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Euclid

Euclid. By: Tristan Mishell.

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Euclid

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  1. Euclid By: Tristan Mishell

  2. Euclid was a Greek mathematician who is best known for his work on the Elements. He is the most prominent mathematician of antiquity best known for his treatise on mathematics in 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 Lived from 330?-275? B.C.

  3. His Writings • Euclid's fame comes from his writings, the best known of which are his treatise entitled, The Elements. He wrote many other works including Data, On Division, Phaenomena, Optics and the lost books Conics and Prisms.

  4. Book Contents 1 Triangles 2 Rectangles 3 Circles 4 Polygons 5 Proportion 6 Similarity 7-10 Number Theory 11 Solid Geometry 12 Pyramids 13 Platonic Solids The Elements Euclid's Elements was used as a text for 2,000 years, and even today a modified version of its first few books forms the basis of high school instruction in plane geometry

  5. The Five Axioms • In The Elements Euclid produced five axioms, a.k.a. postulates. The axioms that Euclid explicitly stated in The Elements were five in number, of which the fifth (parallel axiom) is the most famous due to it's proof being highly controversial until the 19th century. Below are the five axioms: • Euclid's Postulates • 1. A straight Line Segment can be drawn joining any two points. • 2. Any straight Line Segment can be extended indefinitely in a straight Line. • 3. Given any straight Line Segment, a Circle can be drawn having the segment as Radius and one endpoint as center. • 4. All Right Angles are congruent. • 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two Right Angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the Parallel Postulate.

  6. Euclid's Theorem: There is no largest prime number. • Proof • Suppose there was a largest prime number; call it N. Then there are only finitely many prime numbers, because each has to be between 1 and N. Let's call those prime numbers a, b, c, ..., N. Then consider this number: • M = a * b * c * ... * N + 1 • Is this new number M a prime number? We could check for divisibility: • M is not divisible by a, because M / a = b * c * ... * N + 1 / a • M is not divisible by b, because M / b = a * c * ... * N + 1 / b • M is not divisible by c, because M / c = a * b * ... * N + 1 / c • ..... • Hence, M is not divisible by a, b, c, ..., N. Since these are all possible prime numbers, M is not divisible by any prime number, and therefore M is not divisible by any number. That means that M is also a prime number. But clearly M > N, which is impossible, because N was supposed to be the largest possible prime number. Therefore, our assumption is wrong, and thus there is no largest prime number.

  7. Credits www-gap.dcs.st-and.ac.uk/~history/ Mathematicians/Euclid.html http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Euclid.html http://www.shu.edu/html/teaching/math/reals/history/euclid.html http://www.shu.edu/html/teaching/math/reals/logic/proofs/euclidth.html http://users.rcn.com/bobmer.javanet/euklid.htm http://archive.ncsa.uiuc.edu/SDG/Experimental/vatican.exhibit/exhibit/d- mathematics/Greek_math.html

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