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Home , Boethius, Five Ways (Aquinas), Nicole Oresme

The Saga of A Brief History

Boethius (480 – 524)

Boethius became an orphan when he was Smells the Coffee seven years old. He was extremely well educated. Boethius was a Chapter 6 , poet, mathematician, statesman, and (perhaps) martyr.

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Boethius (480 – 524) Boethius (480 – 524)

He is best known as a translator of and This shows Boethius commentator on Greek writings on and calculating with mathematics (, , Nichomachus). Arabic numerals His mathematics texts were the best available competing with and were used for many centuries at a time using an when mathematical achievement in Europe abacus. was at a low point. It is from G. Reisch, Boethius’ Arithmetic taught medieval scholars Margarita about Pythagorean number theory. Philosophica (1508).

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Boethius (480 – 524) Boethius (480 – 524)

Boethius was a main source of material for One of the first musical works to be printed the , which was introduced into was Boethius's De institutione musica, written monasteries and consisted of arithmetic, in the early sixth century. geometry, , and the theory of It was for medieval authors, from around the . ninth century on, the authoritative document on Greek music-theoretical thought and Boethius wrote about the relation of music systems. and science, suggesting that the pitch of a For example, Franchino Gaffurio in Theorica note one hears is related to the frequency of musica (1492) acknowledged Boethius as the sound. authoritative source on .

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Boethius (480 – 524) Gregorian Chant

His writings and were the main The term Gregorian chant is named after works on logic in Europe becoming known (590–604 AD). collectively as Logica vetus. He is credited with arranging a large number Boethius' best-known work is the of choral works, which arose in the early of which was written centuries of in Europe. while he was in prison. Gregorian chant is monophonic, that is, music It looked at the “questions of the of composed with only one melodic line without good and evil, of fortune, chance, or accompaniment. freedom, and of divine foreknowledge.”

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Gregorian Chant Polyphony

As with the melodies of folk music, the chants Although the majority of medieval polyphonic probably changed as they were passed down works are anonymous - the names of the orally from generation to generation. authors were either not preserved or simply never known - there are some composers Polyphony is music where two or more whose work was so significant that their melodic lines are heard at the same time in a names were recorded along with their work. harmony. Hildegard von Bingen (1098 - 1179) Polyphony didn't exist (or it wasn't on record) Perotin (1155 - 1377) until the 11th century. Guillame de Machau (1300 - 1377) John Dumstable (1385 - 1453)

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The Dark Ages The

The Dark Ages, formerly a designation for the With the collapse of the , entire period of the Middle Ages, now refers Christianity became the standard-bearer of usually to the period c.450–750, also known Western civilization. as the . The papacy gradually gained secular authority; monastic communities had the Medieval Europe was a large geographical effect of preserving antique learning. region divided into smaller and culturally By the 8th century, culture centered on diverse political units that were never totally Christianity had been established; it dominated by any one authority. incorporated both traditions and German institutions.

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The Middle Ages The High Middle Ages

The empire created by Charlemagne As Europe entered the period known as the illustrated this fusion. High Middle Ages, the church became the However, the empire's fragile central unifying institution. authority was shattered by a new wave of Militant religious zeal was expressed in the invasions. . Feudalism became the typical social and political organization of Europe. Security and prosperity stimulated intellectual life, newly centered in burgeoning The new framework gained stability from the 11th century, as the invaders became universities, which developed under the Christian. auspices of the church.

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The High Middle Ages The High Middle Ages

From the Crusades and other sources came Christian Europe finally began to assimilate contact with Arab culture, which had the lively intellectual traditions of the Jews preserved works of Greek authors whose and Arabs. writings had not survived in Europe. Translations of ancient Greek texts (and the fine Arabic commentaries on them) into Latin Philosophy, science, and mathematics from made the full range of Aristotelean the Classical and Hellenistic periods were philosophy available to Western thinkers. assimilated into the tenets of the Christian Aristotle, long associated with , was and the prevailing philosophy of adapted by St. to Christian . doctrine.

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The High Middle Ages Thomas Aquinas (1225 – 1272)

Christian values pervaded scholarship St. Thomas Aquinas was an Italian and literature, especially philosopher and literature, but Provencal literature also theologian, , known as reflected Arab influence, and other the Angelic Doctor. flourishing medieval literatures, He is the greatest figure of scholasticism - including German, Old Norse, and philosophical study as Middle English, incorporated the practiced by Christian thinkers in medieval materials of pre-Christian traditions. universities.

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Thomas Aquinas (1225 – 1272) Thomas Aquinas (1225 – 1272)

He is one of the principal saints of the Roman Aquinas's unfinished Theologica , and founder of the system (1265-1273) represents the most complete declared by Pope Leo XIII to be the official statement of his philosophical system. Catholic philosophy. The sections of greatest interest include his St. Thomas Aquinas held that reason and views on the nature of , including the five faith constitute two harmonious realms in ways to prove god's existence, and his which the truths of faith complement those of exposition of natural . reason; both are gifts of God, but reason has an autonomy of its own.

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Natural Law The

Belief that the principles of human conduct Attempts to prove the existence of god have can be derived from a proper understanding been a notable feature of Western of human nature in the context of the philosophy. universe as a rational whole. The Aquinas held that even the divine is The conditioned by reason. The Thus, the provides a non- The moral argument revelatory basis for all human social conduct. The most serious atheological argument is the .

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The Cosmological Argument The Ontological Argument

An attempt to prove the existence of god by Ontological arguments are arguments, for the appeal to contingent facts about the world. conclusion that God exists, from premises which are supposed to derive from some The first of Aquinas's (borrowed source other than observation of the world - from Aristotle's ), begins from the e.g., from reason alone. fact that something is in motion, since Ontological arguments are arguments from everything that moves must have been put nothing but analytic, a priori and necessary premises to the conclusion that God exists. into motion by something else but the series St. claims to derive the of prior movers cannot extend infinitely, there existence of God from the concept of a “being must be a first mover (which is god). than which no greater can be conceived.”

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The Ontological Argument The Ontological Argument

St. Anselm reasoned that, if such a being fails In the 17th century, Rene Descartes to exist, then a greater being – namely, a endorsed a different version of this argument. being than which no greater can be conceived, and which exists – can be In the early 18th century, Gottfried Leibniz conceived. attempted to fill what he took to be a shortcoming in Descartes' view. But this would be absurd, nothing can be greater than a being than which no greater Recently, Kurt Gödel, best known for his can be conceived. incompleteness theorems, sketched a revised So a being than which no greater can be version of this argument. conceived, that is, God, exists.

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The Teleological Argument The Teleological Argument

Based upon an observation of the regularity Let X represent a given species of animal or or beauty of the universe. for a particular organ (e.g. the eye) or a capability of a given species: Employed by Marcus (106-43 BC), X is very complicated and/or purposeful. Aquinas, and William Paley (1743-1805), the The existence of very complex and/or purposeful argument maintains that many aspects of the things is highly improbable, and thus their natural world exhibit an orderly and existence demands an explanation. purposeful character that would be most The only reasonable explanation for the existence naturally explained by reference to the of X is that it was designed and created by an intelligent, sentient designer. intentional design of an intelligent creator.

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The Teleological Argument The Teleological Argument

X was not designed or created by humans, or any Most biologists support the standard theory other Earthly being. of biological evolution, i.e., they reject the Therefore, X must have been designed and third premise. created by a non-human but intelligent and sentient artificer. In other words, Darwin's theory of natural In particular, X must have been designed and selection offers an alternative, non- created by God. teleological account of biological adaptations. Therefore God must exist. In addition, anyone who accepts this line of This argument is very popular today and it is argument but acknowledges the presence of at the core of scientific . imperfection in the natural order is faced with the problem of evil.

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The Moral Argument The Moral Argument

An attempt to prove the existence of god by Society with its various forms of , appeal to presence of moral value in the recognizes the concepts of right and wrong. universe. Where does this uniform impulse come from, There is a universal moral law. if not from God. If there is a universal moral law, then there must The fourth of Aquinas's five ways concludes be a universal moral lawgiver. that god must exist as the most perfect cause Therefore, there must be God. of all lesser goods. Man is an intelligent creature having a argued that postulation of conscience which is based upon an innate god's existence is a necessary condition for moral code. our capacity to apply the moral law. This natural law requires a Law-Giver.

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The Problem of Evil The Problem of Evil

Bad things happen. The presence of evil in the world poses a Whether they are taken to special difficulty for traditional theists. flow from the operation of the world ("natural Since an omniscient god must be aware of evil"), evil, an omnipotent god could prevent evil, result from deliberate human cruelty ("moral and a benevolent god would not tolerate evil, evil"), or simply it should follow that there is no evil. correlate poorly with what seems to be deserved Yet there is evil, from which atheists conclude ("non-karmic evil"). that there is no omniscient, omnipotent, and Such events give rise to basic questions benevolent god. about whether or not life is fair.

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The Problem of Evil Pascal's Wager

The most common theistic defense against (1623 - 1662) the problem, propounded (in different forms) “It makes more sense to believe in God by both Augustine and Leibniz, is to deny than to not believe. If you believe, and the reality of evil by claiming that apparent God exists, you will be rewarded in the cases of evil are merely parts of a larger . If you do not believe, and He whole that embodies greater good. exists, you will be punished for your disbelief. If He does not exist, you have More recently, some have questioned lost nothing either way.” whether the traditional notions of omnipotence and omniscience are coherent. It amounts to hedging your bets.

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The Atheist's Wager Fibonacci (1170-1250)

“It is better to live your life as if there Fibonacci was born are no , and try to make the world in but was a better place for your being in it. If educated in North there is no God, you have lost nothing Africa. and will be remembered fondly by those Fibonacci was you left behind. If there is a benevolent taught mathematics God, He will judge you on your merits in Bugia and and not just on whether or not you traveled widely with believed in Him.” his father.

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Fibonacci (1170-1250) Fibonacci (1170-1250)

He recognized the enormous advantages of Johannes of Palermo, a member of the Holy the mathematical systems used in the Frederick II's court, countries they visited. presented a number of problems as Fibonacci ended his travels around the year challenges to Fibonacci. 1200 and returned to Pisa. Fibonacci solved three of them and put his He wrote a number of important texts, solutions in Flos. including Liber abaci, Practica geometriae, Liber abaci, published in 1202, was based on Flos, and Liber quadratorum. the arithmetic and algebra that Fibonacci accumulated during his travels.

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Fibonacci’s Liber Abaci Fibonacci’s Liber Abaci

It introduced the Hindu-Arabic place-valued A problem in the third section of Liber abaci decimal system and the use of Arabic led to the introduction of the Fibonacci numerals into Europe. numbers and the Fibonacci sequence. The second section of Liber abaci contains a “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many large number of problems about the price of pairs of rabbits can be produced from that goods, how to calculate profit on pair in a year if it is supposed that every transactions, how to convert between the month each pair begets a new pair which various currencies in use at that time, and from the second month on becomes problems which had originated in China. productive?”

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Fibonacci Sequence Fibonacci Numbers

The resulting sequence is: The numbers in the sequence are called Fibonacci numbers. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... We call the numbers the terms of the Notice that each number after the first sequence. two (1 and 1) is the sum of the two Each term can be denoted using subscripts preceding numbers. that identify the order in which the terms appear. That is, 13 is the sum of 8 and 5. We denote the Fibonacci numbers by u1, u2, 55 is the sum of 34 and 21. u3, … , and the n-th term is denoted un.

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Fibonacci Numbers Fibonacci Sequence

So u1=1, u2=1, u3=2 , u4=3 , u5=5, etc. In this case the recurrence equation Note that u3=u1+u2 and u4=u2+u3 and is given at the right. u =1 u5=u3+u4 and this pattern continues for all 1 terms after the first two. The last equation in u =1 Mathematicians write this sequence by the box says that 2 stating the initial conditions, u =1 and u =1, the (n+2)-nd term is 1 2 u = u + u and using a recursive relation which says, in the sum of the n-th n+2 n n+1 an equation, that each term is the sum of its term and the (n+1)- two predecessors. st term.

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Fibonacci Sequence Fibonacci Sequence

When n = 1 for Fibonacci’s sequence of numbers occurs example, this says in many places including that the third term, u1 =1 Pascal’s triangle, the binomial formula, u3, is the sum of the probability, the , the golden first term, u1, and u2 =1 rectangle, plants and nature, and on the the second term, u2, piano keyboard, where one octave contains which is, of course, un+2 = un + un+1 2 black keys in one group, 3 black keys in correct. another, 5 black keys all together, 8 white keys and 13 keys in total.

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Fibonacci Sequence Fibonacci Sequence

Note that the Fibonacci numbers grow Consider the sequence of ratios 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, … formed by dividing without bound, that is, they become each term by the one before it. arbitrarily large, in other words, they go Instead of consistently getting larger, they to infinity. alternate between growing and shrinking! In decimal the sequence is 1, 2, 1.5, 1.666…, In fact the sixtieth term is u60 = 1,548,008,755,920. 1.6, 1.625, 1.615384…, etc. It turns out that the sequence of ratios And u88 = 1,100,087,778,366,101,931. approaches a single target which we can Wow! readily calculate using a clever argument.

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Fibonacci Sequence Fibonacci Sequence

Let’s call the target (or limit as Dividing Fibonacci’s recurrence equation

mathematicians say) L. by un+1 gives Let’s denote the n-th ratio by R . un+2 un un+1 n = + In other words, un+1 un+1 un+1 u which is equal to R = n+1 n u 1 n Rn+1 = +1 Rn

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Fibonacci Sequence Fibonacci Sequence

If we let n approach infinity, then the Using the quadratic formula, we get that last equation becomes 1+ 5 1 L = L = +1 2 L This is called the golden ratio, and it which is equivalent to 2 was known to the ancient Greeks as the L =1+ L most pleasing ratio of the length of a L2 − L −1 = 0 rectangular painting frame to its width.

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The Golden Ratio φ The Golden Ratio φ

Consider a rectangle whose width is 1 So the yellow rectangle and whose length is L. whose width is L – 1 and L whose length is 1 is They assumed that the perfect or similar to the original. golden rectangle has the property that So 1 the removal of a square from it leaves a L 1 (smaller) rectangle that is similar to the = original one. 1 L −1

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The Golden Ratio φ The Golden Ratio φ

Cross multiplying gives The face of the Parthenon in has been seen as a golden rectangle L× L −1 =1×1 () and so have many other facades in L2 − L =1 Greek and Renaissance architecture. L2 − L −1 = 0 The golden ratio appears in many strange places in both the natural world This is the quadratic equation of the and the human world of magnificent Fibonacci ratios. Wow! artistic and scientific achievements.

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The Golden Ratio φ The Golden Ratio φ

Psychologists have shown that the , in The Elements (Book VI, golden ratio subconsciously affects Proposition 30), says that the line AB is many of our choices, such as where to divided in extreme and mean ratio sit as we enter a large auditorium, by C if AB:AC = AC:CB. where to stand on a stage when we We would call it "finding the golden address an audience, etc. etc. section C point on the line".

See Ron Knott’s Fibonacci Numbers and ABC the Golden Section.

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The Golden Ratio φ The Golden Ratio φ Euclid used this phrase to mean the CB AC 1− x x ratio of the smaller part of this line, CB = ⇒ = to the larger part AC (ie the ratio AC AB x 1 CB/AC) is the SAME as the ratio of the larger part, AC to the whole line AB (ie −1+ 5 Solving gives x = is the same as the ratio AC/AB). 2 If we let the line AB have unit length and AC have length x (so that CB is The golden ratio is 1/x = 1.61803398… then just 1–x) then the definition means that See The Golden Ratio

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Fibonacci’s Liber Abaci Fibonacci’s Liber Abaci

Other types of problems in the third section There are also problems involving perfect of Liber abaci include: numbers, the Chinese remainder theorem A spider climbs so many feet up a wall each day and slips back a fixed number each night, how and problems involving the summing many days does it take him to climb the wall. arithmetic and . A hound whose speed increases arithmetically chases a hare whose speed also increases In the fourth section, he deals with irrational arithmetically, how far do they travel before the numbers both with rational approximations hound catches the hare. and with geometric constructions. Calculate the amount of money two people have after a certain amount changes hands and the proportional increase and decrease are given.

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Fibonacci’s Practica geometriae Fibonacci’s Flos

It contains a large collection of geometry In it he gives an accurate approximation to a problems arranged into eight chapters with root of 10x + 2x2+ x3= 20, one of the theorems based on Euclid's Elements and On problems that he was challenged to solve by Divisions. Johannes of Palermo. It includes practical information for surveyors, Johannes of Palermo took this problem from including a chapter on how to calculate the Omar Khayyam's algebra book where it is height of tall objects using similar triangles. solved by means of the intersection a Included is the calculation of the sides of the and a hyperbola. pentagon and the decagon from the diameter Fibonacci proves that the root of the equation of circumscribed and inscribed . is neither an integer nor a fraction, nor the The inverse calculation is also given. square root of a fraction.

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Fibonacci’s Flos Fibonacci’s Liber quadratorum

Without explaining his methods, Fibonacci Liber quadratorum, written in 1225, is then gives the approximate solution in Fibonacci's most impressive piece of work. sexagesimal notation as (1;22,7,42,33,4,40)60 It is a number theory book. 2 3 (This is 1 + 22/60+ 7/60 + 42/60 + ...). In it, he provides a method for finding This converts to the decimal 1.3688081075 Pythogorean triples. which is correct to nine decimal places. Fibonacci first notes that any square number This is a truly remarkable achievement. is the sum of consecutive odd numbers. For example, 1+3+5+7+9=25=52!

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Fibonacci and Pythagorean Triples Fibonacci and Pythagorean Triples

Recall that any odd number is of the form What does this have to do with Pythagorean 2n +1, for some integer n. triples? Well, Fibonacci said, “Ifa the odd numba 2n+1 He noticed that the formula isa squara then you hava Pythagorean triple!” n2+ (2n+1) = (n+1)2 For example, if n = 4, then 2n + 1= 2(4) + 1 = 9 implies that a square plus an odd number then we get the equation 42+ (2×4+1) = (4+1)2 equals the next higher square. which is equivalent to 2 2 2 For example, if n=2 then we get the equation 4 + 3 = 5 22+ (2×2+1) = (2+1)2 which is equivalent to Which gives us the famous (3, 4, 5) Pythagorean triple. 22 + 5 = 32

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Fibonacci’s Liber quadratorum Fibonacci’s Liber quadratorum

Fibonacci also proves many interesting He defined the concept of a congruum, a number theory results including the fact number of the form ab(a + b)(a – b), if a + b is that there is no x, y such that even, and 4 times this if a + b is odd. Fibonacci proved that a congruum must be 2 2 and 2 2 x + y x – y divisible by 24 and he also showed that for x, are both squares. c such that x2 + c and x2 – c are both squares, He also proves that x4 – y4 cannot be a then c is a congruum. square. He also proved that a square cannot be a congruum.

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Fibonacci Numbers Fibonacci Numbers

The Fibonacci Quarterly is a modern Unsolved Problems about Fibonacci journal devoted to studying numbers: mathematics related to this sequence. Are there infinitely many prime Fibonacci Solved Problems: numbers? The only square Fibonacci numbers are 1 Are 1, 8 and 144 the only powers that are and 144! Fibonacci numbers? The only cubic Fibonacci numbers are 1 and 8! The only triangular Fibonacci numbers are 1, 3, 21 and 55!

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Lucas Numbers Edouard Lucas (1842-1891)

Edouard Lucas (1842-1891) gave the name Lucas is also well known for his invention of "Fibonacci Numbers" to the series written the Tower of Hanoi puzzle and other about by Leonardo of Pisa. mathematical recreations. He studied a second series of numbers which The Tower of Hanoi puzzle appeared in 1883 under the name of M. Claus. uses the same recurrence equation but starts with 2 and 1, that is, 2, 1, 3, 4, 7, 11, 18, … Notice that Claus is an anagram of Lucas! His four volume work on recreational These numbers are called the Lucas mathematics Récréations mathématiques has numbers in his honor. become a classic.

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Nicole Oresme (1320-1382) (1320-1382)

A French priest and Oresme was the greatest of the French mathematician. writers of the 14th century. He translated many He wrote Tractatus proportionum, Algorismus of Aristotle’s works proportionum, Tractatus de latitudinibus and questioned formarum, Tractatus de uniformitate et many of the ideas difformitate intensionum, and Traité de la which at that time sphère. were accepted In the Algorismus proportionum is the first without question. use of fractional exponents.

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Nicole Oresme (1320-1382) Nicole Oresme (1320-1382)

In Tractatus de uniformitate, Oresme He was the first to prove Merton’s Theorem, invented a type of coordinate geometry that is, that the distance traveled in a fixed before Descartes, in fact, Descartes time by a body moving under uniform acceleration is the same as if the body moved may have been influenced by his work. at uniform speed equal to its speed at the He proposed the use of a graph for midpoint of the time period. plotting a variable magnitude whose He wrote Questiones Super Libros Aristotelis value depends on another variable. be Anima dealing with the nature, speed and reflection of light.

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Nicole Oresme (1320-1382) Leibniz (1646 – 1716)

Oresme worked on infinite series and was the Leibniz is considered first to prove that the harmonic series 1 + to be one of the 1/2 + 1/3 + 1/4 + … becomes infinite fathers of Calculus. without bound, i.e., it diverges. We will discuss him In Livre du ciel et du monde, he opposed the in further detail theory of a stationary Earth as proposed by later. Aristotle and proposed rotation of the Earth For now, let’s look some 200 years before Copernicus. at his work with He later rejected his own idea. infinite series.

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Leibniz (1646 – 1716) Repeating Decimals

He discovered the following series for π: We can use a clever trick to determine  1 1 1 1 1  the fraction equivalent to a given π = 4×1− + − + − +L repeating decimal.  3 5 7 9 11  Leibniz’s idea out of which his calculus For example, suppose we want to know grew was the inverse relationship of what fraction is equal to 0.666666… sums and differences for sequences of Let x = the decimal. numbers. Then multiply both sides by 10.

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Repeating Decimals Repeating Decimals

Finally, subtract the two equations. If the pattern is two digits, you multiply by 100 instead of 10. 10x = 6.66666666K For example, suppose we want to know x = 0.66666666 K what fraction is equal to 0.45454545… 9x = 6 Let x = 0.45454545… Divide both sides by 9 Multiply both sides by 100, so Thus, x = 6/9 = 2/3. 100x = 45.45454545…

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Repeating Decimals

Subtract the two equations: 100x = 45.45454545K x = 0.45454545K 99x = 45

The solve for x. Thus, x = 45/99 = 5/11.

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