Pythagoras

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Pythagoras

  1. 1. Pythagoras of SamosPythagoras of Samos is often described as the first                                                 pure  is often described as the first                                                 pure mathematician. He is an extremely important                                                figure in the mathematician. He is an extremely important                                                figure in the development of mathematics yet we know                                          relatively little about development of mathematics yet we know                                          relatively little about his mathematical achievementshis mathematical achievements..    Bust of Pythagoras of Samos in the Capitoline Museums, RomeBust of Pythagoras of Samos in the Capitoline Museums, Rome                                                                                         
  2. 2.    IntroductionIntroduction Pythagoras of Samos was Pythagoras of Samos was Ionian Greek philosopher, mathematician, and founder of the Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries information about Pythagoras was written down centuries after he lived, so very little reliable information is known about after he lived, so very little reliable information is known about him. He was born on the island of Samos, and might have him. He was born on the island of Samos, and might have traveled widely in his youth, visiting Egypt and other places traveled widely in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC, he moved to Croton, seeking knowledge. Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious a Greek colony in southern Italy, and there set up a religious sect. His followers pursued the religious rites and practices sect. His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical developed by Pythagoras, and studied his philosophical theories. The society took an active role in the politics of theories. The society took an active role in the politics of Croton, but this eventually led to their downfall. The Croton, but this eventually led to their downfall. The Pythagorean meeting-places were burned, and Pythagoras Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said to have ended his days in was forced to flee the city. He is said to have ended his days in Meta. Meta. 
  3. 3.  Pythagoras made influential contributions to philosophy andPythagoras made influential contributions to philosophy andreligious teaching in the late 6th century BC. He is often reveredreligious teaching in the late 6th century BC. He is often reveredas a great mathematician, mystic and scientist, but he is bestas a great mathematician, mystic and scientist, but he is bestknown for the Pythagorean theorem which bears his name.known for the Pythagorean theorem which bears his name.However, because legend and obfuscation cloud his work evenHowever, because legend and obfuscation cloud his work evenmore than with the other pre-Socratic philosophers, one canmore than with the other pre-Socratic philosophers, one cangive account of his teachings to a little extent, and some havegive account of his teachings to a little extent, and some havequestioned whether he contributed muchquestioned whether he contributed muchto mathematics and natural philosophy. Many of theto mathematics and natural philosophy. Many of theaccomplishments credited to Pythagoras may actually haveaccomplishments credited to Pythagoras may actually havebeen accomplishments of his colleagues and successors. Whetherbeen accomplishments of his colleagues and successors. Whetheror not his disciples believed that everything was related toor not his disciples believed that everything was related tomathematics and that numbers were the ultimate reality ismathematics and that numbers were the ultimate reality isunknown. It was said that he was the first man to call himself aunknown. It was said that he was the first man to call himself aphilosopher, or lover of wisdom and Pythagorean ideasphilosopher, or lover of wisdom and Pythagorean ideasexercised a marked influence on Plato, and through him, allexercised a marked influence on Plato, and through him, allof Western philosophy.of Western philosophy.
  4. 4. • Both Plato and Isocrates affirm that, above all else, Pythagoras wasBoth Plato and Isocrates affirm that, above all else, Pythagoras wasfamous for leaving behind him a way of life.famous for leaving behind him a way of life.Both Iamblichus and Porphyry give detailed accounts of theBoth Iamblichus and Porphyry give detailed accounts of theorganization of the school, although the primary interest of bothorganization of the school, although the primary interest of bothwriters is not historical accuracy, but rather to present Pythagoras aswriters is not historical accuracy, but rather to present Pythagoras asa divine figure, sent by the gods to benefit humankind.a divine figure, sent by the gods to benefit humankind.• ->ISOCRATES ->PLATO->ISOCRATES ->PLATO
  5. 5. • Pythagoras set up an organization which was in some ways a school,Pythagoras set up an organization which was in some ways a school,in some ways a brotherhood (and here it should be noted that sourcesin some ways a brotherhood (and here it should be noted that sourcesindicate that as well as men there were many women among theindicate that as well as men there were many women among theadherents of Pythagoras) and in some ways a monastery. It was basedadherents of Pythagoras) and in some ways a monastery. It was basedupon the religious teachings of Pythagoras and was very secretive. Theupon the religious teachings of Pythagoras and was very secretive. Theadherents were bound by a vow to Pythagoras and each other, for theadherents were bound by a vow to Pythagoras and each other, for thepurpose of pursuing the religious and ascetic observances, and ofpurpose of pursuing the religious and ascetic observances, and ofstudying his religious and philosophical theories. The claim that theystudying his religious and philosophical theories. The claim that theyput all their property into a common stock is perhaps only a laterput all their property into a common stock is perhaps only a laterinference from certain Pythagorean maxims and practices.inference from certain Pythagorean maxims and practices.
  6. 6. Pythagoras TheoremPythagoras Theorem• In mathematics, the Pythagorean theorem or PythagorasIn mathematics, the Pythagorean theorem or Pythagorastheorem is a relation in Euclidean geometry among the three sidestheorem is a relation in Euclidean geometry among the three sidesof a right triangle (right-angled triangle). In terms of areas, itof a right triangle (right-angled triangle). In terms of areas, itstates:states:• In any right triangle, the area of the square whose side isIn any right triangle, the area of the square whose side isthe hypotenuse (the side opposite the right angle) is equal to thethe hypotenuse (the side opposite the right angle) is equal to thesum of the areas of the squares whose sides are the two legs (thesum of the areas of the squares whose sides are the two legs (thetwo sides that meet at a right angle).two sides that meet at a right angle).• The theorem can be written as an equation relating the lengths ofThe theorem can be written as an equation relating the lengths ofthe sides a, b and c, often called the Pythagorean equation:the sides a, b and c, often called the Pythagorean equation:• where c represents the length of the hypotenuse,where c represents the length of the hypotenuse,and a and b represent the lengths of the other two sides.and a and b represent the lengths of the other two sides.
  7. 7. • The Pythagorean theorem is named afterThe Pythagorean theorem is named afterthe Greek mathematician Pythagoras, who by tradition is creditedthe Greek mathematician Pythagoras, who by tradition is creditedwith its discovery and proof, although it is often argued thatwith its discovery and proof, although it is often argued thatknowledge of the theorem predates him. There is evidenceknowledge of the theorem predates him. There is evidencethat Babylonian mathematicians understood the formula, althoughthat Babylonian mathematicians understood the formula, althoughthere is little surviving evidence that they used it in a mathematicalthere is little surviving evidence that they used it in a mathematicalframework.framework.• The theorem has numerous proofs, possibly the most of anyThe theorem has numerous proofs, possibly the most of anymathematical theorem. These are very diverse, including bothmathematical theorem. These are very diverse, including bothgeometric proofs and algebraic proofs, with some dating backgeometric proofs and algebraic proofs, with some dating backthousands of years.thousands of years.• The theorem can be generalized in various ways, including higher-The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects thatdimensional spaces, to spaces that are not Euclidean, to objects thatare not right triangles, and indeed, to objects that are not triangles atare not right triangles, and indeed, to objects that are not triangles atall, but n-dimensional solids. The Pythagorean theorem has attractedall, but n-dimensional solids. The Pythagorean theorem has attractedinterest outside mathematics as a symbol of mathematicalinterest outside mathematics as a symbol of mathematicalabstruseness, mystique, or intellectual power; popular references inabstruseness, mystique, or intellectual power; popular references inliterature, plays, musicals, songs, stamps and cartoons abound.literature, plays, musicals, songs, stamps and cartoons abound.
  8. 8. Done by:Done by:ShaktiShakti

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