Maths in english

Biographies of mathematicians

2012-03-04T05:21:00.001-08:00

1) Georg Cantor (1845 ; 1918) He demonstrated many surprising facts about different kinds of infinities, such as the existence of a one to one function between R and RxR, (or between a straight line and a plane), about which he wrote : "I can see it but I can't believe it.
http://www-history.mcs.st-andrews.ac.uk/Biographies/Cantor.html

2) Pierre de Fermat (1601 ; 1665) He demonstrated many important theorems about the theory of numbers, but probably not his last conjecture : "xⁿ + yⁿ = zⁿ has no integer solution when n ≥ 3", about which he wrote "I have found a very clever solution but I do not have the space to write it in the margin of this book".
http://en.wikipedia.org/wiki/Pierre_de_Fermat

3) Alan Turing (1912 ; 1954) He broke the very difficult code "Enigma" that the german army used during the second world war, and he invented the principle by which the invention of computer has been possible : the "Turing machine". He committed suicide because England condemned homosexuality in the 1950's.
Videos : 1/9 ; 2/9 ; 3/9 ; 4/9 ; 5/9 ; 6/9 ; 7/9 ; 8/9 ; 9/9. A movie.
http://en.wikipedia.org/wiki/Alan_Turing

4) Andrew Wiles (1953 ; …) He demonstrated Fermat's last theorem in 1994, after 7 years of research, whereas many great mathematicians had been trying to solve the conjecture during more than 300 years.

http://www.gap-system.org/~history/Biographies/Wiles.html

The story of stuff

2012-03-02T22:56:00.002-08:00

Story of stuff : the critique :

Money as Debt

2012-01-15T21:09:00.001-08:00

You'll find the full movie with a better image definition here :

http://video.google.com/videoplay?docid=-2550156453790090544

This video is interesting,
but has been a bit criticized in french newspapers and television, as you'll find out here :
http://fr.wikipedia.org/wiki/L%27Argent_Dette

The most important video you'll ever see

2012-01-10T21:28:00.001-08:00

Thebanker's trick.

Bankers mentally calculate the approximative time of doubling of a capital, placed at compound interest, by the following way :« If t is the interest rate (in %), then the capital doubles after 70/t years ».

(Werecall that after n years of placement at the rate t,the value of a capital is multiplied by (1 + t/100)^n ).

a)Prove that, for all x ≥ 0, x - x^2/2 ≤ ln(1+x) ≤ x.

b)Deduce from it a majorant of the mistake in the approximation :

ln(1+x) ≈ x.

c)Justify the banker's trick, for small values of t (t ≤ 14).

d)Find analogous ways to determine mentally the time after which acapital triples, is multiplied by 5, is multiplied by 10.

Some interesting graphics about the increase of the P.I.B. :
http://www.les-crises.fr/historique-pib-france/

1942 : The game of Hex

2012-01-08T09:10:00.000-08:00

(This article is from Pickover's maths book).

Piet Hein (1905-1996), John Forbes Nash, Jr. (1928- ).

Hex is the name of a game for two players which is played on a board formed by hexagonal cells, which usually has the shape of a rhombus of dimension 11x11.

It was invented by the danish poet and mathematician Piet Hein in 1942 and, independently, by the american mathematician John Nash in 1947.

Nash, winner of a Nobel prize of economy, is perhaps better known by the Hollywood movie, A beautiful mind, where his mathematical abilities and his combat against schizophrenia were told. According to the book A beautiful mind, Nash proposed an ideal size of board of 14x14.

The players use coins of different colors (for example red and blue), and they place them alternatively on the hexagonal cells. The objective of the red coins is to form a line which connects two opposite sides of the board. The objective of the blue coins is to form a line which connects the other two opposite sides. The four hexagons at the vertices belong to both players.

Nash discovered that the game can never end in a draw, and that the game is in favor of the player who starts, who can have a winning strategy. A way of making the game more fair is to allow the second player to chose his own color once the first player has made his first move, or after the first three moves.

It has been demonstrated that there is a winning strategy for the player who starts the game for various sizes of the game board. Although the game seems simple, some mathematicians have used it for more profound applications, such as the demonstration of Brouwer's fixed point theorem.

Hein made himself world famous for his designs, poems and mathematical games. When the german invaded Denmark in 1940, he was obliged to move to clandestinity because he was the leader of an anti-nazi group. In 1944, he explained his creative approach : "Art is the solution for all problems which may not be clearly formulated before they have been solved".

See also : Brouwer's fixed point theorem (1909), the strategy of the pig's game (1945), Nash's equilibrium (1950) and instant madness (1966).

Menger's sponge (1926)

2011-12-13T21:06:00.000-08:00

(This article is from Pickover's"Math book")

Karl Menger (1902-1925)

Menger's sponge is a fractal object with an infinite number of "Decays" (caries), the worst nightmare of a dentist. The object was first defined by the austrian mathematician Karl Menger in 1926.

To construct the sponge, we start from a "Mother cube" and we subdivide it in 27 identical smaller cubes. Then, we take off the central cube and the six cubes which have a common face with it. This leaves us with 20 cubes. We repeat the process with the rest of the cubes up to infinity.

The number of cubes increases like 20^n, where n is the number of iterations realized in the mother cube. The second iteration gives us 400 cubes and when we reach the 6th iteration, we have
64 000 000 cubes.

Each face of Menger's sponge is called Sierpinski's carpet. The fractal antennas obtained with Sierpinski's carpets are used as efficient receptors of electromagnetic signals.

Carpets as well as sponges have a few extraordinary properties. For instance, the sponge has an infinite surface and at the same time a volume equal to zero.

According to the Institute for Figuring, with each repetition, Sierpinski's carpet dissolves itself in a foam (mousse) which final structure has a surface equal to zero, although it's perimeter is infinite.

While a line has dimension 1, Sierpinski's carpet has a "fractional" dimension of 1,89. Menger's sponge has a "Hausdorff" dimension of 2,73 (between a plane and a solid). It has been used to visualize a few space-time models similar to foam.

Dr Jeannine Mosely has created a model of Menger's sponge, with 65 000 comercial cards, which weighs approximately 70 kilograms.

(See also Pascal's triangle (1654), Rupert's principle problem (1816), Hausdorff's dimension (1918), Antoine's necklace (1920), and fractals (1975)).

Hypercube (1888)

2011-12-04T07:29:00.001-08:00

This article is taken from Pickover's "Math Book".

Charles Howard Hinton (1853-1907)

I don't know any mathematical subject which has intrigued children as well as adults such as the idea of the fourth dimension, a spacial direction different from those we already know in our every day tridimensional life.

Mathematicians and physicists often use the fourth dimension in their calculations. It is part of some important theories which describe how our universe is constructed.

The hypercube is analogous to the ordinary cube in three dimensions. Sometimes is used the term "Tesseract" to name the hypercube of four dimensions.

As well as a cube may be visualized by "Sliding" a square to the third dimension and observing the shape that the square leaves behind in space, the hypercube is generated by the trace of a cube which moves in the fourth dimension.

Although it's difficult to see a cube which moves in a perpendicular direction perpendicular to its three axes, the computer graphics allow mathematicians to be able to have an intuition objects of more than four dimensions.

Note that a cube is limited by its square faces ; by the same way a hypercube is limited by its cubic faces. We may note the number of vertices, edges, faces and solids of these kind of objects :

Vertices Edges Faces Solids

Point 1 0 0 0

Segment of line 2 1 0 0

Square 4 4 1 0

Cube 8 12 6 1

Hypercube 16 32 24 8

The word tesseract was first used in 1888, in the book "A new era of thought" of the british mathematician Charles Howard Hinton. Hinton, who was bigamous, was also famous for the invention of a set of colored cubes, which, he assured, could be used to see in four dimensions. Hinton's cubes were even used in sessions of spiritism, believing that they could facilitate the contact with the dead.

See also : "The elements" of Euclid (300 B.C.), The problem of Rupert's principle (1816), Klein's bottle (1882), Planilandia (1884), "Philosophy and fun with algebra", of Boole (1909), Rubik's cube (1974) and a perfect magical hypercube (1999).

Fibonacci in nature

2011-11-28T00:56:00.001-08:00

Economía alternativa

2011-11-28T00:35:00.001-08:00

Christian Felber - La Economía del Bien Común from ATTAC.TV on Vimeo.

Hibert's hotel infinity paradox (1925)

2011-11-27T04:39:00.001-08:00

(The following text is from Clifford Pickover's "Math book")

David Hilbert (1862-1943)

Imagine an ordinary hotel of 500 rooms, all of them occupied by some clients. You arrive there one evening and the receptionist tells you that there's no vacancy. You walk away sadly. There's no paradox about that.

Now, imagine a hotel in which there are infinitely many rooms, all of them occupied. Although the hotel is full, the receptionist is able to give you a room. How is it possible ? Later, on the same day, infinitely many persons arrive, and the receptionist is able to make room for all of them, (earning a fortune in doing so).

The german mathematician David Hilbert invented these paradoxes in the 1920s to illustrate the mysterious properties of infinity.

Lets see how we can obtain a room in Hilbert's hotel infinity. When only you arrive and the hotel is full, the receptionist is able to give you a room by asking the client who was in room 1 to go to room 2, the one who was in room 2 to go to room 3, and so on forever, each client moving to the next room. Thus, room 1 is now empty for you.

To be able to receive an infinity of new clients, all the previous ones move to the rooms of even numbers, the one who was in room 1 moving to room 2, the one who was in room 2 moving to room 4, the one who was in room 3 moving to room 6, and so on, every client moving to the room having the number double from his previous room. The receptionist is now able to move all the new clients to the rooms of odd numbers.

Hilbert's hotel infinity paradox may be understood with Cantor's theory of transfinite numbers. Thus, while in a normal hotel, the number of odd numbered rooms is inferior to the total number of rooms, in an infinite hotel, the "number" of odd numbered rooms is not inferior to the total "number" of rooms. (The mathematicians use the term of "cardinality" to refer themselves to the size of these sets of rooms).

The receptionist can even receive infinitely many buses containing infinitely many travelers each, by using all the powers of 2 for the previous clients, all the powers of 3 for the first bus, and all the powers of each different prime numbers for each different bus.

See also Zeno's paradox (445 B.C.), Cantor's transfinite numbers (1874), Peano's axioms (1889) and Hilbert's twenty three problems (1900).

Baccalauréat

2011-06-02T21:56:00.000-07:00

Première partie (10 minutes) :

un exercice de maths de Terminale, donné par le jury : (préparé pendant 20 minutes) :

1) Fonctions
2) Probabilités
3) Statistiques, pourcentages
4) Suites

a) Compréhension du sujet et des questions posées : 2 points
b) Utilisation du vocabulaire spécifique et des symboles mathématiques : 2 points
c) Clarté de l'exposé: aisance, efforts pour enrichir le vocabulaire, aptitude à s’auto corriger dans le domaine de la langue : 2 points
d) Démarche mathématique et capacité à argumenter dans une langue étrangère : 2 points
e) Résolution du problème ou capacité à poser des questions pour le résoudre : 2 points

Deuxième partie : (10 minutes) :

elle consiste en un entretien qui porte sur les travaux effectués pendant l'année scolaire dans la discipline non linguistique et, de manière plus générale dans le cadre de la section européenne. La liste des thèmes étudiés dans la discipline non linguistique est fournie à titre d'information par le candidat le jour de l'interrogation. L'entretien peut également porter sur l'ouverture européenne (partenaires, échanges, clubs, journaux, conférences,...). Il est conduit de manière libre et évite les questions stéréotypées.

a) Réagir à des questions non préparées mais sur un domaine connu, en fonction de compétences linguistiques :
..... Aisance de l'expression : 2 points
..... Prononciation, intonation : 2 points
..... Correction grammaticale : 2 points
b) Donner un avis, une information de nature scientifique, participer à un échange de manière active, sur les thèmes abordés au cours de l'année dans la discipline non linguistique : 4 points

Dimensions

2011-05-25T12:45:00.000-07:00

Sequences

2011-05-25T12:31:00.000-07:00

Sequences 3 : Fibonacci sequence

Sequences 4 : Pascal's triangle

Sequences 5 : golden rectangles

Sequences 6 : continued fractions

Sequences 10 : fractals and chaos

Évariste Galois

2011-05-23T21:53:00.000-07:00

http://scidiv.bellevuecollege.edu/math/galois.html
http://images.math.cnrs.fr/Evariste-Galois-enfance-d-un-genie.html

Born: October 25, 1811 near Paris.

Died: May 31, 1832 in Paris.

In all the history of science there is no completer example of the triumph of crass stupidity over untamable genius than is afforded by the all too brief life of Evariste Galois. (E.T. Bell)

Brilliant. Brash. Unlucky. Died at 20. Galois groups and fields. Galois theory.

Galois was born in the village of Bourg-la-Reine where his father was mayor. His mother was an educated woman and taught Galois at home until he entered school at the age of 12. At first Galois did well in school and won prizes, but by his second year he became bored with the classical studies. His work became mediocre, and he had trouble with the school authorities. He happened on a geometry book by the mathematician Legendre, a difficult book, and quickly mastered it. The algebra textbook used in the school disgusted him and he ignored it. It lacked, he said, the creator's touch of a mathematician, so he went to the masters, Lagrange and Abel. In reports, his teachers described him as "not wicked," "original and queer," "argumentative," "there is only slovenliness and eccentricity in his assigned tasks -- when he deigns to pay any attention to them," and "He is wasting his time here, and all he does is torment his teachers and get into trouble." (Of course, similar reports have been written about thousands of other students who cannot validly claim 'genius' as a defense.)

At 16, Galois took the examinations to enter the prestigious Polytechnique -- and failed. Years later Terquem remarked, "A candidate of superior intelligence is lost with an examiner of inferior intelligence." However, Galois found a mathematics teacher, Louis Richard, and really started studying and doing mathematics. His first paper, on continued fractions, was published when he was 17.

At 18, Galois reapplied to the Polytechnique, and again the examination went badly. Finally, during the oral part of the exam, he lost patience with one of the examiners and threw the eraser at him. It was a hit, but Galois could never apply there again.

At 19, Galois attended the university and wrote 3 original papers on the theory of algebraic equations. He submitted them to the Academy of Sciences for the competition in mathematics. The Secretary of the Academy took them home to read, but then died before writing a report about them and the papers were never found. Galois was understandably upset: "Genius is condemned by a malicious social organization to an eternal denial of justice in favor of fawning mediocrity."

In 1830 the French masses revolted, and Galois was a staunch supporter. The director of the school locked the students in the school during the fighting and then expelled Galois for a public letter he wrote condemning the director. Galois tried to start his own school of mathematics, but got no students, so he joined the National Guard -- "If a carcass is needed to stir up the people, I will donate mine." Galois was jailed for supposedly threatening the King, but was found 'not guilty' by a jury. Finally he was convicted and sentenced to 6 months in jail for "illegally wearing a uniform."

When he was finally released, his last misadventure began. "Thus it happened that he experienced his one and only love affair. In this, as in everything else, he was unfortunate. Galois took it violently and was disgusted with love, with himself, and with his girl." A few days later Galois encountered some of his political enemies and "an affair of honor," a duel, was arranged. Galois knew he had little chance in the duel, so he spent all night writing the mathematics which he didn't want to die with him, often writing "I have not time. I have not time." in the margins. He sent these results as well as the ones the Academy had lost to his friend Auguste Chevalier, and, on May 30, 1832, went out to duel with pistols at 25 paces.

Galois was shot in the intestines, and was taken to the hospital. He comforted his brother with "Don't cry, I need all my courage to die at twenty." He died the day after the duel and was buried in an unmarked, common grave.

Twenty four years after Galois' death, Joseph Liouville edited some of Galois' manuscripts and published them with a glowing commentary. "I experienced an intense pleasure at the moment when, having filled in some slight gaps, I saw the complete correctness of the method by which Galois proves, in particular, this beautiful theorem: In order that an irreducible equation of prime degree be solvable by radicals it is necessary and sufficient that all its roots be rational functions of any two of them."

Galois' complete works fill only 60 pages, but he will be remembered.

Condensed from Men of Mathematics by E. T. Bell (1937), published by Simon and Schuster Pub. Co. DTH

Videos in spanish

2011-03-27T11:05:00.001-07:00

Videos en français

2011-03-20T01:33:00.000-07:00

Comment Google classe les pages web ? Clique ici pour la réponse.
Jean-Paul Delahaye explique dans une video le pagerank des moteurs de recherche :

http://www.canal-u.tv/producteurs/fuscia/dossier_programmes/science_info_lycee_conferences_de_formation_des_professeurs_du_secondaire_en_science_informatique/la_recherche_d_information_les_moteurs_de_recherche_et_le_pagerank

Le célèbre modèle de diffusion d'Ehrenfest : L'IREM de Montpellier a commis un document visible ici :
http://halshs.archives-ouvertes.fr/docs/00/05/45/65/PDF/co24th2.pdf

Une video sur l'empilement des sphères, de Joseph Oesterlé :

http://www.math.jussieu.fr/~kantor/MOSAIQUE/Mosaique-kepler.html

Un article sur les fractales :

http://images.math.cnrs.fr/La-methode-de-Newton-et-son,891.html

Probability

2011-03-10T08:52:00.000-08:00

http://www.philipbrocoum.com/?page_id=91

Long division

2011-01-22T10:32:00.000-08:00

Punk mathematics

2010-12-17T10:15:00.000-08:00

Be careful, there are words in this video which might hurt sensitive ears !

Somme et multiplication des relatifs

2010-11-07T03:15:00.000-08:00

Jobineries
Cargado por dhenin2. - Reportajes sobre tecnología y noticias sobre ciencia.

Multiplication_des_relatifs
Cargado por dhenin2. - Reportajes sobre tecnología y noticias sobre ciencia.

Computer Science Unplugged

2010-10-29T06:50:00.000-07:00

CS Unplugged is a collection of free learning activities that teach Computer Science through engaging games and puzzles that use cards, string, crayons and lots of running around. Videos.

Video math lessons

2010-10-23T11:46:00.000-07:00

Khan Academy: If you're tired of doing searches trying to find algebra videos, you can now start with an extensive listing in one place. Salman Khan has loads of videos teaching algebra and other topics : http://www.khanacademy.org/#Algebra

Creativity

2010-10-22T11:58:00.000-07:00

Symmetry

2010-10-03T01:37:00.000-07:00