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| | BIGpedia - Pythagorean theorem - Encyclopedia and Dictionary Online |
 | | The Pythagorean Theorem is Equivalent to the Parallel Postulate. | | | The theorem is named after and commonly attributed to the 6th century BC Greek philosopher and mathematician Pythagoras, although the facts of the theorem were known by Indian (Baudhayana's and Katyayana's Sulbasutras), Greek, Chinese and Babylonian mathematicians well before he lived. | | | Babylonian tablet demonstrating knowledge of the theorem from 20th century BC |
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http://www.bigpedia.com/encyclopedia/Pythagorean_theorem
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| | Secret Teachings of All Ages: Pythagorean Mathematics |
| | Why the Pythagoreans expressed God as a tetrad is explained in a sacred discourse ascribed to Pythagoras, wherein God is called the Number of Numbers. | | | By the Pythagoreans the heptad--7--was called "worthy of veneration." It was held to be the number of religion, because man is controlled by seven celestial spirits to whom it is proper for him to make offerings. | | | The Pythagoreans taught that the elements of earth, fire, air, and water were permeated by a substance called ether--the basis of vitality and life. |
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http://www.sacred-texts.com/eso/sta/sta16.htm
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| | CATHOLIC ENCYCLOPEDIA: Neo-Pythagorean Philosophy |
| | Next, they interpreted the Pythagorean doctrine in a Platonic sense, when they taught that numbers are the thoughts of God. | | | Their original aim — to save the pagan world from moral and social ruin by the introduction of the religious element into philosophy and into conduct — was, of course, conceived without any reference to the claims of Christianity. | | | Besides, they derived from Oriental religions with which they were in contact at Rome as well as at Alexandria, a highly spiritual notion of God. |
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http://www.newadvent.org/cathen/10745a.htm
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| | Pythagoras and the Pythagoreans |
| | What is known of the Pythagorean school is from a book written by the Pythagorean, Philolaus of Tarentum. | | | Note: Unlike the Babylonnians or Egyptians, the Pythagorean s recognized that this class of numbers was wholly different from the rationals. | | | The whole concept of an eternal world revealed to intellect but not to the senses can be attributed from the teachings of Pythagoras. |
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http://www.math.tamu.edu/~don.allen/history/pythag/pythag.html
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| | Introduction to the Pythagorean Tarot |
| | The choice is between ignorance and enlightenment, between indulgence of the appetites and development of the mind, between worldly and spiritual pursuits, between the quests for earthly success and divine wisdom. | | | Note that Paracelsus reversed sulphur and mercury from the usual correspondences, which are used in the Pythagorean Tarot.] It also represents the meeting of three ways (called Triodos in Greek and Trivium in Latin), a place especially sacred to Hecate, a very important Goddess for Pythagoreans (Opsopaus, "Anc. | | | So much for the theory of divination; see the appendix ("Divinatio") for practical suggestions (spreads, etc.) on the use of the Pythagorean Tarot for divination, meditation and other purposes, as well as for the use of dice casting and similar methods with the Tarot. |
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http://www.cs.utk.edu/~mclennan/BA/PT/Intro.html
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| | Pythagorean - definition of Pythagorean in Encyclopedia |
| | The pentagram (five-pointed star) was an important religious symbol used by the Pythagoreans. | | | In the area of cosmology there is less agreement about what Pythagoras himself actually taught, but most scholars believe that the Pythagorean idea of the transmigration of the soul is too central to have been added by a later follower of Pythagoras. | | | Pythagorean thought was dominated by mathematics, but it was also profoundly mystical. |
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http://encyclopedia.laborlawtalk.com/Pythagorean
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| | Pythagorean Triples |
| | Note that here we use the terms leg, side, hypotenuse as follows: there are two legs and a hypotenuse making the 3 sides of each Pythagorean triangle. | | | There is a fascinating chapter on people with the most amazing ability to do arithmetic calculations in their heads. | | | There is a whole chapter on Pythagorean triangles: The Eternal Triangle. |
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http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html
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| | Pythagorean History |
| | The group was almost cult-like in that it had symbols, rituals and prayers. | | | Not much more is known of his early years. | | | These numerical values, in turn, were endowed with mystical and spiritual qualities. |
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http://www.geom.uiuc.edu/~demo5337/Group3/hist.html
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| | Pythagorean Mysteries |
| | The Pythagorean ethical and political tractates are especially interesting for they are based on the premise that the universal principles of Harmony, Proportion, and Justice govern the physical cosmos, and these writings show how individuals and societies alike attain their peak of excellence when informed by these same principles. | | | Pythagoras, communicating with priests and sages from Persia to Ireland, was perhaps the key figure in this process. | | | For a more detailed prospectus of the Pythagorean Sourcebook, click here. |
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http://members.aol.com/theloego/books
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| | Pythagorean Tarot Review |
| | Opsopaus also rejects the Qabalah from the Major Arcana, claiming there is another, and better, reason why there are 22 trumps: the Greek lettering and numbering system. | | | The back of the book supplies rituals and some complex spreads for tarot divination. | | | The companion book, Guide to the Pythagorean Tarot is weighty with Pythagorean numerology, Qabbalah, Jungian psychology and Greek mythology. |
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http://www.aeclectic.net/tarot/cards/pythagorean/review.shtml
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| | Lesson: Pythagorean Theorem |
| | Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson: | | | The activities and discussions in this lesson address the following NCTM Standard: | | | This lesson teaches the Pythagorean Theorem through the following activities: |
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http://www.shodor.org/interactivate/lessons/pyth.html
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| | Pythagorean Triples Project |
| | The Pythagorean theorem is named for the Greek mathematician Pythagoras, who lived in the 6th century BCE, though the theorem had been known elsewhere for some time before. | | | Section III is more difficult than Section IV, but it is more central to the questions that number theory tends to ask, so it has been given preference in the ordering. | | | Find several examples, and make a conjecture about when this happens. |
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http://www.math.rutgers.edu/~erowland/pythagoreantriples-project.html
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| | PlanetMath: Pythagorean triplet |
| | It follows that there are countably many Pythagorean triplets. | | | This is version 6 of Pythagorean triplet, born on 2001-10-06, modified 2005-12-15. | | | is a Pythagorean triplet if there exists a right triangle whose sides are |
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http://planetmath.org/encyclopedia/PythagoreanTriple2.html
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| | BAIN: A Pythagorean tuning of the diatonic scale |
| | Additionally, the Pythagorean fourth may be derived as the difference between the octave and fifth. | | | The Pythagorean semitone was also called the limma (left over), as it was calculated by the Greeks to be the difference (or amount left over) between a fourth and two whole tones. | | | The Pythagorean semitone may be derived as the difference between the Pythagorean fourth (4/3) and two whole tones (9/8 * 9/8). |
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http://www.music.sc.edu/fs/bain/atmi02/pst
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| | Pythagorean Triples |
| | Relative to the grid of the applet's coordinate axes, pixels are located at the points with rational coordinates with the denominator always equal to the number of pixels between two ticks on either axis. | | | There are infinitely many such numbers and there also exists a way to generate all the triples. | | | This is the well known equation of the unit circle with center at the origin. |
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http://www.cut-the-knot.org/pythagoras/pythTriple.shtml
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| | Pythagorean Theorem |
| | Not only do these numbers satisfy the Pythagorean Theorem, but any multiples of these numbers also satisfy the Pythagorean Theorem. | | | This problem could also be solved using the Pythagorean Triple 3, 4, 5. | | | Since the Pythagorean Theorem is NOT true, this triangle is |
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http://regentsprep.org/Regents/math/fpyth/Pythag.htm
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| | MathSteps: Grade 7: Pythagorean Theorem: What Is It? |
| | Pythagoras (puh thag or us) was a Greek philosopher and mathematician, born in Samos in the sixth century, B.C. He and his followers tried to explain everything with numbers. | | | Pythagorean Triples are groups of three whole numbers that make the Pythagorean Theorem true (and therefore define a true right triangle). | | | We remember him today mainly for his equation relating the lengths of the legs of a right triangle to the length of its hypotenuse. |
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http://www.eduplace.com/math/mathsteps/7/c
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| | fUSION Anomaly. Pythagoras |
| | Members of the order regarded Pythagoras as a demigod and attributed all their doctrines to him. | | | We know little of his life and nothing of his writings; all of our knowledge comes from his followers, the Pythagoreans, a mystical brotherhood he founded at Crotona. | | | The Pythagoreans believed in the transmigration of souls, and the film of their lives had to be completely absorbed and digested in order to detach themselves from it, and to avoid the recurrence of mechanical tendencies and weaknesses, thereby preparing the aspirant for conscious entry into higher worlds. |
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http://fusionanomaly.net/pythagoras.html
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| | Pythagoras, the Father of Numerology. |
| | The Chinese, Japanese, Greek, Hebrews, Egyptians, Phoenicians, early Christians, Mayans, and Incas, all employed some form of numerology to gain a deeper understanding of themselves and the universe. | | | The Pythagorean system, is among the most enduring and popular of all self-help methods ever created. | | | Pythagorean numerology was organized by Greek philosopher and mathematician Pythagoras, who combined the mathematical disciplines of the Arabic, Druid, Phoenician, Egyptian, and Essene sciences. |
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http://www.decoz.com/pythagoras.htm
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| | Exponents Worksheet #2 for Topic 6: Pythagorean Theorem |
| | Use the Pythagorean relationship to see if the triangle with these sidesis a right triangle. | | | Pythagorean triples are three numbers that satisfy the Pythagoreanrelationship. | | | Exponents Worksheet #2 for Topic 6: Pythagorean Theorem |
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http://www.neufeldmath.com/worksheets/html/ex6wk2.html
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| | Practice with Pythagorean Theorem |
| | Answer the following questions dealing with the Pythagorean Theorem. | | | What is the shortest distance, to the nearest tenth of a mile, they must travel to return to their starting point? | | | To get from point A to point B you must avoid walking through a pond. |
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http://regentsprep.org/Regents/math/fpyth/PracPyth.htm
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| | Math Forum: Ask Dr. Math FAQ: Pythagorean Triples |
| | What do Pythagorean triples have to do with Fermat's Last Theorem? | | | Plimpton 322: a remarkable ancient Babylonian tablet on number theory - Zeeman | | | A Pythagorean triple is a set of three positive whole numbers |
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http://mathforum.org/dr.math/faq/faq.pythag.triples.html
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| | Pythagorean.Theorem |
| | This is another place to look to see more about the Pythagorean Theorem. | | | The Pythagorean theorem has had several different proofs many of which are very similar. | | | For more ways of looking at the Pythagorean Theorem, try these: |
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http://jwilson.coe.uga.edu/emt669/Student.Folders/Huberty.Greg/Pythagorean.html
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| | Pythagorean Theorem |
| | The relationship was shown on a 4000 year old Babylonian tablet now known as Plimpton 322. | | | The Pythagorean theorem was first known in ancient Babylon and Egypt (beginning about 1900 B.C.). | | | Corner stakes will then be placed to mark the accurate location of the site. |
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http://www.cs.ucla.edu/~klinger/dorene/math1.htm
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| | The Pythagorean Relationship |
| | Classify triangles according to the measures of their angles. | | | Let A and B denote the other two angles, and a and b the sides opposite them, respectively. | | | right angle, isosceles triangle, scalene triangle, equilateral triangle, right triangle, vertex, angle, side, hypotenuse, square number, square root, Pythagorean relationship |
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http://argyll.epsb.ca/jreed/math8/strand3/3201.htm
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| | Read This: Pythagorean Triangles |
| | What if I told you that you could learn something new about them from this short book? | | | This book, by Wacław Sierpiński, was originally published in 1954 in Warsaw, with the English translation provided by Dr. Ambikeshwar Sharma in 1962. | | | We are also told that it is unknown whether there exists infinitely many Pythagorean triangles in which one arm and the hypotenuse are primes. |
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http://www.maa.org/reviews/pythtriangles.html
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| | Pythagorean tuning - Wikipedia, the free encyclopedia |
| | This discrepancy, of about 23.5 cents, or one quarter of a semitone, is known as a Pythagorean comma. | | | In classical music, this usually means music written prior to the 16th century. | | | The first note in the circle of fifths given here is E flat (equivalent to D#), from which five perfect fifths are tuned before arriving at D, the nominal unison note. |
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http://en.wikipedia.org/wiki/Pythagorean_tuning
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| | Pythagorean triple - Wikipedia, the free encyclopedia |
| | This shows that there are infinitely many primitive Pythagorean triples. | | | There exist infinitely many primitive Pythagorean triples in which one of the legs is the square of a natural number. | | | There exist infinitely many primitive Pythagorean triples whose hypotenuses are squares of natural numbers. |
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http://en.wikipedia.org/wiki/Pythagorean_triple
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| | Pythagorean Theorems - Some 'Not So Familiar' Implications |
| | The latter equation is known as the Pythagorean Identity in trigonometry. | | | (Note: There are many different proofs of the Pythagorean Theorem, including one by U. President J.A. Garfield in 1876. | | | This is the traditional Pythagorean result, both algebraically and geometrically. |
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http://contracosta.edu/math/pythagoras.htm
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| | Pythagorean Theorem and its many proofs |
| | It is known that the Pythagorean Theorem is Equivalent to Parallel Postulate. | | | The book is a collection of 367 proofs of the Pythagorean Theorem and has been republished by NCTM in 1968. | | | Dunham [Mathematical Universe] cites a book The Pythagorean Proposition by an early 20th century professor Elisha Scott Loomis. |
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http://www.cut-the-knot.org/pythagoras/index.shtml
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| | Pythagorean theorem -- Encyclopædia Britannica |
| | He founded the Pythagorean brotherhood, a group of his followers whose beliefs and ideas were rediscovered during the Renaissance and contributed to the development of mathematics and Western rational... | | | This belief was shaken, however, by the discovery that the diagonal of a unit square (that is, a square whose sides have a length of 1) cannot be expressed as a rational number. | | | Initially, the Pythagoreans believed that all things could be measured by the discrete natural numbers (1, 2, 3, …) and their ratios (ordinary fractions, or the rational numbers). |
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http://www.britannica.com/eb/article-9343833
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| | The Pythagorean Theorem |
| | The proof of the Pythagorean Theorem that was inspired by a figure in this book was included in the book Vijaganita, (Root Calculations), by the Hindu mathematician Bhaskara. | | | However, since his work on similarity was not to be until Books V and VI, it was necessary for him to come up with another way to prove the Pythagorean Theorem. | | | It is even said that the man who divulged the secret was drowned at sea. |
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http://jwilson.coe.uga.edu/emt669/Student.Folders/Morris.Stephanie/EMT.669/Essay.1/Pythagorean.html
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| | Pythagorean Physics - Writings by Todd Matthews Kelso |
| | Pythagorean Physics follows an axiomatic system that starts with definitions and proceeds step by step from there in a logical fashion that provides meaning in a way that other approaches can not. | | | Pythagorean Physics postulates the existence of a basic unit of matter, the Pythagorean atom. | | | This practice can add credence to ideas that should be challenged. |
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http://home.att.net/~zei/TMKelso
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| | Some Pythagorean Texts |
| | "The Pythagoreans, according to Aristoxenus, practiced the purification (catharsis) of the body by medicine, and of the soul by music." | | | For it is not lawful for one who partakes in these rites to be buried in woolen clothes. | | | And so the Pythagoreans used to invoke the Tetrad as their most binding oath: `By him that gave to our generation the Tetractys, which contains the fount and root of eternal nature...'" |
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http://www.csun.edu/~hcfll004/pythag.html
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| | Pythagorean Triplets |
| | There are some simple rules for determining a subset of Pythagorean triplets. | | | It was originally pointed out to me by Matthew Q. Boeke that there are Pythagorean triplets in which the a side is an even number. | | | With the rules for Pythagorean triplets, where a = 2pq, a must be even. |
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http://www.friesian.com/pythag.htm
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| | Pythagoras Theorem and Fibonacci Numbers |
| | Pythagoras` desire was to find the mathematical harmonies of all things. | | | A Pythagorean Triangle is a right angled triangle which sides are whole numbers. | | | That is one side, a, of the Pythagorean Triangle: |
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http://milan.milanovic.org/math/english/Pythagoras/Pythagoras.html
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| | The Exhortation to Philosophy |
| | In this work, the second volume of his "Pythagorean encyclopedia," Iamblichus the Neoplatonist (fourth century A.D.) describes the nature of philosophic life and shows how the path of philosophia leads from the realm of Becoming to the world of Being. | | | This work is also an introduction to the study of Plato and contains Iamblichus's commentaries on The Golden Verses of Pythagoras and some select Pythagorean aphorisms. |
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http://www.phanes.com/exhphi.html
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| | Pythagorean Theorem Problems |
| | This is a hands-on exercise for you to convince yourself that the Pythagorean theorem works. | | | The Pythagorean Triples were described with Tip number 1. | | | There is a simple formula that gives all the Pythagorean triples. |
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http://www.arcytech.org/java/pythagoras/problems.html
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| | PlanetMath: Pythagorean theorem |
| | This is version 15 of Pythagorean theorem, born on 2001-10-06, modified 2005-05-15. | | | Cosines law is a generalization of Pythagorean theorem for any triangle. | | | It implies that the converse of Pythagorean theorem also holds: if the sides of a triangle satisfy |
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http://planetmath.org/encyclopedia/PythagorasTheorem.html
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| | Pythagorean Theorem |
| | Before the proofs, it is important for students to see the theorem as it is worded. | | | However 1000 years later, between the years of 580-500 BC, Pythagoras of Samos was the first to prove the theorem. | | | With the Pythagorean theorem being such a popular topic, it is no wonder high school students study the theorem. |
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http://www.ms.uky.edu/~lee/ma502/pythag/pythag.htm
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| | SparkNotes: Special Triangles: The Pythagorean Theorem |
| | This is a physical interpretation of the Pythagorean Theorem. | | | One of the most interesting and well-known formulas in math is the Pythagorean Theorem, which only holds true for right triangles. | | | The three sides of a right triangle can be of any length, provided that they obey the laws of the Pythagorean Theorem. |
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http://www.sparknotes.com/math/geometry2/specialtriangles/section5.rhtml
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| | The Pythagorean Theorem |
| | Thus, implicit in this particular constructivist approach to the Pythagorean theorem is the notion that the student should build his or her own knowledge by "eyeballing" right angles. | | | Some may argue that it doesn't really matter which method is used to teach the Pythagorean theorem - i.e., that both methods lead to the same result. | | | Yet, in the name of constructivism, we seem to be encouraging a generation of children to erect this pillar of mathematical knowledge on just such a basis. |
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http://www.mathematicallycorrect.com/pythag.htm
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| | Pythagorean Tuning and Medieval Polyphony - Table of Contents |
| | While our focus here is on the music of medieval Europe, the concept of a tuning based on a series of twelve notes in perfect fifths also plays an important part in other world musical traditions, for example in Chinese theory and practice. | | | Readers interested in the practical details of Pythagorean tuning are encouraged to jump directly from Section 2 to Section 4. | | | Section 4 explores some aspects of the tuning in more detail, while Section 5 considers its relationship to other systems of just intonation as well as alternative approaches such as equal temperament. |
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http://www.medieval.org/emfaq/harmony/pyth.html
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| | Cool math .com - Trigonometry Lessons - The Pythagorean Identities |
| | This page shows the derivations of the three Pythagorean Identities. | | | Cool math.com - Trigonometry Lessons - The Pythagorean Identities | | | A "derivation" means that we need to create this from scratch - or, at least, from other things that we know. |
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http://www.coolmath.com/pythagoreanidentities.htm
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| | Pythagorean Tuning - Basic concepts |
| | In fact, Pythagorean tuning is described in the medieval sources as being based on four numbers: 12:9:8:6. | | | More specifically, it is a form of just intonation based on the numbers 3 and 9. | | | As mentioned above, Pythagorean tuning defines all notes and intervals of a scale from a series of pure fifths with a ratio of 3:2. |
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http://www.medieval.org/emfaq/harmony/pyth2.html
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| | Pythagorean comma Definition Information Explanation Review WikiCity.com - Wikipedia Free Encyclopedia, Free ... |
| | When you ascend by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging 12 times, you eventually reach a note around seven octaves above the note you started on, which, when lowered to the same octave as your starting point, is 23.46 centss higher than the initial note. | | | This interval, 531441:524288 or approximately 1.0136:1, is called a Pythagorean comma. | | | This interval has serious implications for the various tuning schemes of the chromatic scale, because in western music, 12 perfect fifths and seven octaves are treated as the same interval. |
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http://www.wikicity.com/wikipedia/p/py/pythagorean_comma.html
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| | Pythagorean Th. |
| | One of the most interesting areas is the "Infinite Descent" which is a style of proof in which we start with the smallest number with a certain property, and then prove that there is a smaller number with that property -- therefore there is really no number with that property. | | | A "Pythagorean Triple" is a set of three numbers a,b,c such that a²+b²=c². | | | Half of the difference of the two odd numbers is the other square to form the sum, 5. |
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http://mcraefamily.com/MathHelp/Pythag.htm
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| | Pythagorean Theorem |
| | Read section 21.4 in your textbook on the Pythagorean Theorem | | | Are there other relationships related to the Pythagorean Theorem? | | | You need to know how the Pythagorean Theorem is applied when finding the distance between two points. |
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http://www.gowcsd.com/master/ghs/math/furman/pythagor/pythag.htm
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| | ez5-3 Pythagorean Theorem & its Converse (Prentice Hall Geometry) |
| | This section introduces the Pythagorean Theorem and its converse. | | | A Pythagorean Triple is a group of three whole numbers that satisfies the equation | | | Theorem 5-5 The Converse of the Pythagorean Theorem |
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http://www.e-zgeometry.com/classph/sec5/5.3/5.3.htm
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| | Pythagorean Tarot |
| | See also the collected Verses on the Trumps (which also appear in the individual descriptions of the trumps). | | | The Pythagorean Tarot book (480 pages) and 78-card deck set has been published by Llewellyn, ISBN 1-56718-449-9. | | | Special Announcement: I will be giving a presentation on the Pythagorean Tarot at the International Tarot Society World Tarot Congress in Chicago, May 10-12, 2002. |
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http://www.cs.utk.edu/~mclennan/BA/PT/PT.html
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