21. Fermat's Last Theorem Fermat s last theorem (also called Fermat s great theorem) is one of themost famous theorems in the history of mathematics. It states that http://www.xasa.com/wiki/en/wikipedia/f/fe/fermat_s_last_theorem.html

Fermat's last theorem Wikipedia Fermat's last theorem (also called Fermat's great theorem ) is one of the most famous theorems in the history of mathematics . It states that: There are no positive natural numbers a b , and c such that in which n is a natural number greater than 2. The 17th-century mathematician Pierre de Fermat wrote about this in in his copy of Claude-Gaspar Bachet 's translation of the famous Arithmetica of Diophantus ': "I have discovered a truly remarkable proof but this margin is too small to contain it". This statement is significant because all the other theorems proposed by Fermat were settled, either by proofs he supplied, or by rigorous proofs found afterwards. Mathematicians were long baffled, for they were unable either to prove or to disprove it. The theorem was therefore not the last that Fermat conjectured, but the last to be proved . The theorem is generally thought to be the mathematical result that has provoked the largest number of incorrect proofs. For various special exponents n , the theorem had been proved over the years, but the general case remained elusive. In

22. Ceva's And Menelaus's Theorems Now is not an problem to prove many famous theorems stating that certain cevianshave common point, for example that medians (altitude, internal bisectors) are http://www.math.uci.edu/~mathcirc/math194/lectures/advanced3/node2.html

Next: Homework problems Up: Advanced Geometry III Previous: The nine-point circle Ceva's and Menelaus's Theorems The line segment joining a vertex of a triangle to any given point on the opposite side is called a cevian . Thus, if X Y and Z are points on the respective sides BC CA and AB of triangle ABC , the segments AX BY and CZ are cevians. This term comes from the name of the Italian mathematician Giovanni Ceva, who published in 1678 the following very useful theorem: Ceva's Theorem If three cevians AX BY and CZ , one through each vertex of a triangle ABC , are concurrent, then Conversely, if this equation holds for points X Y and Z on the three sides, then these three point are concurrent. (We say that three lines or segments are concurrent if they all pass through one point) Figure 2: Ceva's theorem Proof. Given the concurrence we can use that the areas of the triangles with equal altitudes are proportional to the bases of the triangles. Referring to Figure , we have Similarly, Now, if we multiply these, we find Conversely, suppose that the first two cevians meet at

23. Theorem - Encyclopedia Article About Theorem. Free Access, No Registration Neede Click the link for more information. for a list of famous theorems andconjectures. Gödel s incompleteness theorem. preview not available. http://encyclopedia.thefreedictionary.com/theorem

Dictionaries: General Computing Medical Legal Encyclopedia Theorem Word: Word Starts with Ends with Definition A theorem is a statement which can be proven In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. Proofs employ logic but usually include some amount of natural language which of course admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of informal logic. In the context of proof theory, where purely Click the link for more information. true within some logical Roughly speaking, logic is the study of prescriptive systems of reasoning, that is, systems proposed as guides for how people (as well, perhaps, as other intelligent beings/machines/systems) ought to reason. Logic says which forms of inference are valid and which are not. Traditionally, logic is studied as a branch of philosophy, but it can also be considered a branch of mathematics and Computer Science. How people actually reason is usually studied under other headings, including cognitive psychology. Logic is traditionally divided into deductive reasoning, concerned with what follows logically from given premises, and inductive reasoning, concerned with how we can go from some number of observed events to a reliable generalization.

24. Theorem - Encyclopedia Article About Theorem. Free Access, No Registration Neede . Click the link for more information. for a list of famous theorems andconjectures. Gödel s incompleteness theorem. preview not available. http://encyclopedia.thefreedictionary.com/Theorem

Dictionaries: General Computing Medical Legal Encyclopedia Theorem Word: Word Starts with Ends with Definition A theorem is a statement which can be proven In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. Proofs employ logic but usually include some amount of natural language which of course admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of informal logic. In the context of proof theory, where purely Click the link for more information. true within some logical Roughly speaking, logic is the study of prescriptive systems of reasoning, that is, systems proposed as guides for how people (as well, perhaps, as other intelligent beings/machines/systems) ought to reason. Logic says which forms of inference are valid and which are not. Traditionally, logic is studied as a branch of philosophy, but it can also be considered a branch of mathematics and Computer Science. How people actually reason is usually studied under other headings, including cognitive psychology. Logic is traditionally divided into deductive reasoning, concerned with what follows logically from given premises, and inductive reasoning, concerned with how we can go from some number of observed events to a reliable generalization.

25. Read This: Proofs From THE BOOK The section on Combinatorics includes chapters (20) Pigeonhole and double counting,(21) Three famous theorems on finite sets, (22) Cayley s formula for the http://www.maa.org/reviews/thebook.html

Read This! The MAA Online book review column Proofs from THE BOOK by Martin Aigner and Günter M. Ziegler Reviewed by Mary Shepherd In the preface of Proofs from THE BOOK , we read that "Paul Erdös often talked about The Book, in which God maintains the perfect proofs of mathematical theorems." As I read through the preface to this book, I began to ask some questions. What constitutes a "beautiful" proof? How about a "perfect" proof? Is there any such thing as a "perfect" proof? I don't know the answer to these questions. This book was inspired by Erdös and contains many of his suggestions. It was to appear in March, 1998 as a present to Erdös on his 85th birthday, but he died in the summer of 1997, so he is not listed as a co-author. Instead the book is dedicated to his memory. In the Number Theory section, the chapters are: (1) Six proofs of the infinity of primes, (2) Bertrand's postulate, (3) Binomial coefficients are (almost) never powers, (4) Representing numbers as sums of two squares, (5) Every finite division ring is a field, and (6) Some irrational numbers. I found most of these chapters to be somewhat difficult, requiring some background in algebra and analysis and even topology to be easily understandable. My favorite of these chapters was (4) because of the simplicity of statement of this theorem by Fermat, and the use of geometry to help visualize part of the solution. There was also a series of annoying but minor errors in chapter (6) in the reductions of the fractions on page 31.

26. Disproving Statements There are many famous conjectures and famous theorems that were conjectures formany years (The 4 color theorem and Fermat s Last Theorem, for example). http://www.math.csusb.edu/notes/proofs/pfnot/node3.html

Next: Types of Proof Up: NOTES ON METHODS OF Previous: Definitions and Theorems Disproving Statements Some conjectures are false. Verifying that a conjecture is false is often easier than proving a conjecture is true. Despite that, showing a conjecture is false may have its own challenges and usually requires a deep knowledge of the subject. Some statements are often shown to be false by a counter examples . Such statements have the form ``For all x in X conclusion ''. This is shown to be false by finding one element of X which does not satisfy the conclusion. As an example, consider the statement ``For all prime numbers p p +1 is prime''. This statement is true for the primes 2, 3 and 5. It is also true for the primes 11 and 23. However, the statement is an assertion for all primes. Clearly the statement is not true for the prime 7 (since 15 = 3 5) and we have obtained a counter example to the statement. For finite sets, statements of the form such as ``There exists an x such that conclusion '' can be refuted by example. By testing each element of the set and showing no element satisfies the conclusion we have shown the statement is false through an

27. The Pythagorean Theorem As on the of the most famous theorems in mathematics, the Pythagorean theorem isnamed after Greek mathematician Pythagoras of Samos who lived about 500 BCE http://educ.queensu.ca/~fmc/december2003/Pythagorean.html

A theorem is basically a generalization that can be demonstrated to be true. As on the of the most famous theorems in mathematics, the Pythagorean theorem is named after Greek mathematician Pythagoras of Samos who lived about 500 B.C.E., not because he discovered it but rather because he offered the first real proof for it. Long before Pythagoras, around 1800 B.C.E., the Babylonians knew and used this theorem, but they did not have a proof for it, meaning that they could not explain why it worked. A thousand years before Pythagoras, the Egyptians and the Chinese also knew the theorem, with Egyptian surveyors using ropes that were knotted in segments of equal length which they wrapped around stakes in the ground to create perfect right angle enabling them to lay out square corners for land surveying and on the pyramids they built. Try proving the Pythagorean Theorem yourself! This famous algebraic formula concerns the length of the sides of a right-angled triangle. The formula is a2 + b2 = c2, where c is the hypotenuse (the side opposite the right angle, which is always the longest side). Materials: graph paper makers or coloured pencils ruler pencil scissors Directions: Draw a right-angled triangle on piece of graph paper. Name the sides a, b, and c (c being the hypotenuse). Measure the length of side a. On another piece of paper, draw a square with sides the same length as a. Repeat this for sides b and c. Colour the squares and cut them out. Fit them beside the appropriate sides of your right-angled triangle.

28. Math 460 (Senior Seminar) Home Page and colorful characters who were mathematicians but most of all it s a wellwrittenpresentation of twelve interesting and famous theorems in mathematics. http://userpages.wittenberg.edu/bshelburne/Math460HomePage.htm

Math 460 - Senior Seminar - Home Page Journey through Genius Instructor Brian Shelburne 329-E BDK Science Class Meetings Th 2:10 - 3:40 - Room 327 Textbook Journey Through Genius by W. Dunham Course Objectives This is a capstone course for mathematics majors. Its purpose is to let you think about and reflect on what mathematics is and to tie together your years of studying mathematics. Dunham's book, Journey Through Genius , covers the story of mathematics from the 5th century B.C.E. up to the 20th Century C.E. by looking at some famous problems and theorems, and the mathematicians who worked on them. The book is many things. It's a selective history of mathematics, it's a look at some of the famous and colorful "characters" who were mathematicians but most of all it's a well-written presentation of twelve interesting and famous theorems in mathematics. Through the twelve theorems, Dunham presents his idea of what makes a theorem great! . The book is well written, fun to read and it will give you a deeper appreciation of the unique endeavor we call mathematics . Enjoy!

29. The Mathematical Association - Supporting Mathematics In Education Pythagoras Theorem is one of the most famous theorems in mathematics.Although it is commonly named after Pythagoras, it was known http://www.m-a.org.uk/resources/publications/books/are_you_sure_learning_about_p

20 May 2004 Home Contact Us Join the MA Whats New ... Site Map Search: M-A WWW Education Primary 11 to 16 Post 16 Higher ... Feedback to QCA Resources Conferences Local Activities Periodicals Professional Development ... MA Library Association President's Page Organisation Rules Regulations ... Links site by acumedia Are You Sure? Learning About Proof Sample Pages The pages shown here are © The Mathematical Association; their formats have been adapted for display on the Internet. Title Page Introduction Are you sure? Learning about Proof Edited by Doug French and Charlie Stripp The Mathematical Association A Book of Ideas for Teachers of Upper Secondary School Students Introduction Contents Contents Chapter 1 Why Proof? Is it True? Types of Proof What is Proof? Learning about Proof Chapter 2 Geometry and Proof The Angles of a Triangle Pythagoras' Theorem The Angles of a Polygon The Five Platonic Solids The Circle Theorems Three Polygons at a Point Trigonometric Picture Proofs Area of a Trapezium Some Proofs with Vectors Chapter 3 Number Would You Believe It?

30. W. H. Freeman Publishers - Geometry Reasoning 2.1 Conditional Statements 2.2 Definitions 2.3 Direct Proof 2.4 IndirectProof 2.5 A Deductive System 2.6 Some famous theorems of Geometry Chapter 3 http://www.whfreeman.com/highschool/book.asp?disc=&id_product=2001003117&compTyp

31. Journey Through Genius: The Great Theorems Of Mathematics Dunham investigates and explains, in easyto-understand language andsimple algebra, some of the most famous theorems of mathematics. http://www.programming-reviews.com/Journey_Through_Genius_The_Great_Theorems_of_

Journey Through Genius: The Great Theorems of Mathematics Journey Through Genius: The Great Theorems of Mathematics by Authors: William Dunham Released: August, 1991 ISBN: 014014739X Paperback Sales Rank: List price: Our price: You save: Book > Journey Through Genius: The Great Theorems of Mathematics > Customer Reviews: Average Customer Rating: Journey Through Genius: The Great Theorems of Mathematics > Customer Review #1: math history through great theorems For a detailed description of the chpaters in this work, look at the detailed review by Shard here at Amazon. I found this book well written and authoritative and learned a few things about Euler and number theory that I hadnt known from my undergraduate and graduate training in mathematics. Yet I did not give the book five stars. There are a couple of omissions that I find reduce it to a four star rating. My main objection is the slighting of Evariste Galois. Galois was the great French mathematician who died in a duel at the early age of 21 in the year 1832. Yet, in his short life he developed a theory of abstract algebra seemingly unrelated to the great unsolved questions about constructions with straight edge and compass due to the Greeks and yet his theory resolved many of these questions. I was very impressed in graduate school when I learned the Galois theory and came to realize that problems such as a solution to the general 5th degree equation by radicals and the trisection of an arbitrary angle with straight edge and compass were impossible.

32. Wilson Stothers' Cabri Pages Classical theorems. Here are some Cabri *.fig files written to illustratefamous theorems of geometry. In each case, the screen shows http://www.maths.gla.ac.uk/~wws/cabripages/classic0.html

Classical theorems Here are some Cabri *.fig files written to illustrate famous theorems of geometry. In each case, the screen shows how the figure looks in Cabri, but you can't drag the points around! Desargues Theorem La Hire's Theorem Pappus's Theorem Brianchon's First Theorem ... Conics Menu

33. Summer 2003 -- Proofs From The Book Tentative order of topics for the third week. (May 2730, 2003) chap 22 Threefamous theorems on finite sets. Three famous theorems on finite sets. http://www.math.fsu.edu/~bellenot/class/su03/book/

Summer 2003 `B' Term Proofs from the Book MAT 5932-24 Meets MTWRF 9:30-10:50 112 MCH Instructor Steve Bellenot (bellenot at math.fsu.edu) 850.644.7189 OFFICE HOURS MTWR 11:00-11:30am Text: Proofs From the Book, by Martin Aigner and Gunter Ziegler Amazon link to the book. The course Syllabus The on-line gradebook Links Quizzes and Exercises . Here are some old Activities. Chapters spoken for include ch 5 Goce; and ch20 Fazhe; Tentative order of topics for the five week. (Jun 9-13, 2003) chap 4 Representing numbers as the sums of two squares chap 8 Lines in the plane and decompositions of graphs chap 18 On a lemma of Littlewood and Offord chap 29 Turan's graph theorem Tentative order of topics for the fourth week. (Jun 2-6, 2003) chap 13 Every large set has an obtuse angle chap 32 Probability makes counting (sometimes) easy. chap 24 Caylay's formula for the number of trees chap 17 A Theorem of Polya on polynomials Tentative order of topics for the third week. (May 27-30, 2003) chap 22 Three famous theorems on finite sets. chap 19 Cotangent and the Herglotz trick chap 2 Bertrands postulate (there is always a prime between n and 2n).

34. The Living Mathematics Project Pythagoras Theorem, An animated proof of one of the most famous theorems of geometry.Dudeny s dissection, Yet another animated proof of Pythagoras theorem. http://www.sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/java.html

SunSITE @ UBC http://SunSITE.UBC.CA/ The Living Mathematics Project ``Constructing a new medium for the communication of Mathematics'' A sk any mathematician , and you'll be told - Mathematics is a dynamic, living subject. But for many people, going beyond the static images and formulas of current mathematics texts requires an effort in creative visualization which is often beyond their means. The Living Mathematics Project hosted at SunSITE UBC is working to apply recent advances in computer programming languages and the technology of the World Wide Web to construct a new medium for the communication of Mathematics. Local developments: Please let us know what you think about these projects. If you are using them for interesting projects, or if you can think of a way they might be improved, we'd love to hear from you. Copycat Jim Morey's Educational Game - A JavaCup winner Fourier Series Explore the representation of functions by Fourier series Catenary Animation - an amusing property of the Catenary. Pythagoras' Theorem An animated proof of one of the most famous theorems of geometry.

35. KET | Watch | TV Schedules TVG Continues the examination of mathematical proof with a look at several geometricor algebraic proofs of one of the most famous theorems in mathematics http://www.ket.org/cgi-plex/watch/?date=2004-04-13 18:00&cd=1&

36. E.W. Dijkstra Archive: On One Of Cayley's Theorems (EWD 677) connecting N labeled nodes. This is ascertained by one of Cayley sfamous theorems, of which this note offers a new proof. Remark. http://www.cs.utexas.edu/users/EWD/transcriptions/EWD06xx/EWD677.html

On one of Cayley's theorems A (finite) graph consists of a (finite) set of nodes with at most one edge between any two nodes. The so-called "multiplicity" of a node is defined as the number of edges of which that node is an endpoint. A connected graph without cycles is called a "tree"; a tree with N nodes has N-1 edges. In a tree the nodes with a multiplicity =1 are called its "leaves". A tree with at least 2 nodes has at least 2 leaves. In the following "a labeled tree" is a tree in which each node is labeled with a distinct integer. From a labeled tree with at least 2 nodes we can remove the leaf with the lowest number, together with the edge connecting it to the rest of the tree; the remaining graph is again a labeled tree. Hence this action can be repeated until the tree has been reduced to a single node; that remaining node is the one with the maximum number. For instance, the tree would give rise to the following sequence - in the order from left to right - of edge removals; each time we have written the number of the leaf being removed in the upper line: We remark that the right-most value of the bottom line is always the highest node number that the top line is always a permutation of the remaining node numbers that the number of times that a value occurs in such a scheme equals the multiplicity of the corresponding node.

37. Oxford University Press Part II ranges widely through related topics, including mapcolouring on surfaceswith holes, the famous theorems of Kuratowski, Vizing, and Brooks, the http://www.oup.com/ca/isbn/0-19-851062-4

/local_assets/ca Click here for quick links About OUP Contact Us Search the Catalogue Services and Resources Site Map How to Order OUP Worldwide Home OUP Canada Home Education K-12 Higher Education English as a Second Language General and Reference Book Information Online Order Form Search the catalogue Features Table of Contents Graphs, Colourings and the four-colour theorem Robert A. Wilson , Professor of Group Theory, The University of Birmingham Price: $ 58.50 CDN ISBN: 0-19-851062-4 Publication date: February 2002 OUP UK 150 pages, numerous figures, 156 mm x 234 mm There is an alternative edition (Cloth) Ordering Customers in Canada can place an order using our online order form Over 100 diagrams illustrating and clarifying definitions and proofs, etc Contains exercises in every chapter. Introductory and well paced explanations of the proof of the four-colour theorem. Suitable for any level from late undergraduate upwards. Description The four-colour theorem is one of the famous problems of mathematics, that frustrated generations of mathematicians from its birth in 1852 to its solution (using substantial assistance from electronic computers) in 1976. The theorem asks whether four colours are sufficient to colour all conceivable maps, in such a way that countries with a common border are coloured with different colours. The book discusses various attempts to solve this problem, and some of the mathematics which developed out of these attempts. Much of this mathematics has developed a life of its own, and forms a fascinating part of the subject now known as graph theory.

38. Liverpool Pure Maths: Dynamics Group to ergodic theory. A number of famous theorems in mathematics arein fact ergodic theorems. We mention a few. Dirichlet s theorem http://www.liv.ac.uk/maths/PURE/MIN_SET/CONTENT/RESEARCH_GROUPS/dynam.html

Pure Mathematics Dynamics Group Department Home Page Find someone? Pure home page Dynamical Systems Members of the Research Group Toby Hall Kit Nair Mary Rees Mary Rees works in Complex Dynamics. She has particular interest in the variation of dynamics in parameter spaces, and is engaged on a (long term) detailed study of the parameter space of quadratic rational maps. Toby Hall works in Topological Dynamics, with particular interest in surface homeomorphisms. Recently he has been working with Andre de Carvalho on pruning theory - which describes the controlled destruction of the dynamics of surface homeomorphisms - and its relation to horseshoe creation and the Henon family. A: subsequence ergodic theorems, their proofs and applications; B: distribution modulo one and its relation to ergodic theorems; C: the failure of ergodic theorems and its relation to diophantine approximation and hyperbolic dynamics; D: Glasner sets, toplogical dynamics and exponential sums; E: the existance of invariant measures for maps of the interval;

39. Livre Proofs From The Book - M. Aigner, G. Ziegler - - Librairie Eyrolles counting; Three famous theorems on finite sets; Shuffling cards; http://www.eyrolles.com/Sciences/Livre/9783540404606/livre-proofs-from-the-book.

40. Joel N. Franklin programming. The book concludes with easy, elementary proofs of thefamous theorems of Brouwer, of Kakutani, and of Schauder. These http://ec-securehost.com/SIAM/CL37.html

new books author index subject index series index Purchase options are located at the bottom of the page. The catalog and shopping cart are hosted for SIAM by EasyCart. Your transaction is secure. If you have any questions about your order, contact harris@siam.org Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems Joel N. Franklin Classics in Applied Mathematics 37 Many advances have taken place in the field of combinatorial algorithms since Methods of Mathematical Economics first appeared two decades ago. Despite these advances and the development of new computing methods, several basic theories and methods remain important today for understanding mathematical programming and fixed-point theorems. In this easy-to-read classic, readers learn Wolfe's method, which remains useful for quadratic programming, and the Kuhn-Tucker theory, which underlies quadratic programming and most other nonlinear programming methods. In addition, the author presents multiobjective linear programming, which is being applied in environmental engineering and the social sciences. The book presents many useful applications to other branches of mathematics and to economics, and it contains many exercises and examples. The advanced mathematical results are proved clearly and completely. By providing the necessary proofs and presenting the material in a conversational style, Franklin made

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