Boethius ( ) Europe Smells the Coffee. Boethius ( ) Boethius ( ) Boethius ( ) Boethius ( ) Lewinter & Widulski 1

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1 Boethis (480 54) Erope Smells the Coffee Chapter 6 Lewinter & Widlski The Saga of Mathematics Boethis became an orphan when he was seven years old. He was extremely well edcated. Boethis was a philosopher, poet, mathematician, statesman, and (perhaps) martyr. Lewinter & Widlski The Saga of Mathematics Boethis (480 54) He is best known as a translator of and commentator on Greek writings on logic and mathematics (Plato, Aristotle, Nichomachs). His mathematics texts were the best available and were sed for many centries at a time when mathematical achievement in Erope was at a low point. Boethis Arithmetic taght medieval scholars abot Pythagorean nmber theory. Lewinter & Widlski The Saga of Mathematics 3 Boethis (480 54) This shows Boethis calclating with Arabic nmerals competing with Pythagoras sing an abacs. It is from G. Reisch, Margarita Philosophica (508). Lewinter & Widlski The Saga of Mathematics 4 Boethis (480 54) Boethis was a main sorce of material for the qadrivim, which was introdced into monasteries and consisted of arithmetic, geometry, astronomy, and the theory of msic. Boethis wrote abot the relation of msic and science, sggesting that the pitch of a note one hears is related to the freqency of sond. Lewinter & Widlski The Saga of Mathematics 5 Boethis (480 54) One of the first msical works to be printed was Boethis's De instittione msica, written in the early sixth centry. It was for medieval athors, from arond the ninth centry on, the athoritative docment on Greek msic-theoretical thoght and systems. For example, Franchino Gaffrio in Theorica msica (49) acknowledged Boethis as the athoritative sorce on msic theory. Lewinter & Widlski The Saga of Mathematics 6 Lewinter & Widlski

2 Boethis (480 54) His writings and translations were the main works on logic in Erope becoming known collectively as Logica vets. Boethis' best-known work is the Consolations of Philosophy which was written while he was in prison. It looked at the qestions of the natre of good and evil, of fortne, chance, or freedom, and of divine foreknowledge. Lewinter & Widlski The Saga of Mathematics 7 Gregorian Chant The term Gregorian chant is named after Pope Gregory I ( AD). He is credited with arranging a large nmber of choral works, which arose in the early centries of Christianity in Erope. Gregorian chant is monophonic, that is, msic composed with only one melodic line withot accompaniment. Lewinter & Widlski The Saga of Mathematics 8 Gregorian Chant As with the melodies of folk msic, the chants probably changed as they were passed down orally from generation to generation. Polyphony is msic where two or more melodic lines are heard at the same time in a harmony. Polyphony didn't exist (or it wasn't on record) ntil the th centry. Polyphony Althogh the majority of medieval polyphonic works are anonymos - the names of the athors were either not preserved or simply never known - there are some composers whose work was so significant that their names were recorded along with their work. Hildegard von Bingen (098-79) Perotin (55-377) Gillame de Macha ( ) John Dmstable ( ) Lewinter & Widlski The Saga of Mathematics 9 Lewinter & Widlski The Saga of Mathematics 0 The Dark Ages The Dark Ages, formerly a designation for the entire period of the Middle Ages, now refers sally to the period c , also known as the Early Middle Ages. Medieval Erope was a large geographical region divided into smaller and cltrally diverse political nits that were never totally dominated by any one athority. Lewinter & Widlski The Saga of Mathematics The Middle Ages With the collapse of the Roman Empire, Christianity became the standard-bearer of Western civilization. The papacy gradally gained seclar athority; monastic commnities had the effect of preserving antiqe learning. By the 8th centry, cltre centered on Christianity had been established; it incorporated both Latin traditions and German instittions. Lewinter & Widlski The Saga of Mathematics Lewinter & Widlski

3 The Middle Ages The empire created by Charlemagne illstrated this fsion. However, the empire's fragile central athority was shattered by a new wave of invasions. Fedalism became the typical social and political organization of Erope. The new framework gained stability from the th centry, as the invaders became Christian. Lewinter & Widlski The Saga of Mathematics 3 The High Middle Ages As Erope entered the period known as the High Middle Ages, the chrch became the nifying instittion. Militant religios zeal was expressed in the Crsades. Secrity and prosperity stimlated intellectal life, newly centered in brgeoning niversities, which developed nder the aspices of the chrch. Lewinter & Widlski The Saga of Mathematics 4 The High Middle Ages From the Crsades and other sorces came contact with Arab cltre, which had preserved works of Greek athors whose writings had not srvived in Erope. Philosophy, science, and mathematics from the Classical and Hellenistic periods were assimilated into the tenets of the Christian faith and the prevailing philosophy of scholasticism. Lewinter & Widlski The Saga of Mathematics 5 The High Middle Ages Christian Erope finally began to assimilate the lively intellectal traditions of the Jews and Arabs. Translations of ancient Greek texts (and the fine Arabic commentaries on them) into Latin made the fll range of Aristotelean philosophy available to Western thinkers. Aristotle, long associated with heresy, was adapted by St. Thomas Aqinas to Christian doctrine. Lewinter & Widlski The Saga of Mathematics 6 The High Middle Ages Christian vales pervaded scholarship and literatre, especially Medieval Latin literatre, bt Provencal literatre also reflected Arab inflence, and other florishing medieval literatres, inclding German, Old Norse, and Middle English, incorporated the materials of pre-christian traditions. Lewinter & Widlski The Saga of Mathematics 7 Thomas Aqinas (5 7) St. Thomas Aqinas was an Italian philosopher and theologian, Doctor of the Chrch, known as the Angelic Doctor. He is the greatest figre of scholasticism - philosophical stdy as practiced by Christian thinkers in medieval niversities. Lewinter & Widlski The Saga of Mathematics 8 Lewinter & Widlski 3

4 Thomas Aqinas (5 7) He is one of the principal saints of the Roman Catholic Chrch, and fonder of the system declared by Pope Leo XIII to be the official Catholic philosophy. St. Thomas Aqinas held that reason and faith constitte two harmonios realms in which the trths of faith complement those of reason; both are gifts of God, bt reason has an atonomy of its own. Thomas Aqinas (5 7) Aqinas's nfinished Smma Theologica (65-73) represents the most complete statement of his philosophical system. The sections of greatest interest inclde his views on the natre of god, inclding the five ways to prove god's existence, and his exposition of natral law. Lewinter & Widlski The Saga of Mathematics 9 Lewinter & Widlski The Saga of Mathematics 0 Natral Law Belief that the principles of hman condct can be derived from a proper nderstanding of hman natre in the context of the niverse as a rational whole. Aqinas held that even the divine will is conditioned by reason. Ths, the natral law provides a nonrevelatory basis for all hman social condct. Lewinter & Widlski The Saga of Mathematics The Existence of God Attempts to prove the existence of god have been a notable featre of Western philosophy. The cosmological argment The ontological argment The teleological argment The moral argment The most serios atheological argment is the problem of evil. Lewinter & Widlski The Saga of Mathematics The Cosmological Argment An attempt to prove the existence of god by appeal to contingent facts abot the world. The first of Aqinas's five ways (borrowed from Aristotle's Metaphysics), begins from the fact that something is in motion, since everything that moves mst have been pt into motion by something else bt the series of prior movers cannot extend infinitely, there mst be a first mover (which is god). The Ontological Argment Ontological argments are argments, for the conclsion that God exists, from premises which are spposed to derive from some sorce other than observation of the world - e.g., from reason alone. Ontological argments are argments from nothing bt analytic, a priori and necessary premises to the conclsion that God exists. St. Anselm of Canterbry claims to derive the existence of God from the concept of a being than which no greater can be conceived. Lewinter & Widlski The Saga of Mathematics 3 Lewinter & Widlski The Saga of Mathematics 4 Lewinter & Widlski 4

5 The Ontological Argment St. Anselm reasoned that, if sch a being fails to exist, then a greater being namely, a being than which no greater can be conceived, and which exists can be conceived. Bt this wold be absrd, nothing can be greater than a being than which no greater can be conceived. So a being than which no greater can be conceived, that is, God, exists. Lewinter & Widlski The Saga of Mathematics 5 The Ontological Argment In the 7th centry, Rene Descartes endorsed a different version of this argment. In the early 8th centry, Gottfried Leibniz attempted to fill what he took to be a shortcoming in Descartes' view. Recently, Krt Gödel, best known for his incompleteness theorems, sketched a revised version of this argment. Lewinter & Widlski The Saga of Mathematics 6 The Teleological Argment Based pon an observation of the reglarity or beaty of the niverse. Employed by Marcs Cicero (06-43 BC), Aqinas, and William Paley ( ), the argment maintains that many aspects of the natral world exhibit an orderly and prposefl character that wold be most natrally explained by reference to the intentional design of an intelligent creator. The Teleological Argment Let X represent a given species of animal or for a particlar organ (e.g. the eye) or a capability of a given species: X is very complicated and/or prposefl. The existence of very complex and/or prposefl things is highly improbable, and ths their existence demands an explanation. The only reasonable explanation for the existence of X is that it was designed and created by an intelligent, sentient designer. Lewinter & Widlski The Saga of Mathematics 7 Lewinter & Widlski The Saga of Mathematics 8 The Teleological Argment X was not designed or created by hmans, or any other Earthly being. Therefore, X mst have been designed and created by a non-hman bt intelligent and sentient artificer. In particlar, X mst have been designed and created by God. Therefore God mst exist. This argment is very poplar today and it is at the core of scientific creationism. Lewinter & Widlski The Saga of Mathematics 9 The Teleological Argment Most biologists spport the standard theory of biological evoltion, i.e., they reject the third premise. In other words, Darwin's theory of natral selection offers an alternative, nonteleological accont of biological adaptations. In addition, anyone who accepts this line of argment bt acknowledges the presence of imperfection in the natral order is faced with the problem of evil. Lewinter & Widlski The Saga of Mathematics 30 Lewinter & Widlski 5

6 The Moral Argment An attempt to prove the existence of god by appeal to presence of moral vale in the niverse. There is a niversal moral law. If there is a niversal moral law, then there mst be a niversal moral lawgiver. Therefore, there mst be God. Man is an intelligent creatre having a conscience which is based pon an innate moral code. This natral law reqires a Law-Giver. Lewinter & Widlski The Saga of Mathematics 3 The Moral Argment Society with its varios forms of government, recognizes the concepts of right and wrong. Where does this niform implse come from, if not from God. The forth of Aqinas's five ways concldes that god mst exist as the most perfect case of all lesser goods. Immanel Kant arged that postlation of god's existence is a necessary condition for or capacity to apply the moral law. Lewinter & Widlski The Saga of Mathematics 3 The Problem of Evil Bad things happen. Whether they are taken to flow from the operation of the world ("natral evil"), reslt from deliberate hman crelty ("moral evil"), or simply correlate poorly with what seems to be deserved ("non-karmic evil"). Sch events give rise to basic qestions abot whether or not life is fair. The Problem of Evil The presence of evil in the world poses a special difficlty for traditional theists. Since an omniscient god mst be aware of evil, an omnipotent god cold prevent evil, and a benevolent god wold not tolerate evil, it shold follow that there is no evil. Yet there is evil, from which atheists conclde that there is no omniscient, omnipotent, and benevolent god. Lewinter & Widlski The Saga of Mathematics 33 Lewinter & Widlski The Saga of Mathematics 34 The Problem of Evil The most common theistic defense against the problem, proponded (in different forms) by both Agstine and Leibniz, is to deny the reality of evil by claiming that apparent cases of evil are merely parts of a larger whole that embodies greater good. More recently, some have qestioned whether the traditional notions of omnipotence and omniscience are coherent. Pascal's Wager Blaise Pascal (63-66) It makes more sense to believe in God than to not believe. If yo believe, and God exists, yo will be rewarded in the afterlife. If yo do not believe, and He exists, yo will be pnished for yor disbelief. If He does not exist, yo have lost nothing either way. It amonts to hedging yor bets. Lewinter & Widlski The Saga of Mathematics 35 Lewinter & Widlski The Saga of Mathematics 36 Lewinter & Widlski 6

7 The Atheist's Wager It is better to live yor life as if there are no Gods, and try to make the world a better place for yor being in it. If there is no God, yo have lost nothing and will be remembered fondly by those yo left behind. If there is a benevolent God, He will jdge yo on yor merits and not jst on whether or not yo believed in Him. Fibonacci (70-50) Fibonacci was born in Italy bt was edcated in North Africa. Fibonacci was taght mathematics in Bgia and traveled widely with his father. Lewinter & Widlski The Saga of Mathematics 37 Lewinter & Widlski The Saga of Mathematics 38 Fibonacci (70-50) He recognized the enormos advantages of the mathematical systems sed in the contries they visited. Fibonacci ended his travels arond the year 00 and retrned to Pisa. He wrote a nmber of important texts, inclding Liber abaci, Practica geometriae, Flos, and Liber qadratorm. Lewinter & Widlski The Saga of Mathematics 39 Fibonacci (70-50) Johannes of Palermo, a member of the Holy Roman emperor Frederick II's cort, presented a nmber of problems as challenges to Fibonacci. Fibonacci solved three of them and pt his soltions in Flos. Liber abaci, pblished in 0, was based on the arithmetic and algebra that Fibonacci accmlated dring his travels. Lewinter & Widlski The Saga of Mathematics 40 Fibonacci s Liber Abaci It introdced the Hind-Arabic place-valed decimal system and the se of Arabic nmerals into Erope. The second section of Liber abaci contains a large nmber of problems abot the price of goods, how to calclate profit on transactions, how to convert between the varios crrencies in se at that time, and problems which had originated in China. Fibonacci s Liber Abaci A problem in the third section of Liber abaci led to the introdction of the Fibonacci nmbers and the Fibonacci seqence. A certain man pt a pair of rabbits in a place srronded on all sides by a wall. How many pairs of rabbits can be prodced from that pair in a year if it is spposed that every month each pair begets a new pair which from the second month on becomes prodctive? Lewinter & Widlski The Saga of Mathematics 4 Lewinter & Widlski The Saga of Mathematics 4 Lewinter & Widlski 7

8 The reslting seqence is:,,, 3, 5, 8, 3,, 34, 55,... Notice that each nmber after the first two ( and ) is the sm of the two preceding nmbers. That is, 3 is the sm of 8 and is the sm of 34 and. Fibonacci Nmbers The nmbers in the seqence are called Fibonacci nmbers. We call the nmbers the terms of the seqence. Each term can be denoted sing sbscripts that identify the order in which the terms appear. We denote the Fibonacci nmbers by,, 3,, and the n-th term is denoted n. Lewinter & Widlski The Saga of Mathematics 43 Lewinter & Widlski The Saga of Mathematics 44 Fibonacci Nmbers So =, =, 3 =, 4 =3, 5 =5, etc. Note that 3 = + and 4 = + 3 and 5 = and this pattern contines for all terms after the first two. Mathematicians write this seqence by stating the initial conditions, = and =, and sing a recrsive relation which says, in an eqation, that each term is the sm of its two predecessors. In this case the recrrence eqation is given at the right. The last eqation in the box says that the (n+)-nd term is the sm of the n-th term and the (n+)- st term. = = n+ = n + n+ Lewinter & Widlski The Saga of Mathematics 45 Lewinter & Widlski The Saga of Mathematics 46 When n = for example, this says that the third term, 3, is the sm of the first term,, and the second term,, which is, of corse, correct. = = n+ = n + n+ Fibonacci s seqence of nmbers occrs in many places inclding Pascal s triangle, the binomial formla, probability, the golden ratio, the golden rectangle, plants and natre, and on the piano keyboard, where one octave contains black keys in one grop, 3 black keys in another, 5 black keys all together, 8 white keys and 3 keys in total. Lewinter & Widlski The Saga of Mathematics 47 Lewinter & Widlski The Saga of Mathematics 48 Lewinter & Widlski 8

9 Note that the Fibonacci nmbers grow withot bond, that is, they become arbitrarily large, in other words, they go to infinity. In fact the sixtieth term is 60 =,548,008,755,90. And 88 =,00,087,778,366,0,93. Wow! Consider the seqence of ratios /, /, 3/, 5/3, 8/5, 3/8, /3, formed by dividing each term by the one before it. Instead of consistently getting larger, they alternate between growing and shrinking! In decimal the seqence is,,.5,.666,.6,.65,.65384, etc. It trns ot that the seqence of ratios approaches a single target which we can readily calclate sing a clever argment. Lewinter & Widlski The Saga of Mathematics 49 Lewinter & Widlski The Saga of Mathematics 50 Let s call the target (or limit as mathematicians say) L. Let s denote the n-th ratio by R n. In other words, n+ Rn= n Dividing Fibonacci s recrrence eqation by n+ gives n+ n n+ = + n+ n+ which is eqal to Rn+ = R + n n+ Lewinter & Widlski The Saga of Mathematics 5 Lewinter & Widlski The Saga of Mathematics 5 If we let n approach infinity, then the last eqation becomes L = + L which is eqivalent to L = + L L L = 0 Lewinter & Widlski The Saga of Mathematics 53 Using the qadratic formla, we get that + 5 L = This is called the golden ratio, and it was known to the ancient Greeks as the most pleasing ratio of the length of a rectanglar painting frame to its width. Lewinter & Widlski The Saga of Mathematics 54 Lewinter & Widlski 9

10 Consider a rectangle whose width is and whose length is L. They assmed that the perfect or golden rectangle has the property that the removal of a sqare from it leaves a (smaller) rectangle that is similar to the original one. So the yellow rectangle whose width is L and whose length is is similar to the original. So L = L L Lewinter & Widlski The Saga of Mathematics 55 Lewinter & Widlski The Saga of Mathematics 56 Cross mltiplying gives L ( L ) = L L = L L = 0 This is the qadratic eqation of the Fibonacci ratios. Wow! Lewinter & Widlski The Saga of Mathematics 57 The face of the Parthenon in Athens has been seen as a golden rectangle and so have many other facades in Greek and Renaissance architectre. The golden ratio appears in many strange places in both the natral world and the hman world of magnificent artistic and scientific achievements. Lewinter & Widlski The Saga of Mathematics 58 Psychologists have shown that the golden ratio sbconsciosly affects many of or choices, sch as where to sit as we enter a large aditorim, where to stand on a stage when we address an adience, etc. etc. See Ron Knott s Fibonacci Nmbers and the Golden Section. Lewinter & Widlski The Saga of Mathematics 59 Eclid, in The Elements (Book VI, Proposition 30), says that the line AB is divided in extreme and mean ratio by C if AB:AC = AC:CB. We wold call it "finding the golden section C point on the line". A Lewinter & Widlski The Saga of Mathematics 60 C B Lewinter & Widlski 0

11 Eclid sed this phrase to mean the ratio of the smaller part of this line, CB to the larger part AC (ie the ratio CB/AC) is the SAME as the ratio of the larger part, AC to the whole line AB (ie is the same as the ratio AC/AB). If we let the line AB have nit length and AC have length x (so that CB is then jst x) then the definition means that Lewinter & Widlski The Saga of Mathematics 6 CB AC Solving gives = AC x x = AB x + x = The golden ratio is /x = See The Golden Ratio Lewinter & Widlski The Saga of Mathematics 6 5 Fibonacci s Liber Abaci Other types of problems in the third section of Liber abaci inclde: A spider climbs so many feet p a wall each day and slips back a fixed nmber each night, how many days does it take him to climb the wall. A hond whose speed increases arithmetically chases a hare whose speed also increases arithmetically, how far do they travel before the hond catches the hare. Calclate the amont of money two people have after a certain amont changes hands and the proportional increase and decrease are given. Fibonacci s Liber Abaci There are also problems involving perfect nmbers, the Chinese remainder theorem and problems involving the smming arithmetic and geometric series. In the forth section, he deals with irrational nmbers both with rational approximations and with geometric constrctions. Lewinter & Widlski The Saga of Mathematics 63 Lewinter & Widlski The Saga of Mathematics 64 Fibonacci s Practica geometriae It contains a large collection of geometry problems arranged into eight chapters with theorems based on Eclid's Elements and On Divisions. It incldes practical information for srveyors, inclding a chapter on how to calclate the height of tall objects sing similar triangles. Inclded is the calclation of the sides of the pentagon and the decagon from the diameter of circmscribed and inscribed circles. The inverse calclation is also given. Lewinter & Widlski The Saga of Mathematics 65 Fibonacci s Flos In it he gives an accrate approximation to a root of 0x + x + x 3 = 0, one of the problems that he was challenged to solve by Johannes of Palermo. Johannes of Palermo took this problem from Omar Khayyam's algebra book where it is solved by means of the intersection a circle and a hyperbola. Fibonacci proves that the root of the eqation is neither an integer nor a fraction, nor the sqare root of a fraction. Lewinter & Widlski The Saga of Mathematics 66 Lewinter & Widlski

12 Fibonacci s Flos Withot explaining his methods, Fibonacci then gives the approximate soltion in sexagesimal notation as (;,7,4,33,4,40) 60 (This is + /60+ 7/60 + 4/ ). This converts to the decimal which is correct to nine decimal places. This is a trly remarkable achievement. Fibonacci s Liber qadratorm Liber qadratorm, written in 5, is Fibonacci's most impressive piece of work. It is a nmber theory book. In it, he provides a method for finding Pythogorean triples. Fibonacci first notes that any sqare nmber is the sm of consective odd nmbers. For example, =5=5! Lewinter & Widlski The Saga of Mathematics 67 Lewinter & Widlski The Saga of Mathematics 68 Fibonacci and Pythagorean Triples Recall that any odd nmber is of the form n +, for some integer n. He noticed that the formla n + (n+) = (n+) implies that a sqare pls an odd nmber eqals the next higher sqare. For example, if n= then we get the eqation + ( +) = (+) which is eqivalent to + 5 = 3 Fibonacci and Pythagorean Triples What does this have to do with Pythagorean triples? Well, Fibonacci said, Ifa the odd nmba n+ isa sqara then yo hava Pythagorean triple! For example, if n = 4, then n + = (4) + = 9 then we get the eqation 4 + ( 4+) = (4+) which is eqivalent to = 5 Which gives s the famos (3, 4, 5) Pythagorean triple. Lewinter & Widlski The Saga of Mathematics 69 Lewinter & Widlski The Saga of Mathematics 70 Fibonacci s Liber qadratorm Fibonacci also proves many interesting nmber theory reslts inclding the fact that there is no x, y sch that x + y and x y are both sqares. He also proves that x 4 y 4 cannot be a sqare. Lewinter & Widlski The Saga of Mathematics 7 Fibonacci s Liber qadratorm He defined the concept of a congrm, a nmber of the form ab(a + b)(a b), if a + b is even, and 4 times this if a + b is odd. Fibonacci proved that a congrm mst be divisible by 4 and he also showed that for x, c sch that x + c and x c are both sqares, then c is a congrm. He also proved that a sqare cannot be a congrm. Lewinter & Widlski The Saga of Mathematics 7 Lewinter & Widlski

13 Fibonacci Nmbers The Fibonacci Qarterly is a modern jornal devoted to stdying mathematics related to this seqence. Solved Problems: The only sqare Fibonacci nmbers are and 44! The only cbic Fibonacci nmbers are and 8! The only trianglar Fibonacci nmbers are, 3, and 55! Lewinter & Widlski The Saga of Mathematics 73 Fibonacci Nmbers Unsolved Problems abot Fibonacci nmbers: Are there infinitely many prime Fibonacci nmbers? Are, 8 and 44 the only powers that are Fibonacci nmbers? Lewinter & Widlski The Saga of Mathematics 74 Lcas Nmbers Edoard Lcas (84-89) gave the name "Fibonacci Nmbers" to the series written abot by Leonardo of Pisa. He stdied a second series of nmbers which ses the same recrrence eqation bt starts with and, that is,,, 3, 4, 7,, 8, These nmbers are called the Lcas nmbers in his honor. Edoard Lcas (84-89) Lcas is also well known for his invention of the Tower of Hanoi pzzle and other mathematical recreations. The Tower of Hanoi pzzle appeared in 883 nder the name of M. Clas. Notice that Clas is an anagram of Lcas! His for volme work on recreational mathematics Récréations mathématiqes has become a classic. Lewinter & Widlski The Saga of Mathematics 75 Lewinter & Widlski The Saga of Mathematics 76 Nicole Oresme (30-38) A French priest and mathematician. He translated many of Aristotle s works and qestioned many of the ideas which at that time were accepted withot qestion. Lewinter & Widlski The Saga of Mathematics 77 Nicole Oresme (30-38) Oresme was the greatest of the French writers of the 4 th centry. He wrote Tractats proportionm, Algorisms proportionm, Tractats de latitdinibs formarm, Tractats de niformitate et difformitate intensionm, and Traité de la sphère. In the Algorisms proportionm is the first se of fractional exponents. Lewinter & Widlski The Saga of Mathematics 78 Lewinter & Widlski 3

14 Nicole Oresme (30-38) In Tractats de niformitate, Oresme invented a type of coordinate geometry before Descartes, in fact, Descartes may have been inflenced by his work. He proposed the se of a graph for plotting a variable magnitde whose vale depends on another variable. Nicole Oresme (30-38) He was the first to prove Merton s Theorem, that is, that the distance traveled in a fixed time by a body moving nder niform acceleration is the same as if the body moved at niform speed eqal to its speed at the midpoint of the time period. He wrote Qestiones Sper Libros Aristotelis be Anima dealing with the natre, speed and reflection of light. Lewinter & Widlski The Saga of Mathematics 79 Lewinter & Widlski The Saga of Mathematics 80 Nicole Oresme (30-38) Oresme worked on infinite series and was the first to prove that the harmonic series + / + /3 + /4 + becomes infinite withot bond, i.e., it diverges. In Livre d ciel et d monde, he opposed the theory of a stationary Earth as proposed by Aristotle and proposed rotation of the Earth some 00 years before Copernics. He later rejected his own idea. Lewinter & Widlski The Saga of Mathematics 8 Leibniz (646 76) Leibniz is considered to be one of the fathers of Calcls. We will discss him in frther detail later. For now, let s look at his work with infinite series. Lewinter & Widlski The Saga of Mathematics 8 Leibniz (646 76) He discovered the following series for π: π = L Leibniz s idea ot of which his calcls grew was the inverse relationship of sms and differences for seqences of nmbers. Repeating Decimals We can se a clever trick to determine the fraction eqivalent to a given repeating decimal. For example, sppose we want to know what fraction is eqal to Let x = the decimal. Then mltiply both sides by 0. Lewinter & Widlski The Saga of Mathematics 83 Lewinter & Widlski The Saga of Mathematics 84 Lewinter & Widlski 4

15 Repeating Decimals Finally, sbtract the two eqations. 0x = K x = K 9x = 6 Divide both sides by 9 Ths, x = 6/9 = /3. Repeating Decimals If the pattern is two digits, yo mltiply by 00 instead of 0. For example, sppose we want to know what fraction is eqal to Let x = Mltiply both sides by 00, so 00x = Lewinter & Widlski The Saga of Mathematics 85 Lewinter & Widlski The Saga of Mathematics 86 Repeating Decimals Sbtract the two eqations: 00x = K x = 99x = K The solve for x. Ths, x = 45/99 = 5/. Lewinter & Widlski The Saga of Mathematics 87 Lewinter & Widlski 5