Aristotle's Organon

Aristotle was a Greek philosopher and scientist, a pupil of Plato, and teacher of Alexander the Great. The Organon (Instrument) refers to a collection of six of Aristotle's work in logic: Categories, Prior Analytics, De Interpretatione, Posterior Analytics, Sophistical Refutations, and Topics. Andronicus of Rhodes (c. 428-348 B.C.) determined the ordering of the six works around 40 B.C. Although Plato (c. 428-348 B.C.) and Socrates (c. 470-399 B.C.) delved into logical themes, Aristotle actually systematized the study of logic, which dominated scientific reasoning in the Western world for 2,000 years.
The goal of Organon is not to tell readers what it is true, but rather to give approaches to how to investigate the truth and how to make sense of the world. The primary tool in Aristotle's tool kit is the syllogism, a three-step argument, such as "All women are mortal; Cleopatra is a woman; therefore, Cleopatra is mortal." If the two premises are true, we know that the conclusion must be true. Aristotle also made a distinction between particulars and universals (general categories). Cleopatra is a particular term. Woman and mortal are universal terms. When universal are used, they are preceded by "all", "some", or "no." Aristotle analyzed many possible kinds of syllogisms and showed which of them are valid.
Aristotle also extended his analysis to syllogisms that involved modal logic - that is, statements containing words "possibly" or "necessarily." Modern mathematical logic can depart from Aristotle's methodologies or extend his work into other kinds of sentences structures, including ones that express more complex relationships or ones that involve more than one quantifier, such as "No women like all women who dislike some women." Nevertheless, Aristotle's systematic attempt at developing logic is considered to be one of humankind's greatest achievements, providing an early impetus for fields of mathematics that are close in partnership with logic and even influencing theologians in their quest to understand reality.  

Platonic Solids

A Platonic Solid is a convex multifaceted 3-D object whose faces are all identical polygons, with sides of equal length and angles of equal degrees. A Platonic solid also has the same number of faces meeting at every vertex. The best-known example of a Platonic solid is the cube, whose faces are six identical squares.
The ancient Greeks recognized and proved that only five Platonic solids can be constructed: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. For example, the icosahedron has 20 faces, all in a shape of equilateral triangles.
Plato described the five Platonic solids in Timaeus in around 350 B.C. He was not only awestruck by their beauty and symmetry, but he also believed that the shapes described the structures of four basic elements thought to compose the cosmos. In particular, the tetrahedron was the shape that represented the fire, perhaps because of the polyhedron's sharp edges. The octahedron was air. Water was made up of icosahedra, which are smoother than the other Platonic solids. Earth consisted of cubes, which look sturdy and solid. Plato decided that God used the dodecahedron for arranging the constellations in the heavens.
Pythagoras of Samos - the famous mathematician and mystic who lived in the time of Buddha and Confucius, around 550 B.C. - probably knew of three of the five Platonic solids (the cube, tetrahedron, and dodecahedron). Slightly rounded versions of the Platonic solids made of stone have been discovered in areas inhabited by the late Neolithic people of Scotland at least 1,000 years before Plato. The German astronomer Johannes Kepler (1571-1630) constructed models of Platonic solids nested within one another in an attempt to describe the orbits of planets about the sun. Although Kepler's theories were wrong, he was one of the first to insist on a geometrical explanation for celestial phenomena.

Quadrature of the Lune

Ancient Greek mathematicians were enchanted by the beauty, symmetry, and order of geometry.Succumbing to this passion, Greek mathematician Hippocrates of Chios demonstrated how to construct a square with an area equal to a particular lune. A lune is a crescent-shaped area, bounded by two concave circular arcs, and this Quadrature of the Lune is one of the earliest-known proofs in mathematics In other words, Hippocrates demonstrated that the areas of these lunes could be expressed exactly asa rectilinear area, or "quadrature."
For the ancient Greeks, finding the quadrature meant using a straightedge and compass to construct a square equal in area to a given shape. If such construction is possible, the shape is said to be "quadrable" ( or "squarable"). The Greeks had accomplished the quadrature of polygons, but curved forms were more difficult. In fact, it must have seemed unlikely, at first, that curved objects could be quadrable at all.
Hippocrates is also famous for compiling the first-known organized work on geometry, nearly a century before Euclid. Euclid may have used some of Hippocrates' ideas in his own work, Elements. Hippocrates' writings were significant because they provided a common framework upon which other mathematicians could build.
Hippocrates' lune quest was actually part of a research effort to achieve the "quadrature of the circle" - that is, to construct a square with the same area as the circle. Mathematicians have tried to solve the problem of "squaring the circle" for more than 2,000 years, until Ferdinand von Lindemman in 1882 proved that it is impossible. Today, we know that only five types of lunes are quadrable. Three of these were discovered by Hippocrates, and two more were found in the mid-1700s.

Zeno's Paradox

For more than a thousand years, philosophers and mathematician have tried to understand Zeno's paradoxes, a set of riddles that suggest that motion should be impossible or that it is an illusion. Zeno was a pre-Socratic Greek philosopher from southern Italy. His most famous paradox involves the Greek hero Alchilles and a slow tortoise that Achilles can never overtake during a race once the tortoise is given a head start. In fact, the paradox seems to imply that you can never leave the room you are in.  In order to reach the door, you must first travel half the distance there. You;ll also need to continue to half the remaining distance, and half again, and so on. You won't reach the door in a finite number of jumps! Mathematically one can represent this limit of an infinite sequence of actions as the sum of the series (1/2 + 1/4 + 1/8 + ...). One modern tendency is to attempt to resolve Zeno's paradox by insisting that this sum of infinite series 1/2 + 1/4 + 1/8 is equal to 1. If each step is done in half as much time, the actual time to complete the infinite series is no different than the real time required to leave the room.
However, this approach may not provide a satisfying resolution because it does not explain how one is able to finish going through an infinite number of points, one after another. Today, mathematicians make use of infinitesimal (unimaginably tiny quantities that are almost but not quite zero) to provide a microscopic analysis of the paradox. Coupled with a branch of mathematics called nonstandard analysis and, in particular, internal set theory, we may have resolved the paradox, but debates continues. Some have also argued that if space and time are discrete, the total number of jumps in going from one point to another must be finite.

Pythagoras Founds Mathematical Brotherhood

Around the year 530 B.C., the Greek mathematician Pythagoras moved to Croton, Italy, to teach mathematics, music, and reincarnation. Although many of Pythagoras's accomplishments may actually being due to his disciples, the ideas of his brotherhood  influenced both numerology and mathematics for centuries. Pythagoras is usually credited with discovering mathematical relationships relevant to music harmonies. For example, he observed that vibrating strings produce harmonious sounds when the ratios of the length of the strings are whole numbers. He also studied triangular numbers ( based on patterns of dots in a triangular shape) and perfect numbers (integers that are the sum of their proper positive divisors). Although the famous theorem that bears his name, a^2 + b^2 + c^2 for a right  triangle with legs a, b and hypothenuse c, may have been known to the Indians and Babylonians much earlier, some scholars have suggested that Pythagoras or his students were among the first Greeks to prove it.
To Pythagoras and his followers, numbers were like gods, pure and free from material change. Worship of the numbers 1 though 10 was kind of polytheism for the Pythagoreans. They believed that numbers were alive, with a telepathic form of consciousness. Humans could relinquish their three-dimensional lives and telepatize with those numbers by using various forms of meditation.
Some of these seemingly odd ideas are not foreign to modern mathematicians who often debate whether mathematics is a creation of the human mind or if it's simply a part of the universe, independent of human thought. To the Pythagoreans, mathematics was an ecstatic revelation. Mathematical and Theological blending flourished under the Pythagoreans and eventually affected much of the religious philosophy in Greece, played a role in religion of the Middle Ages, and extended to philosopher Immanuel Kant in modern times.  Bertrand Russell mused that if it were not for Pythagoras, theologians wouldn't not have so frequently sought logical proofs of God and immortality.

Go

Go is a two-player board game that originated in China around 2000 B.C. The earliest written references to the game are from the earliest Chinese work narrative history, Zuo Zhuan ( Chronicles of Zuo), which describes a man in 548 B.C. who played the game. The game spread to Japan, where it became popular in the thirteenth century. Two players alternately place black and white stones on intersections of a 19x19 playing board. A stone or a group of stones is captured and removed if it's tightly surrounded by stones of the opposing color. The objective is to control a larger territory than one's opponent.
Go is complex for many reasons, including its large game board, multifaceted strategies, and huge numbers variations in possible games. After taking symmetry into account here, there are 32,940 opening moves, of which 992 are considered to be strong ones. The number of possible board configurations is usually estimated to be one of the order 10^172, with about 10^768 possible games. Typical games between talented players consist of about 150 moves, with an average of about 250 choices per move. While powerful chess software is capable of defeating top chess players, the best Go program often lose to skillful children.
Go-playing computers find it difficult to "look ahead" in the game to judge outcomes because many more reasonable moves must be considered in Go than in chess. The process of evaluating favorability of a proposition is also quite difficult because a difference of a single unoccupied grid point can affect large group of stones.
In 2006, two Hungarians researchers reported that an algorithm called UCT (for Upper Confidence bounds applied to Trees) could compete with professional Go players, but only on 9x9 boards. UCT helps the computer focus its search on the most promising move.

Pythagorean Theorem and Triangles

Today, young children sometimes first hear of the famous Pythagorean theorem from the mouth of the Scarecrow, when he finally gets a brain in MGM's 1939 film version of The Wizard of Oz. Alas, the Scarecrow's recitation of the famous theorem is completely wrong!
The Pythagorean theorem states that for any triangle, the square of the hypothenuse c is equal to the sum of the squares on the two (shorter) "leg" lengths a and b - which is written a^2 + b^2 = c^2. The theorem has more published proofs than any other, and Elisha Scott Loomie's book Pythagorean proposition contained 367 proofs.
Pythagorean triangles (PTs) are right triangles with integer sides. The " 3-4-5" PT - with legs of length 3 and 4, and a hypothenuse of length 5 - is the only PT with three sides as consecutive numbers and the only triangle with integers sides, the sum of whose sides (12) is equal to double its area (6).  After the 3-4-5 PT, the next triangle with consecutive leg lengths 21-20-29. The tenth such triangle is much larger: 27304197-273004196-38613965.
In 1643, French mathematician Pierre de Fermat (1601-1665) asked for a PT, such that both the hypothenuse c and the sum (a + b) had values that were square numbers. it was startling to find that the smallest three numbers satisfying these conditions are 4,565,486,027,761, 1, 061, 652,293,520, and 4,687,298,610,289. It turns out that the second such triangle would be so "large" that if its numbers were represented as feet, the triangle's legs would project from Earth to beyond sun!
Although Pythagoras is often credited wit the formulation of the Pythagorean theorem, evidence suggests that the theorem was developed by Hindu mathematician Baudhayana centuries earlier around 800 B.C. in his book Baudhayana Sulba Sutra. Pythagorean triangles were probably known  even earlier to Babylonians.

Les des

Les dés

Imaginez un monde sans nombres aléatoires. Dans les années 1940, la génération de nombres statistiques  aléatoires était essentielle parce qu’il était important pour les physiciens de simuler des explosions thermonucléaires. Aujourd'hui, de nombreux systèmes informatiques utilisent les nombres aléatoires pour sélectionner de façon objective des échantillons d’électeurs potentiels.

A l’origine, les dés, fabriqués à partir d'astragale (un os de cheville) d'ongulés, constituèrent l'une des premières méthodes de production de nombres aléatoires. Dans les anciennes civilisations, les dieux étaient censés contrôler le résultat des lancers de dés. Ainsi, les dés étaient invoqués pour prendre des décisions cruciales, depuis la sélection des dirigeants jusqu’au partage de propriétés à l’occasion d’héritage. Aujourd'hui encore, la métaphore de Dieu contrôlant les dés est ordinaire, comme le prouve la citation de l’astrophysicien Stephen Hawking : «non seulement Dieu joue aux dés, mais Il nous embrouille parfois en les jetant là où ils ne peuvent pas être vus."

Les plus anciens dés connus ont été exhumés avec un jeu de backgammon vieux de 5.000 ans près de la ville légendaire de Burnt au sud de l'Iran. La ville qui présente quatre étapes successives de civilisations détruites par des incendies, a été abandonnée en 2100 avant JC. Sur ce même site, les archéologues ont également découvert le plus ancien œil artificiel connu qui donnait un regard hypnotique au visage d'une antique prêtresse ou devineresse.

Pendant des siècles, les lancers de dés ont été utilisés pour enseigner les probabilités. Pour un seul lancer d’un dé à n côtés avec un nombre différent sur chaque face, la probabilité d'obtenir une valeur est 1 / n. La probabilité d’obtenir une séquence particulière de i nombres est 1 / n ^ i. Par exemple, les chances d’obtenir un 1 suivi d’un 4 sur un dé traditionnel est 1/6 ^ 2 = 1/36. En utilisant deux dés traditionnels, la probabilité d’obtenir une somme donnée est le nombre de façons d’obtenir cette somme divisée par le nombre total de combinaisons, c’est à dire la somme divisée par le nombre total de combinaisons. C’est pourquoi une somme égale à 7 est beaucoup plus plus facile à obtenir qu’une somme égale à 2.

Tic Tac Toe

The game of Tic Tac Toe (TTT) is among humanity's best-known and most ancient games. Although the precise date of TTT with its modern rules may be relatively recent, archeologists can trace what appear to be "three-in-a-row games" to ancient Egypt around 1300 B.C., and I suspect that similar kinds of games originated at the very dawn of human societies. For TTT, two players, O and X, take turns making their symbols in the spaces of a 3x3 grid. The player who first places three of his own marks on a horizontal, diagonal, or vertical row wins. A draw can always be obtained for the 3x3 board.
In ancient Egypt, during the time of the great pharaohs, boar games played an important role in everyday life, and TTT-like games are known to have been played during these ancient days. TTT may be considered an "atom" upon which the molecules of more advanced games of position were built through the centuries. With the slightest variations and extensions, the simple game of TTT becomes a fantastic challenge requiring a significant time to master.
Mathematicians and puzzles aficionados have extended TTT to larger boards, higher dimensions, and strange playing surfaces such as rectangular or square boards that are connected to their edges to form a torus (doughnut shape) or Kline bottle (a surface with just one side).
Consider some TTT curiosities. Players can place their Xs and Os on a TTT board in 9! = 362,880 ways. There are 255,168 possible games in TTT when considering all possible games that end in 5, 6, 7, 8, and 9 moves. In the early 1980s, computer geniuses Danny Hillis, Brian Silverman, and friends built a Tinkertoy computer that played TTT. The device was made of 10,000 Tinkertoy parts. In 1998, researchers and students at the University of Toronto created a robot to play three-dimensional (4 x 4 x4) TTT with a human.

Le Papyrus de Rhind

Lundi 19 novembre 2012,

Traduit de l'anglais par Hervé....mon ami.

Le Papyrus de Rhind

Le Papyrus de Rhind est considéré comme la plus importante source d‘informations concernant les mathématiques des anciens Égyptiens. Ce rouleau, de 30 cm de haut sur 5,5 mètres de long, a été retrouvé dans une tombe de Thèbes sur la rive Est du Nil. Le scribe Ahmès l’a écrit en hiératique, une écriture dérivée du système hiéroglyphique. Cet écrit ayant été réalisé vers 1650 avant J.C. environ, Ahmès est l’individu le plus anciennement référencé de l'histoire des mathématiques ! Le rouleau contient également les symboles mathématiques les plus anciens connus à ce jour pour des opérations mathématiques : le signe “+” est en effet désigné par une paire de jambes allant vers le nombre à ajouter.

En 1858, l'égyptologue et avocat écossais Alexander Henry Rhind visitait l’Egypte pour soigner des problèmes de santé quand il acheta le parchemin sur un marché à Louxor. Le British Museum de Londres a acquis le rouleau en 1864.

Ahmès écrivit que le papyrus permet un "calcul précis afin d’enquêter sur les choses, et d’avoir la connaissance de toutes choses et de tous mystères ... en toute sécurité ". Le contenu du rouleau traite de problèmes mathématiques intégrant des fractions, des progressions arithmétiques, l'algèbre et la géométrie pyramidale, ainsi que des cas mathématiques utiles à l'arpentage, la construction et la comptabilité. Le problème qui m'intrigue le plus est le problème 79 dont l'interprétation a d'abord été déroutante.

Aujourd'hui, beaucoup interprètent le problème 79 comme un puzzle qui peut être traduit par «Sept maisons contiennent sept chats. Chaque chat tue sept souris. Chaque souris a mangé sept épis. Chaque épi aurait produit sept heqats (1 heqat ou hekat = 4.8 litres) de le blé. Combien cela fait-il d’heqats ?". Fait intéressant, ce mème casse-tête indestructible, impliquant le chiffre 7 et des animaux, semble avoir persisté pendant des milliers d'années ! En effet, nous observons quelque chose de similaire dans le Liber Fibonacci Abati (Livre des calculs), publié en 1202, et plus tard dans le puzzle de St. Ives, un poème pour enfants en vieil anglais qui fait référence à 7 chats.

Rhind Papyrus

The Rhind Papyrus is considered to be the most important known source of information concerning ancient Egyptians mathematics. This scroll, about a foot (30 cm) high and 18 feet (5.5 meters) long, was found in a tomb in Thebes on the east bank of the river Nile. Ahmes, the scribe, wrote it in hieratic, a script related to the hieroglyphic system. Given that the writing occurred in around 1650 B. C., this makes Ahmes the earliest-known individual in the history of mathematics! The scroll also contains the earliest-known symbols for mathematical operations - plus is denoted by a pair of legs walking toward the number to be added.
In 1858, Scottish lawyer and Egyptologist Alexander Henry Rhind had been visiting Egypt for health reasons when he bought the scroll in a market in Luxor. The British Museum in London acquired the scroll in 1864.
Ahmes wrote that the scroll gives an "accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secures." The content of the scroll concerns mathematical problems involving fractions, arithmetic progressions, algebra, and pyramid geometry, as well as practical mathematics useful for surveying, building, and accounting. The problem that intrigues me the most is Problem 79, the interpretation of which was initially baffling.
Today, many interpret Problem 79 as a puzzle, which may be translated as "Seven houses contain seven cats. Each cat kills seven mice. Each mouse had eaten seven ears of grain. Each ear of grain would have produced seven hekats (measures) of wheat. What is the total number of all of these?" Interestingly, this indestructible puzzle meme, involving the number 7 and animals, seems to have persisted through thousands of years! We observe something quite similar in Fibonacci's Liber Abati (Book of Calculation), published in 1202, and later in the St. Ives puzzle, an Old English children's rhyme involving 7 cats.

Plimpton 322

Plimpton 322 refers to a mysterious Babylonian clay tablet featuring numbers in cuneiform script in a table of 4 columns and 15 rows. Eleanor Robson, a historian of science, refers to it as "one of the world's most famous mathematical artifacts." Written around 1800 B. C., the table lists Pythagoreans triples - that is, whole numbers that specify the side lengths of right triangles that are solutions to the Pythagorean theorem a^2 + b^2 = c^2. For example, the numbers 3, 4, and 5 are a Pythagorean triple. The fourth column in the table simply contains the row number. Interpretations vary as to the precise meaning of the numbers in the table, with some scholars suggesting that the numbers were solutions for students studying algebra or trigonometry-like problems.
Plimpton 322 is named after New York publisher George Plimpton who, in 1922, bought the tablet for $10 dollars from a dealer than donated the tabMesopotamia, the fertile valley of the Tigris and Euphrates rivers, which is now located in Iraq. To put the era into perspective, the unknown scribe who generated Plimpton 322lived within about a century of King Hammurabi, famous for his set of laws that included "an eye for an eye, a tooth for a tooth." According to biblical history, Abraham, who is said to have led his people west from the city of Ur on the bank of the Euphrates into Canaan, would have been another near contemporary of the scribe.
The Babylonians wrote on wet clay by pressing a stylus or wedge into the clay. In the Babylonian number system, the number 1 was written with a single stroke and the numbers 2 through 9 were written by combining multiples of a single stroke.

Magic Squares

Legends suggest that magic squares originated in China and were first mentioned in a manuscript from the time of Emperor Yu, around 2200 B.C. A magic square consists of N^2 boxes, called cells, filed with integers that are all different. The sums of the numbers in the horizontal rows, vertical columns, and main diagonal are equal.
If the integers in a magic square are the consecutive numbers from 1 to N^2, the square is said to be of the Nth order, and the magic number, or sum of each row, is a constant equal to N(N^2  + 1)/2.
As far back as 1693, the 880 different fourth-order magic squares were published posthumously in Des quassez ou tables magiques by Bernard Frenicle de Bessy, an eminent amateur french mathematician and one of the leading magic squares of all time.
We've come a long way from the simplest 3 x 3 magic squares venerated by civilizations of almost every period or continent, from the Mayans Indians o the Hasua people of Africa. Today, mathematicians study these magic objects in high dimensions - for example, in the form of fourth-dimensional hypercubes that have magic sums within all appropriate directions.

Dice

Imagine a world without random numbers. In the 1940s, the generation of statistically random number was important because was important to physicists simulating thermonuclear explosions, and today, many computer networks employ random numbers to select unbiased samples of potential voters.
Dice, originally made  from anklebones of roofed animals, were one of the earliest means for producing random numbers. In ancients civilizations, the gods were believed to control the outcome of dice tosses; thus, dice were relied upon to make crucial decisions, ranging from the selection of rulers to the division of property in an inheritance. even today, the metaphor of God controlling dice  is common, as evidence by astrophysicist Stephen Hawking's quote, "Not only God play dice, but He sometimes confuses us by throwing them where they can't be seen."
the oldest known dice were excavated together with a 5,000 year backgammon set from the legendary Burnt City in southeastern Iran. The city represents four stages of civilization that were destroyed by fires before being abandoned in 2100 B.C. At this same site, archeologists also discovered the earliest known artificial eye, which once stared out hypnotically from the face of an ancient female priestess or soothsayer.
For centuries, dice rolls have been used to teach probability. For a single roll of an n-sided die with a different number for each face, the probability of rolling any value is 1/n. The probability of ruling a particular sequence of i numbers is 1/n^i. For example, the chances of rolling a 1 followed by 4 on a traditional die is 1/6^2 = 1/36. Using two traditional dice, the probability of throwing any given sum is the number of ways to throw that sum divided by the total number of combinations, which is a sum divided by the total number of combinations, which is why a sum of 7 is much more likely than a sum of 2.

Quipu

The ancient incas used quipus (pronounced "key-poos"), memory banks made of strings and knots, for storing numbers. Until recently, the oldest-known quipus dated from about A.D.650. However, in 2005, a quipu from the Peruvian coast city of Caral was dated about 5,000 years ago. The incas of south America had a complex civilization with common state religion and a common language. Although they did not have writing, they kept extensive records encoded by logical numerical system on quipus, which varied in complexity from the three to around three thousand cords. Unfortunately, when the Spanish came to South America, they saw the strange quipus and thought they were works of the Devil. The Spanish destroyed thousands of them in the name of God, and today only 600 quipus remain. Knots types and positions, cord directions, cord levels, and color and spacing represent numbers mapped to real-world objects. Different knot groups were used for different powers of 10. The knots were probably used to record human and material resources and calendar information. The quipus may have contained more information such as construction plans, dance patterns, and even aspects of inca history. The quipu is significant because it dispels the notion that mathematics flourishes only after a civilization has developed writing; however, societies can reach advanced states without ever having developed written records. Interesting, today there are computer systems whose file managers are called quipus, in honor of this very useful ancient device. One sinister application of the quipusby the incas was as a death calculator. Yearly quotas of adults and children were ritually slaughter, and this enterprise was planned using a quipu. Some quipus represented the empire, the cords referred to roads and the knots to sacrificial victims.

Ishango bone

In 1960, Belgium geologist and explorer Jean de Heinzelin de Braucourt (1920-1998) discovered a baboon bone with markings in what is today the Democratic Republic of Congo. The Ishango bone, with it's sequences of notches, was first thought to be a simple tally stick used by a Stone Age African. However, according to some scientists, the marks suggest a mathematical prowess that goes beyond counting of objects. The bone was found in Ishango, near the headwaters of the Nile River, the home of a large population of upper Paleolithic people prior to a volcanic eruption that buried the area. One column of marks on the bone begins with three notches that double to six notches. Four notches double to eight. Ten notches halve to five. This may suggest a simple understanding of doubling and halving. Even more striking is the fact that numbers in other columns are all odd (9, 11, 13, 17, 19, 21). One column contains the prime numbers between 10 and 20, and the numbers in each column sum to 60 or 48, both multiplies 12. A number of tally sticks have been discovered that predate the Ishango bone. For example, the Switzerland Lebombo bone is a 37,000-yard baboon fibula with 29 notches. A 32,000-year old tibia with 57 notches, grouped in fives, was found in Czechoslovakia. Although quite speculative, some have hypnothized that the markings on the Ishango bone form a kind of lunar calendar for a Stone Age woman who kept their menstrual cycles, giving rise to the slogan "menstruation created mathematics." even if the Ishango was a simple bookkeeping device, these tallies seems to sets us apart from animals and represent the first steps to symbolic mathematics. The full mystery of the Ishango bone can't be solved until other similar bones are discovered.

Knots

The use of knots may predate modern humans (Homo Sapiens). For example, seashells colored with ocher, pierced with holes, and dated to 82,000 years ago have been discovered in a Moroccan cave. Other archeological evidence suggests much older bead use in humans. The piercing implies the use of cords and the use of a knot to hold the objects to a loop, such as a necklace.
The quintessence of ornamental knots is exemplified by The Book of Kells, an ornately illustrated Gospel Bible, produced by Celtic monks in about A.D. 800. In modern times, the study of knots, such as the trefoil knot with three crossings, is part of a vast branch of mathematics dealing with closed twisted loops. In 1914, German mathematician Max Denh (1878-1952) showed that the trefoil knot's mirror images are not equivalent.
For centuries, mathematicians ave tried to develop ways to distinguish tangles that look like knots (called unknots) from true knots and to distinguish true knots from one another. Over the years, mathematicians have created seemingly endless tables of distinct knots. So far, more than 1.7 million nonequivalent knots with pictures containing 16 or fewer crossings have been identified.
Entire conferences are devoted to knots today. Scientists study knots in fields ranging from molecular genetics - to help us understand how to unravel a loop of DNA - to particle physics, in an attempt to represent the fundamental nature of elementary particles.
Knots have been crucial to the development of civilization, where they have been used to tie clothing, to secure weapons to the body, to create shelters, and to permit the sailing of ships and world exploration. Today, knot theory in mathematics has become so advanced that mere mortals find it challenging to understand its most profound applications. In a few millennia, humans have transformed knots from simple necklace ties to models of the very fabric of reality.

Cicada-generated prime numbers

Cicadas are winged insects that evolved around 1.8 million years ago during the Pleistocene epoch, when glaciers advanced and retreated across North America. Cicadas of the genus Magicicada spend most of their lives below ground, feeding on the juices of plant roots, and then emerge, mate, and die quickly. These creatures display a startling behavior: Their  emergence is synchronized with periods of years that are usually the prime numbers 13 and 17. (A prime number is an integer such as 11, 13, and 17 that has only two integers divisors: 1 and itself.) During the spring of their 13th or 17th year, these periodical cicadas construct an exit tunnel. Sometimes more than 1.5 million individuals emerge in a single acre; this abundance of bodies may have survival value as they overwhelm predators such as birds that cannot possibly eat them all at once.
Some researchers have speculated that the evolution of prime-number life cycles occurred so that the creatures increased their chances of evading shorter-lived predators and parasites. For example, if the cicadas had a 12-year life cycles, all predators with life cycles of 2, 3, 4, or 6 years might more easily find the insects. Mario Markus of the Max Planck Institute for Molecular Physiology in Dortmund, Germany, and his co-workers discovered that these kind of prime-nimber cycles arise naturally from evolutionary mathematical models of interactions between predator and prey. In order to experiment, they first assigned random life-cycle durations to their computer-simulated populations. After some time, a sequence of mutations always locked the synthetic cicadas into a stable prime-number cycle.
Of course, this research is still in it's infancy and many questions remain. What is special about 13 and 17? What predators or parasites have actually existed to drive the cicadas to these periods? Also, a mystery remains as to why, of the 1,500 cicada species worldwide, only a small number of genus Magicicada are known to be periodical.

Primates count

Around 60 million years ago, small, lemur-like primates had evolved in many areas of the world, and 30 million years ago, primates with monkeylike characteristics existed. Could such creatures count? The meaning of counting by animals is a highly contentious issue among animal behavior experts. However, many scholars suggest that animals have some sense of number. H. kalmus writes in his nature article "Animals as mathematicians": There is now little doubt that some animals such as squirrels or parrots can be trained to count... Counting faculties have been reported in squirrels, rats, and for pollinating insects. Some of these animals and others can distinguish numbers in otherwise similar visual patterns, while others can be trained to recognize and even to reproduce sequences of acoustic signals. A few can even be trained to recognize and even to reproduce sequences of acoustic signals. A few can even be trained to tap out the number of the elements (dots) in a visual pattern... The lack of the spoken numeral and the written symbol makes many people reluctant to accept animals as mathematicians. Rats have been shown to "count" by performing an activity the correct number of times in exchange for a reward. Chimpanzees can press numbers on a computer that match numbers of bananas in a box. Gesture Matsuzawa of the primate research institute at Kyoto university in Japan taught a chimpanzee to identify numbers from 1 to 6 by pressing the appropriate computer key when she was shown a certain number of objects on the computer screen. Michael Beran, a research scientist at Georgia state university in Atlanta, Georgia, trained chimps to use a computer screen and joystick. The screen flashed a numeral and then a series of dots, and the chimps had to match the two. One chimp learned numerals 1 to 7, while another managed to count to 6. When the chimps were tested again after a gap of three years, both chimps were able to match numbers, but with double the error rate.

Ant odometer

Ants are social insects that evolved from vespoid wasps in the mid-Cretaceous period, about 150 million years ago. After the rise of flowering plants, about 100 million years ago, ants diversified into numerous species. The Sahara desert ant travels immense distances over sandy terrain, often completely devoid of landmarks, as it searches for food. These creatures are able to return to their nests using a direct route rather than by retracing their outbound path. Not only they judge directions, using light from the sky for orientation, but they also appear to have a built-in "computer" that functions like a pedometer that counts their steps and allows them to measure exact distances. An ant may travel as far as 160 feet (about 50 meters) until it encounter a dead insect, whereupon it tears a piece to carry directly back to its nest, accessed via a hole often less than a millimeter in diameter. By manipulating the leg lengths of ants to give them longer and shorter strides, a research team of German and Swiss scientists discovered that ants "count" steps to judge distance. For example, after ants had reached their destination, the legs were lengthened by adding stilts or shortened by partial amputation. The researchers then returned the ants so that the ants could start on their journey back to the nest. Ants with the stilts traveled too far and passed best entrance, while those with the amputated legs did not reach it. However, if the ants started their journey from their nest with the modified legs, they were able to compute the appropriate distances. This suggests that stride length is the crucial factor. Moreover, the highly sophisticated computer in the ant's brain enables the ant to compute a quantity related to the horizontal projections of its path so that it doesn't not become lost even if the sandy landscape develops hills and valleys during its journey.

Proof by working backwards(3)

The expression (x^4)/4 + (x + 1)^3 > ((x +1)^4)/4 for all real numbers x >= -1.

(Scratch work)

x^4 + 4(x^2 +2x +1)(x +1) > (x +1)^2(x^2 +2x +1)

x^4 + 4(x^3 +2x^2 +x +x^2 +2x +1) > (x+1)(x^3 +3x^2 +3x +1)

x^4 + 4(x^3 +3x^2 +3x +1) > x^4 +4x^3 +6x^2 +4x +1

x^4 + 4x^3 +12x^2 + 12x + 4 > x^4 + 4x^3 + 6x^2 + 4x + 1

6x^2 + 6x + 3 > 0

3x^2 +3x +1 > 0

Proof. Let f(x) = 3x^2 + 3x + 1, then f '(x) = 6x + 3 which implies that f attains its minimum value at x = -1/2. So for all x > -1,

f(x) >= 3/4 - 3/2 +1 = 1/4 > 0

Therefore we have that

3x^2 +3x +1 > 0 =>

6x^2 + 6x +3 > 0 =>

x^4 +  4x^3 +12x^2 + 12x +4 > x^4 +4x^3 +6x^2 +4x +1 =>

x^4 + 4(x^3 +3x^2 +3x +1) > x^4 +4x^3 +6x^2 +4x +1 =>

x^4 +4(x^3 +2x^2 + x + x^2 + 2x +1) > (x + 1)(x^3 + 3x^2 + 3x +1) =>

x^4 + 4(x^2 +2x + 1)(x + 1) > (x + 1)^2 (x^2 +2x + 1) =>

(x^4)/4 + (x + 1)^3 > ((x + 1)^4)/4

QED

Proof by working backwards(2)

(2) If n^3 + 5n + 6 is divisible by 3 for some integer n, then (n + 1)^3 + 5(n + 1) + 6 is divisible by 3.

(Scratch work)

(n + 1)^3 + 5(n + 1) + 6 = (n + 1)(n^2 + 2n +1) +5n + 5 + 6
                                     
                                       = n^3 + 2n^2 + n + n^2 + 2n + 1 + 5n + 11

                                       = n^3 + 5n + 6 + 3n^2 + 3n + 6

                                       = n^3 + 5n + 6 + 3(n^2 + n + 2)


Proof. Assume that n^3 + 5n + 6 is divisible by 3 for some integer n. Then

3 | (n^3 + 5n + 6) =>

3 | [(n^3 + 5n + 6) + 3(n^2 +n + 2)] =>

3 | [n^3 + 5n + 6 + 3n^2 + 3n + 6] =>

3| [n^3 + 2n^2 +n + n^2 + 2n + 1 + 5n + 11] =>

3| [(n + 1)(n^2 + 2n + 1) + 5n +5 +6] =>

3| [(n + 1)^3 + 5(n + 1) + 6]

QED

Proof by working backwards(1)

Write clear and complete proofs by working backwards for the following mathematical statement.
(1) The expression x + 9/x >= for all real numbers x > 0.
(Scratch work)

x + 9/x >= 6

x^2 + 9 >= 6x

x^2 - 6x + 9 >= 0

(x - 3)^2 >= 0

Proof. Assume that x is a real number with x > 0. Then notice that

(x -3)^2 >= 0

x^2 - 6x +9 >= 0

x^2 + 9 >= 6x

x + 9/x >= 6x.

QED

Proof by cases(4)

If a and b  are real numbers.

Case 1: Assume that a >= 0, b >= 0. Then |a| = a, |b| = b, and |a - b| = a - b. So in this case
||a| - |b|| = |a - b| =< |a - b|.

Case 2: Assume that a >= 0, and b < 0. Then |a| = a and |b| = -b.

-Sub-case 1: Assume that |a| >= |b|, then 0=< a + b =< a =< a - b.
So |a + b| < |a - b| giving us

||a| - |b|| = |a + b| < |a - b|.

-Sub-case 2: Assume that |b| > |a|, then we have both 0 < -b =< a - b, and b < a + b < 0. So it follows that |a + b| < |a - b| giving us

||a| - |b|| = |a + b| < |a - b|

Case 3: Assume that a < 0, and b >= 0. Then |a| = - a and |b| = b.
-Sub-case 1: Assume that |a| >= |b|, then 0 < - a =< b - a = |a - b|, and a=< a + b =< 0. So that |a + b| =< |a - b|. It follows that

||a| - |b|| = |- a - b| = |a + b| =< |a - b|.

-Sub-case 2: Assume that |b| > |a|, then a - b =< - a - b< 0. This gives us |- a - b| =< |a - b|. So again
||a| - |b|| = |- a - b| =< |a - b|.

Case 4: Assume that a < 0, and b < 0. Then |a| = - a and |b| = - b.
Then in this case
||a| - |b|| = |- a + b| = |a - b| =< |a - b|.

So in all cases ||a| - |b|| =< |a - b|

QED

Proofs by Case (3)

(3) If n is an even integer, then n = 4j or n = 4j - 2 for some integer j.

Proof. Assume that n is an even integer. Then n = 2k for some integer k.
Case 1: Assume that k is even. Then k = 2p for some integer p. So in this case

n = 2k = 2(2p) = 4p

and the results holds.

Case 2: Assume that k is odd. Then k = 2p + 1 for some integer p. So in this case

n = 2k = 2(2p + 1) = 4p + 2 + 4 - 4 = 4(p - 1) - 2 = 4j - 2

where j = p - 1 is an integer, and therefore the result holds.

So in all cases n = 4j or n = 4j - 2 for some integer j.

QED

Proofs by Case (2)

Write clear and complete proofs by cases for the mathematical statement.

(2) If x is a real number, then |x - 1| + |x + 5| >= 6.

Proof. Assume that x is a real number. Consider the following cases:

Case 1: Assume that x  < -5. In this case x - 1 < 0, and x + 5 < 0. So |x -1| = -(x - 1) = 1 - x, and |x + 5| = -(x + 5) = -5 - x. Then

|x - 1| + |x + 5| = 1 - x - 5 - x = -4 - 2x > -4 -2(-5) = -4 + 10 = 6.

So in this case |x - 1| + |x + 5| >= 6.

Case 2: Assume that -5 =< x < 1. In this case x - 1 < 0, and x + 5 >= 0.

So |x - 1| = -(x - 1) = 1 - x, and |x + 5| = x + 5. Then

|x - 1| + |x + 5| = 1 - x + x + 5 = 6 >= 6.

Case 3: Assume that x >= 1. In this case x - 1 < 0, and x + 5 >= 0. So

|x - 1| = x -1 and |x + 5| = x + 5. Then

|x - 1| + |x+5| = x - 1 + x + 5 = 2x + 4 >= 2(1) + 4 = 6

So in all cases |x - 1| + |x + 5| >= 6.

QED

Proofs by Case (1)

Write clear and complete proofs by case for the following statement.

(1) If x is a real number, then |x + 3| - x > 2.

Proof. Assume that x is a real number.

- Case 1: Assume that x >= -3, so that x + 3 >= 0. In this case |x + 3| = x + 3. So we have

              |x + 3| - x = x + 3 - x = 3 > 2.

- Case 2: Now assume that x < - 3, so that x + 3 < 0. In this case |x + 3| = -(x + 3) = - 3 - x. So we have

|x + 3| - x = - 3 - x -x = - 2x - 3 > -2(-3) -3 = 6 - 3 = 3 > 2.

So in all cases |x+3| - x > 2.

QED

Direct Proofs (4)

(4) If m is odd, then m^2 + 1 is even.

Proof. Assume that m is odd. Then m = 2k + 1 for some integer k. So
       
          m^2 + 1 = (2k +1)^2 + 1
                        = 4k^2 + 4k + 1 + 1
                        = 4k^2 + 4k +2
                        = 2(2k^2 + 2k + 1)
                        = 2q

where q = 2k^2 + 2k + 1 is an integer. So m^2 + 1 is even.

QED

Basic Proofs Methods II

Two-part proof of P <=> Q
(i) Show P => Q by any method.
(ii) Show Q => P by any method.
Therefore, P => Q

Example. Let a be a prime, and b and c be positive integers. Prove that a divides the product bc if and only if a divides b or a divides c.

Proof.
(i) Suppose a is a prime and a divides bc. By the Fundamenta Theorem of Arithmetic, b and c may be written uniquely as products of primes: b = p1p2p3...pkq1q2q3...qr. Since a is a prime and a divides bc, a is one of the prime factors of bc. Thus, either a = pi for some i or a = qj for some j. If a = pi, then a divides b. If a = qj, then a divides c. Therefore, either a divides b or a divides c.
(ii) Suppose a divides b or a divides c.If a divides b, then a divides bc. If a divides c, then again a divides bc. In either case, a divides bc.

QED

Number Theory-Divisibility (1)

Proper divisor. We say that a is a proper divisor of b if a | b and a < b.

For example, 3 is a proper divisor of 6, but 6 is not a proper divisor of 6. Note that for any b, all divisors of b are proper except for b itself.

Nontrivial divisor. We say that a is a nontrivial divisor of b if a | b and 1 < a < b.

For example, the nontrivial divisors of 6 are 2 and 3. Note that 1 is a proper divisor of every large integer, but 1 has no trivial divisors. The following theorem states some properties of divisibility.

Theorem 2.1
Properties of Divisibility
D1: 1 | a and a | a for all a.
D2: If a | b, then a =< b
D3: If a | b and b | a, then a = b.
D4: If a | b and b | c, then a | c.
D5: For all c, a | b if and only if ac | bc.
D6: If a | b and c | d, then ac | bd
D7: If a | b and a | c, then a | (bx + cy) for all x, y

Proof of D2: If a | b, then by proper division, b = ka for some k >= 1, so ka >=a, that is b >=a.
QED

Proof D4: If a | b and b | c, then b = ka and c = jb for some k and j. Therefore c = k(ka) = (jk)a, so a | c.
QED

Proof of D7: If a |b and a | c, then b = ka and c = ja for some j, k. Therefore, bx + cy = (ka)x + (ja)y =
(kx + jy)a, so a | (bx + cy).
QED

Remarks
In order to prove that m = n, it is sometimes convenient to use D3, that is, to prove that m | n and n | m.

Number Theory-Divisibility

Definition 2.1 Divisibility
We say that a divides b, and we write a|b if b = ka for some k.
For example, 3|12, since 12 = 4 * 3, also 8|40, since 40 = 5 * 8. If a|b, we might also say "a is a divisor of b", "a is a factor of b", or " "b is a multiple of a". If a does not divide b, then we write a |/ b. For example, 3 |/ 14; also, 8 |/ 42.

Logic-Basic Proof Methods II

Proof by contraposition P => Q
Suppose ~Q.
     .
     .
     .
Therefore, ~P ( via direct proof)
Thus, ~Q => ~P.
Therefore, P => Q.

QED

Example. Let m be an integer. Prove that if m^2 is odd, then m is odd.
In this example the symbol "^" means power.

Proof. Supposed m is not odd. [Suppose ~Q.] Then m is even. Thus, m = 2k for some integer k. Then m^2 = (2k)^2 = 4k^2 = 2 (2k^2). Since m^2 is twice the integer  2k^2, m^2 is even. [Deduce ~P.] Thus, if m is even, then m^2 is even; so, by contraposition, if m^2 is odd, then m is odd.

Example. Let x and y be real numbers such that x , 2y. [The second assumption (~Q) begins our proof by contraposition.]  then 2y - x > 0 and 3x - y > 0. Therefore, (2y - x)(3x - y) = 7x - 3x^2 -2y^2 > 0. Hence, 7xy > 3x^2 + 2y^2. We have shown that if 3x > y, then 7xy > 3x^2 + 2y^2. We conclude that if 7xy <=(equal sign) 3x^2 + 2y^2, then 3x <=(equal sign) y.

QED

Example. Prove that the graphs of y = x^2 + x + 2 and y = x - 2 do not intersect.

Proof. Suppose that the graphs of y = x^2 + x + 2 and y = x - 2 do intersect at some point (a,b0. [Suppose ~P.] Since (a,b) is a point on both graphs, b= a^2 + a +2 and b = a -2. Therefore, a-2 = a^2 +a +2, so a^2 = -4. Thus, a^2 < 0. But a is a real number, so a >=0. This is impossible..[ The statement a^2 < 0 ^(and) a^2 >= 0 is a contradiction.] Therefore, the graphs do not intersect.

QED

Proof of P by Contradiction
Suppose ~P.
     .
     .
     .
Therefore, Q.
Hence, Q ^ ~Q, a contradiction.
Thus, P.

QED

Example.
Prove that square root(2), or sqrt(2) is an irrational number.

Proof. Suppose that sqrt(2) is a irrational number. [Assume ~P.] Then sqrt(20 = s/t, where s and t are integers. Thus, 2 = s^2/t^2, and 2t^2 = s^2. Since s^2 and t^2 are squares, s^2 contains an even number of 2's as prime factors [This is our Q statement.], and t^2 contains an even number of 2's. But then 2t^2 contains an odd number of 2's as factors. Since 2t^2 = s^2, s^2 has an odd number of 2's. [This statement ~Q.] This is a contradiction, because s^2 cannot have  both an even and an odd number of 2's as factors. We conclude that sqrt(2) is irrational.

QED

Logic-Basic Proof methods I

A few of the basic tautologies we shall refer to are

P v ~ P    Excluded middle
(P => Q) <=> (~Q => ~P)   Contrapositive
P v (Q v R) <=> (P v Q) v R   Associative
P ^ (Q ^ R <=> (P ^ Q) ^ R    Associative
P ^ (Q v R) <=> (P ^ Q) v (P ^ R)   Distributive
P v (Q ^ R) <=> (P v Q) ^ (P v R)   Distributive
(P <=> Q) <=> (P => Q) ^ (Q => P)   Biconditional
~(P => Q) <=> P ^ ~Q   Denial of implication
~(P ^ Q)<=> ~P v ~Q    De morgan's law
~(P v Q) <=> ~P ^ ~Q   De morgan's law
P <=> (~P => (Q ^ ~ Q))  Contradiction
[(P => Q) ^ (Q => R)] => (P => R)   Transitivity
[P ^ (P => Q) => Q    Modus Poenus

Direct Proof of P => Q

Assume P.
     .
     .
     .
Therefore, Q.
Thus P => Q

QED

Example. Suppose a, b, and c are integers. Prove that if a divides b and b divides c, then a divides c.

Proof. [For a direct proof, we assume the antecedent, which is "a divides b and b divides c." Our goal is to derive the consequent, "a divides c," as our last step.] Suppose a, b, and c are integers, and that a divides b and b divides c. [We now rewrite these assumptions by using the definition of "divides," so that we have some equation to work with.] Then b = ak for some integer k, and c = bj for some integer j. [To show that a divides c, we have to write c as a multiple of a.] Therefore, c = bj = (ak)j = a(kj), so a divides b and b divides c, a divides c.

QED

Direct Proofs (3)

Write clear and complete direct proofs for each of the following mathematical statements.

If m is even and n is odd, then mn + 3n is odd.

Proof. Assume that m is even and n is odd. So m = 2k and n = 2j + 1 for some integers k and j. Then
           mn + 3n = (2k)(2j + 1) + 3(2j + 1)
                         = 4kj + 2k + 6j + 3
                         = 4kj + 2k + 6j + 2 + 1
                         = 2(2kj + k + 3j + 1) + 1
                         = 2q + 1

where q = 2kj + k + 3j + 1 is an integer. So mn + 3n is odd.

QED

Direct Proofs (2)

Write clear and complete direct proofs for the following mathematical statement.
(2) If x | a and y | b, then a = xm and b = yn for some integers m and n. Then

                                       ab = (xm)(yn)
                                           = xy(mn)
                                           = (xy)q

where q = mn is an integer. Thus xy | ab as desired.

QED

Truth Tables (5)

(5) He will fail the biochemistry exam unless he studies all week.

P := He fails the biochemistry exam.
Q := He studies all week.

So our statement is P unless Q which is equivalent to (~Q) => P or

If he does not study all week, then he will fail the biochemistry exam. or
If he passes the biochemistry exam, then he studied all week.

Truth Tables (4)

(4) I will buy a new car only if I win the lottery.

P := I will buy a new car.
Q := I will win the lottery.

So our statement is P only if Q which is equivalent to P => Q or

If I buy a new car, then I won the lottery. or
If I don't win the lottery, then I won't buy a new car.

Truth tables (3)

Rewrite each of the following sentences in symbolic propositional form. Then write each sentence in conditional (If...then...) form.

(3) She is not tall or she does not have brown eyes.

P := She is tall.
Q := She has brown eyes.

So our statement is (~P) v (~Q) which is equivalent to (~(~Q)) => (~P) or
Q => (~P) or

If she has brown eyes, then she is not tall. or
If she is tall, then she does not have brown eyes.

Truth tables-(2)

Rewrite each of the following sentences in symbolic propositional form. Then write each sentence in conditional (if...then...) form.
(2) The food is cold or the food is bad.

P := The food is cold.
Q := The food is bad.

So our statement is P v Q which is equivalent to P v(~(~Q)) or
(~Q) => P or

If the food is not bad, then the food is cold. or
If the food is not cold, then the food is bad.

Truth Tables-(1)

Example 1: Rewrite each of the following sentences in symbolic propositional form. Then write each sentence in conditional (If...then...) form.

(1) It will rain or it will not hail.
P := It will rain.
Q := It will hail.
So our statement is P v (~Q) which is equivalent to Q => P or
If it hails, then it will rain. or
If it does not rain, then it will not hail.

Direct Proofs

Write clear and complete direct proofs for each of the following mathematical statements.
(1) If x | a and x | b, then x | (a^2 - b^2).
Proof. Assume that x | a and x | b, then   we know that a = xm, and b = xn for some integers m and n. Now
a^2 - b^2 = (xm)^2 - (xm)^2
               = x^2m^2 - x^2n^2
               = x(xm^2 - xn^2)
               = xq
where q = (xm^2 - xn^2) is an integer. Therefore x| (a^2 - b^2).

QED