { } brackets were used by Vieta around l')93. Descartes used the square root symbol V, which is an ■ l.ihoration on the letter r for radix, or root; but he wrote vc for the ■ n he root. 81 TAMING THE INFINITE LURE OF Fill. UNKNOWN To see how different Renaissance algebraic notation was from ours, here is a short extract from Cardano's Ars Magna: 5p: R m:15 5m: R m:15 25m:m:15 qd. est 40 In modern notation this would read: (5 + V-15)(5- V-15) = 25 - (-15) = 40 So here we see p: and m: for plus and minus, R for 'square root', and 'qd. est' abbreviating the Latin phrase 'which is'. He wrote qdratu aeqtur 4 rebus p:32 where we would write x2 = 4x + 32 and therefore used separate abbreviations 'rebus' and 'qdratu' for the unknown (thing) and its square. Elsewhere he used R for the unknown, Z for its square and C for its cube. An influential but little-known figure was the Frenchman Nicolas Chuquet, whose book Triparty en la Science de Nombres of 1484 discussed three main mathematical topics: arithmetic, roots and unknowns. His notation for roots was much like Cardano's, but he started to systematize the treatment of powers of the unknown, by using superscripts for exponents. He referred in ilit" first four powers of the unknown as premier, champs, cubiez .iihI champs de champs. For what we would now write as 6x, 4x2 and Sx ; he used .6.1, .4.2 and .5.3. He also used zeroth and negative powers, writing .2.0 and .3.'-™ where we would write 2x° and i\ 1 In short: he used exponential notation (superscripts) for powers of die unknown, but had no explicit symbol for the mm I m. >wn itself. That omission was supplied by Descartes. His notation was ■ry similar to what we use nowadays, with one exception. Where I would write 5 + 4x + 6x2 +1 lx3 + 3x4 I Jescartes wrote 5 + 4x + 6xx +1 lx3 + 3x+ 111.u is, he used xx for the square. Occasionally, though he used < Newton wrote powers of the unknown exactly as we do in rw, including fractional and negative exponents, such as x3/1 for Bi square root of x3. It was Gauss who finally abolished xx in ■your of x2; once the Grand Master had done this, everyone else I. illi iwed suit. The logic of species klgebra began as a way to systematize problems in arithmetic, but M i lie time ofVieta it had acquired a life of its own. Before Vieta, |i braic symbolism and manipulation were viewed as ways to state mkI carry out arithmetical procedures, but numbers were still the iii.iiii point. Vieta made a crucial distinction between what he ■lied the logic of species and the logic of numbers. In his view, mi .ilgebraic expression represented an entire class (species) of mi limetical expressions. It was a different concept. In his 1591 In Irtem Anah/ticam Isagoge (Introduction to the Analytic Art) he ■ 111.lined that algebra is a method for operating on general forms, I lereas arithmetic is a method for operating on specific numbers. II i is may sound like logical hair-splitting, but the difference in 11h- point of view was significant. To Vieta, an algebraic calculation •.mil as (in our notation) (2x + 3y) - (x + y) = x + 2y 83 •0 TAMING THE INFINITE LURE OF THE UNKNOWN What algebra does for us The leading consumers of algebra in the modern world are scientists, who represent nature's regularities in terms of algebraic equations. These equations can be solved to represent unknown quantities in terms of known ones. The technique has become so routine that no one notices they're using algebra. Algebra was very nearly applied to archaeology in one episode of Time Team, when the intrepid TV archaeologists wanted to work out how deep a mediaeval well was. The first idea was to drop something down it, and time how long it took to reach the bottom. It took six seconds. The relevant algebraic formula here, neglecting the speed of sound, is s =