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.
domingo, 23 de dezembro de 2012
sábado, 22 de dezembro de 2012
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.
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.
terça-feira, 27 de novembro de 2012
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.
segunda-feira, 19 de novembro de 2012
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.
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.
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.
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.
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