61. The Pythagorean Theorem ancient civilizations. This famous theorem is named for the Greekmathematician and philosopher, Pythagoras. Pythagoras founded http://jwilson.coe.uga.edu/emt669/Student.Folders/Morris.Stephanie/EMT.669/Essay

Department of Mathematics Education J. Wilson, EMT 669 The Pythagorean Theorem by Stephanie J. Morris The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students. The Pythagorean Theorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea. The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that:

62. Essay 1 -- Pythagorean Theorem Behnaz Rouhani. The Pythagorean theorem is one of the earliest theorems. This famoustheorem is named after the Greek mathematician and Philosopher, Pythagoras. http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002/Rouhani/Essay1/pythagorastheorem

Pythagorean Theorem by Behnaz Rouhani The Pythagorean theorem is one of the earliest theorems. This famous theorem is named after the Greek mathematician and Philosopher, Pythagoras. Although the theorem was named after him but there is evidence that the Babylonians knew this relationship some 1000 years earlier. Plimpton 322, a Babylonian mathematical tablet dated back to 1900 B.C., contains a table of Pythagorean triples. The Chou-pei, an ancient Chinese text, also provides the evidence that the Chinese knew about the Pythagorean theorem many years before Pythagorean. The Pythagorean Theorem is Pythagoras' most famous mathematical contribution. According to the legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that square root of 2 is irrational and, therefore cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. The Pythagorean theorem states that: "The area of the square built on the hypotenuse of a right triangle is equal to the sum of the squares on the remaining two sides." According to the Pythagorean Theorem, the sum of the areas of the red and yellow squares is equal to the area of the purple square.

63. Fermat, Pierre De After many talented mathematicians over the period of hundreds of years failed toprove it, this famous theorem was recently proved by Andrew Wiles of Princeton http://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib

Fermat, Pierre de (16011665) Fermat was born to a prosperous family in France. He studied the classics and mastered Latin, Greek, Italian, and Spanish. O ne of the seventeenth century’s greatest mathematicians, Fermat hesitated to publish his work and rarely wrote complete descriptions even for his own use. Most of his work was reported in correspondence with fellow mathematicians, Gassendi, Huygens , and Mersenne. Fermat was one of the co-founders, along with Descartes, of analytic geometry. He benefited from reading Viète's works. Fermat's book Ad locos planos et solidos isagoge Introduction to Plane and Solid Loci ) contained a more direct and clearer system than Descartes La g om trie Fermat is probably most famous for his work in number theory. His famous unproved “last theorem” (that a n = b n + c n After many talented mathematicians over the period of hundreds of years failed to prove it, t his famous theorem was recently proved by Andrew Wiles of Princeton. Fermat's name slipped into relative obscurity until the late 1800s, and it was from an edition of his works published at the turn of the century that the true importance of his many achievements became clear. Besides his work in physics and number theory, Fermat realized the concept that the area under a curve could be viewed as the limit of sums of rectangle areas (as we do today) and also developed a method for finding the centroids of shapes bounded by curves in the plane.

64. WQO Theory For example, a famous theorem of WQO theory is Kruskal s theorem Finite treesare WQO under embeddability. We now consider a famous theorem of BQO theory. http://www.math.psu.edu/simpson/cta/problems/node8.html

65. Chapter 11, 12 Pythagoras And His Theorem 97. 12. A famous THEOREM. Great mathematical discoveries were attributed tothis strange teacher, the most famous being the Pythagorean Theorem. http://www.anselm.edu/homepage/dbanach/pyth1.htm

Selections from Julia E. Diggins String, Straightedge, and Shadow Viking Press, New York 1965. (Illustrations by Corydon Bell) Now back in print for purchase at http://wholespiritpress.com/string.htm 11. PYTHAGORAS AND HIS FOLLOWERS The early story of Greek geometry is strangely different from its founding in Miletus . Most of what we know is a mixture of myth and magic, shapes and rules, all revolving around the fabulous figure of Pythagoras. The "divine" Pythagoras-that was what he was called, not only after his death but even in his own lifetime. For the latter part of the 6th century B.C. was still a time of superstition. The Ionian "physiologists" had only tried to find an orderly pattern in nature. Most men continued to believe that gods and spirits moved in the trees and the wind and the lightning. And cults were popular all over the Greek world-"mysteries," they were called-that promised to bring their members close to the gods in secret rites. Some were even headed by seers. Pythagoras was one of these. A native of the island of

66. Fermat's Last Theorem In this way the famous Last theorem came to be published. It was found by Samuelwritten as a marginal note in his father s copy of Diophantus s Arithmetica. http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html

Fermat's last theorem Number Theory Index History Topics Index Pierre de Fermat died in 1665. Today we think of Fermat as a number theorist, in fact as perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was in fact a lawyer and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous article written as an appendix to a colleague's book. There is a statue of Fermat and his muse in his home town of Toulouse: (Click it to see a larger version) Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat 's letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel written as a marginal note in his father's copy of Diophantus 's Arithmetica Fermat's Last Theorem states that x n y n z n has no non-zero integer solutions for x y and z when n Fermat wrote I have discovered a truly remarkable proof which this margin is too small to contain.

67. Geometry Miracle http//www.cutthe-knot.com/triangle/morley/index.html A discussionof Morley s famous theorem and the research of which it was a tiny part. http://www.directory.net/Science/Math/Geometry/

Geometry Directory: Guide to Geometry sites on the internet. Search Engines: Google Yahoo MSN FindWhat ... City Guides Geometry Science Math Geometry Categories Algebraic Geometry Computational Geometry Differential Geometry Events ... Solid Geometry Websites Problem of the Minimum Rotation Surface http://home.ural.ru/~iagsoft/pmrs_j.html Java applet demonstrating the catenary. Geometry in Action http://www.ics.uci.edu/~eppstein/geom.html Includes collections from various areas in which ideas from discrete and computational geometry meet real world applications. The Geometry Junkyard http://www.ics.uci.edu/~eppstein/junkyard/ Usenet clippings, web pointers, lecture notes, research excerpts, papers, abstracts, programs, problems, and other stuff related to discrete and computational geometry. Erich's Packing Center http://www.stetson.edu/~efriedma/packing.html Graphics and links for various packing, tiling and covering problems. Tim Lister's Hyperbolic Geometry http://mcs.open.ac.uk/tcl2/none/none.html Cabri constructions for the demonstration of the basic concepts of hyperbolic geometry in the Poincare disc model. Bette Veteto's Homepage http://www.people.memphis.edu/~brveteto/

68. Geometry Links Morley s Miracle A discussion of Morley s famous theorem and the researchof which it was a tiny part. 5 proofs are given including http://www.edinformatics.com/geometry.htm

EDinformatics Home Page Geometry Resources the entire directory only in Math/Geometry Top Science Math : Geometry Algebraic Geometry Computational Geometry Differential Geometry Events ... Topology See also: Kids and Teens: School Time: Math: Trigonometry Science: Math: Algebra: Geometric Algebra Science: Math: Recreations: Specific Numbers Sites: - Volunteer tutors help you work through your geometry problems for free. ArXiv Front: MG Metric Geometry - Metric geometry section of the mathematics e-print arXiv. ArXiv Front: SG Symplectic Geometry - Symplectic geometry section of the mathematics e-print arXiv. Bette Veteto's Homepage - History of Mathematics, very strong on geometry. Bob's Pages - A collection of interactive geometry applets. Topics include polyhedra, Poncelet's porism, Soddy's hexlet, Mandelbrot set, Steiner porism, Pappus's chain, Repulsion polyhedra and stereo pictures. Center of Points in 2- or Higher-dimensional Space - It provides important formulations for finding a center of points in 2 or higher dimensional space. It can be used in drawing delauney triagulation and voronoi diagram in two or higher dimensions. It also describes how many points are required in defining a center in n-dimensional space.

69. The Prime Number Theorem distribution of the primes. The theorem which we shall eventually proveis the famous theorem of Hadamard and de la Vallée Poussin http://www.maths.ex.ac.uk/~mwatkins/zeta/pnt.htm

the prime number theorem "Some order begins to emerge from this chaos when the primes are considered not in their individuality but in the aggregate; one considers the social statistics of the primes and not the eccentricities of the individuals." P.J. Davis and R. Hersh, The Mathematical Experience , Chapter 5 hyperlinked proof outline Eric Weisstein's notes on the prime number theorem Ilan Vardi's "Introduction to Analytic Number Theory" including useful notes on the PNT Chapter 5 of P.J. Davis and R. Hersh's The Mathematical Experience , dealing with the PNT A proof of the prime number theorem involving Fourier transforms (summarised by Jonas Wiklund) Although not its primary subject matter, this article by A. Granville contains a beautifully succinct explanation of the PNT, in a historical context. "The prime number theorem obtained by statistical methods" a heuristic argument from What is Mathematics? by Courant and Robbins A new way to visualise the Prime Number Theorem - approximate logarithmic spirals generated from the distribution of primes N. Wiener's The Fourier Integral and Certain of its Applications , section 17, "The Prime-Number Theorem as a Tauberian Theorem" begins: "The present section and the three following will be devoted to the application of Tauberian theorems to the problem of the distribution of the primes. The theorem which we shall eventually prove is the famous theorem of Hadamard and de la Vallée Poussin..."

70. I-MATH Congruent Triangles After an introduction to the Pythagorean Theorem (often called the most famous theoremin mathematics ), students may be interested in exploring the internet http://www.k12.hi.us/~mathappl/im12tri.htm

I-MATH Triangles Please do the GSP Lab Files, Chapter 04 Congruent Triangles and Chapter 06 Right Triangles, in connection with this web page. A triangle is the simplest polygon, with the least number of sides. Congruent triangles are triangles that are the same size and the same shape. This is easily seen when one looks at them from a transformational point of view. In the following diagrams, each pair of triangles is congruent and "connected" by a transformation. When teaching congruent triangles, I find it helps students in finding the congruent triangles, and the correspondence, if they have some experience with transformations, especially visually and on GSP. I do not like to use the complicated transformation terminology, but that is a personal opinion. I find that students understand transformations easily and quickly with a few visual examples. I use a sheet of plastic "transparency", and either the chalkboard, whiteboard or an overhead projector. I draw a triangle on the board or overhead projector screen, trace it onto the plastic transparency, then "flip" it (for reflection), use a compass point to "rotate" it, and simply slide it to demonstrate translation. An interesting note: when we get to similar triangles, we will see that the fourth transformation, "dilation" is a very useful concept to use in teaching similarity concepts and correspondence.

71. Mod Ex07 Logic Cl Rem 7. CVICENIE Z PREDMETU LOGIKA PRE pred Prime Prime(p) p = 2 \ax(Div(x,p) - x = 1 \/ x = p) rem \para We willnow prove the famous theorem of \ft Euclid \end asserting that to any numer http://delo.dcs.fmph.uniba.sk/2inf/clfanklub/cl2/ex/blank/ex07.cl

mod Ex07 logic: 'cl' rem 7. CVICENIE Z PREDMETU LOGIKA PRE INFORMATIKOV 2 SUBOR

72. Lakehead University - Agora - January 2004 another proof to the famous theorem was just another pattern, pushing the frontierof mathematics beyond its existing level, widening its base of knowledge. http://agora.lakeheadu.ca/pre2004/jan2004/02.html

Lakehead Mathematician Discovers New Proof for Pythagorean Theorem Dr. Medhat Rahim, Faculty of Education at Lakehead University, recently published new proof of famous Pythagorean theorem using a unique method By Tiina Ahokas Dr. Medhat Rahim is a mathematics specialist at Lakehead’s Faculty of Education. And now he has come up with a unique proof of an ancient theorem. “A new proof of the Pythagorean using a compass and unmarked straight edge,” was published in the January-February 2003 issue of International Journal of Mathematical Education in Science and Technology. Pythagoras was a Greek philosopher, religious teacher, and musician, well noted as the first pure mathematician. He is particularly remembered for his famous geometry theorem, now known as the Pythagorean theorem, from approximately 532 BC. Although the theorem was known to the Babylonians 1000 years earlier, he may have been the first to prove it. Evidence for this exists in the form of cuneiform clay tablet text, known as Plimpton 322; part of the Plimpton collection resides at Butler library of Columbia University in New York. Pythagoras founded an influential society of disciples. Men and women in the society were treated equally–an unusual concept at the time–and all property was held in common. Members of the society practiced the master's teachings, a religion of tenets. Pythagoras was interested in the principles of mathematics, the concept of number, the concept of a triangle or other mathematical figure, and the abstract idea of a

73. Lakehead University: Agora: January 2004 Adding another proof to the famous theorem was just another pattern, pushing thefrontier of mathematics beyond its existing level, widening its base of http://agora.lakeheadu.ca/pre2004/jan2004/print.htm

ATAC ~ Future of the North Campaign Update for AGORA By Susan Wright, Office of Development The ATAC ~ Future of the North Campaign continues in full swing. With government funding of $29.5 million and private sector donations totaling $4.6 million in place, Lakehead University needs $9.9 million to fund the $44 million dollar Centre. Lakehead’s fundraising volunteer teams and staff continue to contact potential donors, both within the University and externally, for these dollars. ATAC Tours Upcoming Events A number of community groups have shown interest in using ATAC facilities for their community meetings. These include both the Port Arthur and Fort William Gyro Clubs, and St. Joseph’s Care Group. These groups will have their dinner meetings in the faculty lounge and then a tour of the ATAC building. With groups such as these we hope to widen awareness in the community of what Lakehead University and ATAC have to offer. Lakehead Mathematician Discovers New Proof for Pythagorean Theorem Dr. Medhat Rahim, Faculty of Education at Lakehead University, recently published new proof of famous Pythagorean theorem using a unique method

74. Red Lion Stencils Famous Olde World Santas - Beautifully Detailed Theorems Click on image for larger view, scroll down to see more! The ChristmasHolidays bring lots of fun and merriment. Homes are decorated http://www.redlionstencils.com/olde world santas.htm

Home Welcome to the World of Decorative Stenciling! Directory Free-Form Designs Theorems Retail Store Locator ... Cart Click on image for larger view, scroll down to see more! The Christmas Holidays bring lots of fun and merriment. Homes are decorated with sprigs of holly, pine boughs, the smell of gingerbread, and colorful decorations. Our Santas have become a collection for many of our patrons. They are used to create welcome signs, hung by the front door to greet the guests. Many have used them for coordinating Holiday linens, and others have had fun putting them on T-shirts and giving them as gifts. They have been featured in magazine articles as seen in this photo of Woman's Day, special Christmas Edition! Remember that we show colors, but the choice for your project is your preference! OWS-91 Santa with Tree 4½" x 9" 4-layers Add to my cart! OWS-92 Santa with Sack 4½" x 8" 4-layers

75. Famous Computer Scientists Shannon s fundamental and very famous Theorem gives an upper bound to the capacityof a link, in bits per second (bps), as a function of the available http://www.comp.leeds.ac.uk/roger/Famous/eponymy_content.html

Abigailisms abigail@delanet.com. An Abigailism is a response to a question containing basic cognitive errors posed to a newsgroup (in particular, comp.lang.perl.misc ). An example: Q. Do I need to test Perl scripts on a web server or can it be done offline? A. Do I need to have a bell on my bicycle, or can I ride it wearing a hat? Ada Ada Byron, Lady Lovelace (1815-1852). The Ada language was named after her. She was an assistant of Babbage, and arguably the first ever computer programmer. Web material on her abounds. Armstrong's Axioms William W. Armstrong. Armstrong's Axioms are rules of inference for functional dependencies. AWK Alfred Aho, Peter Weinberger, Brian Kernighan. A Unix-based text/handling/macro programming language. Backus-Naur form John Backus, Peter Naur A formal metasyntax used to express context-free grammars. Bernoulli drive Daniel Bernoulli (1700-1782). Just like a disk drive but faster and more rugged - however not as fast as a hard drive. Bernstein's Conditions Arthur Bernstein.

76. History Of Computing Science: Blaise Pascal By the age of sixteen, Pascal had written his Treatise on Conic Sections, which includedhis famous theorem of hexagons (Pascal s Theorem), and presented it to http://lecture.eingang.org/pascal.html

Blaise Pascal Next Index Prev Blaise Pascal At the age of fourteen, Pascal started to attend Mersenne's meetings. Mersenne belonged to the religious order of the Minims, and his cell in Paris was a frequent meeting place for Fermat, Pascal, Gassendi, and others. By the age of sixteen, Pascal had written his Treatise on Conic Sections , which included his famous theorem of hexagons (Pascal's Theorem), and presented it to Mersenne. Already the young Pascal was on equal footing with some of the great scientific minds of his day. Perhaps the most famous of these religious works is , a collection of personal thoughts on human suffering and faith in God. "Pascal's wager" claims to prove that belief in God is rational with the following argument: "If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing." Later in his life, additional studies in geometry, hydrodynamics, and hydrostatic and atmospheric pressure led him to invent the syringe and hydraulic press, and to discover Pascal's law of pressure. He worked on conic sections and produced important theorems in projective geometry. In correspondence with Fermat, he helped lay the foundation for the theory of probability. Finally, his last work was on the cycloid, the curve traced by a point on the circumference of a rolling circle.

77. Blaise Pascal Pascal s famous theorem, also known as the Mystic Hexagram, states If any six sided,six angled figure is inscribed in any conic section, and the sides of the http://math.berkeley.edu/~robin/Pascal/theorem.html

Life Accomplishments Bibliography Back to the front page Pascal's Theorem Julia Chew Pascal's favorite mathematical topic to study, geometry, led to the formulation of Pascal's theorem. This states that pairs of opposite sides of a hexagon inscribed in any conic section meet in three collinear points. Pascal published this as Essai pour les Coniques Pascal's interest in conic sections most likely came about from his love of geometry and his association with Desargues, who was a great contributor to the study of conics. In his Essai , Pascal expresses his gratitude to the teachings and writings of Desargues. Pascal's study of conics follows from the theory of Desargues. He uses many of the theorems introduced in Desargues writings. The generation of conic sections is described by Pascal in the first portion of his work. A cone is constructed with a circle, a fixed point in space not in the plane of the circle, and a straight line through the point and the circumference of the circle. The line moves around the circumference of the circle, generating a conic surface. (Davidson) The surface of the two cones generated stretch out to infinity. Next, Pascal introduces a plane that also extends infinitely. Thus the two figures will meet with the plane cutting the cone. It can do this in six different ways producing six different sections: a point, a straight line, an angle, a closed figure (a circle or ellipse), a parabola, or a hyperbola. (Davidson) Next Pascal describes the properties of what he calls the

78. NewsScan Publishing Inc. - NewsScan Daily Archives About his most famous theorem, known as Fermat s last theorem, Fermat wrote inthe margin of his copy of the works of the 3rd century BC Greek geometer http://www.newsscan.com/cgi-bin/findit_view?table=honorary_subscriber&id=668

79. Rinton Press - Publisher In Science And Technology often beaten in the annual CADE theoremproving contests, but it has the most mathematicalresults of any prover (although the most famous theorem-prover result http://www.rintonpress.com/books/review/myers.html

Automated Reasoning with OTTER John A. Kalman (U of Auckland) with a foreword by Larry Wos (Argonne) 552 pages 10x7 inches w/ CDROM Feb 2001 Hardcover ISBN 1-58949-004-5 US$88 a book review published in The Bulletin of Symbolic Logic, Vol.8 No.3 (Sept, 2002) John Arnold Kalman. Automated Reasoning with Otter With a foreword by Larry Wos Rinton Press, Princeton 2001, XV + 536 pp. + CD-ROM John Kalman’s Automated reasoning with Otter is a complete user’s guide for OTTER. William McCune’s OTTER is the most widely used automated theorem-proving program. It is the current offering of the Argonne automated reasoning group led by L. Wos. OTTER and its resolution-based variants are often beaten in the annual CADE theorem-proving contests, but it has the most mathematical results of any prover (although the most famous theorem-prover result, the completeness of Robbins’s axiomatization of Boolean algebras belongs to McCune’s EQP). This book’s author, John Kalman, is noted for solving open problems about equivalential calculus and condensed detachment using OTTER and OTTER’s predecessor ITP. McCune’s OTTER 3.0 reference manual and guide (report ANL-94/6, available through

80. Graduate Papers Proofs are given of the impossibility of trisecting an angle with ruler and compass,and also the famous theorem of Galois, that quintic polynomials cannot be http://www.math.waikato.ac.nz/GradHBook/GraduateCourses.html

Graduate Papers Modules and papers at the graduate level may include lectures, practical work, special readings, assignments and projects. Most of the modules and papers listed below are taught each year, but some may be taught every second year. Normally a module is taught and examined in one semester (12 weeks of lectures), and counts as one half-paper for degree credit. In addition, there are dissertations and theses available. The appropriate choice will be worked out in consultation with each student. 0654.591 Dissertation (1 paper) 0654.592 Dissertation (2 papers) 0654.593 Dissertation (3 papers) 0654.594 Thesis (4 papers) Descriptions of Papers Metric, Normed and Hilbert Spaces The foundations of modern abstract analysis including metric spaces, topological spaces, convergence and continuity, compactness, normed spaces, equivalence of norms on Euclidean spaces, spaces of continuous functions, Banach spaces, Hilbert spaces, projections and complete orthonormal sets. Measure and Ergodic Theory This paper will present some basic topics from measure theory, and reinforce their importance with some applications in probability and ergodic theory. The syllabus will be drawn from: sigma-algebras and measure; functions, measure and integration; integral convergence theorems; Lebesgue measure; measure spaces and convergence theorems; invariant measures in probability and ergodic theory; Birkhoff's ergodic theorem; iterated function systems; construction of invariant measures for simple dynamical systems; Markov chain models in dynamics.

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