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euclidz geometry


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euclidz geometry

  1. 1. Euclid's Geometry
  2. 2. Index <ul><li>What is Euclid's Geometry? </li></ul><ul><li>Difference between Axioms and Postulates. </li></ul><ul><li>All Axioms and Postulates. </li></ul><ul><li>Different Mathematicians and their contribution towards Mathematics. </li></ul>
  3. 3. What is Euclid’s Geometry 1.1 <ul><li>Euclidean geometry,  the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician  Euclid  ( c.  300 BCE). In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools. Indeed, until the second half of the 19th century, when  non-Euclidean geometries  attracted the attention of mathematicians,  geometry  meant Euclidean geometry. </li></ul>
  4. 4. What is Euclid’s Geometry 1.2 <ul><li>It is the most typical expression of general mathematical thinking. Rather than the memorization of simple algorithms to solve equations by rote, it demands true insight into the subject, clever ideas for applying theorems in special situations, an ability to generalize from known facts, and an insistence on the importance of proof. In Euclid’s great work, the  Elements , the only tools employed for geometrical constructions were the ruler and the compass—a restriction retained in elementary Euclidean geometry to this day. </li></ul>
  5. 5. Difference between Axioms and Postulates. <ul><li>Postulates - The assumptions that were specific to geometry are called postulates. </li></ul><ul><li>Axioms - The assumptions that are used throughout mathematics and not specifically linked to geometry are called Axioms. </li></ul>
  6. 6. All Axioms and Postulates. <ul><li>Axioms:- </li></ul><ul><li>Things that equal the same thing also equal one another. </li></ul><ul><li>If equals are added to equals, then the wholes are equal. </li></ul><ul><li>If equals are subtracted from equals, then the remainders are equal. </li></ul><ul><li>Things that coincide with one another equal one another. </li></ul><ul><li>The whole is greater than the part. </li></ul><ul><li>  Things which are double of the same things are equal to one and other. </li></ul><ul><li>Things which are halves of the same things are equal to one another. </li></ul>
  7. 7. Postulates <ul><li>1. A straight line segment can be drawn joining any two points. </li></ul><ul><li>2. Any straight line segment can be extended indefinitely in a straight line. </li></ul><ul><li>3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. </li></ul><ul><li>4. All right angles are congruent. </li></ul><ul><li>5. if a straight line falling on 2 straight lines makes the interior angles on the same side of it taken together less than 2 right angles, then 2 straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than 2 right angles. </li></ul>
  8. 8. Euclid’s 1 to 4 Postulates
  9. 9. 5 th Postulate
  10. 10. Different Mathematicians and their contribution towards Mathematics. <ul><li>Euclid of Alexandria </li></ul><ul><li>Srinivasa Ramanujan </li></ul><ul><li>Rene Descartes </li></ul><ul><li>Aryabhatta </li></ul><ul><li>Thales </li></ul>
  11. 11. Euclid of Alexandria <ul><li>Euclid of Alexandria  is the most prominent mathematician of antiquity best known for his treatise on mathematics  The Elements . The long lasting nature of  The Elements  must make Euclid the leading mathematics teacher of all time. However little is known of Euclid's life except that he taught at Alexandria in Egypt. </li></ul>
  12. 12. Srinivasa Ramanujan <ul><li>Srinivasa Ramanujan was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series. </li></ul><ul><li>Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic. </li></ul>
  13. 13. Rene Descartes <ul><li>Rene Descartes was a great philosopher and thinker, many overlook his contribution to math because of his overwhelming additions to the field of philosophy, however we would like to point out this mans work on mathematics so that he gets even more credit to his name. By the way he passed away from a cold, away from him native France, and could have probably made an even bigger impact on  modern science if he had not passed away in a relatively early age. </li></ul>
  14. 14. Aryabhatta <ul><li>Aryabhatta is a renowned mathematician and astronomer of ancient India. He was born in 476 AD in Kerala. He studied at the University of Nalanda. One of his major work was Aryabhatiya written in 499 AD. The book dealt with many topics like astronomy, spherical trigonometry, arithmetic, algebra and plane trigonometry. He jotted his inventions in mathematics and astronomy in verse form. The book was translated into Latin in the 13th century. Through the translated Latin version of the Aryabhattiya, the European mathematicians learned how to calculate the areas of triangles, volumes of spheres as well as how to find out the square and cube root.  </li></ul>
  15. 15. Thales <ul><li>Thales, an engineer by trade, was the first of the Seven Sages, or wise men of Ancient Greece. Thales is known as the first Greek philosopher, mathematician and scientist. He founded the geometry of lines, so is given credit for introducing abstract geometry. </li></ul><ul><li>Thales is credited with the following five theorems of geometry: </li></ul><ul><li>A circle is bisected by its diameter. </li></ul><ul><li>Angles at the base of any isosceles triangle are equal. </li></ul><ul><li>If two straight lines intersect, the opposite angles formed are equal. </li></ul><ul><li>If one triangle has two angles and one side equal to another triangle, the two triangles are equal in all respects. (See Congruence) </li></ul><ul><li>Any angle inscribed in a semicircle is a right angle. This is known as Thales' Theorem. </li></ul>