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# MODERN GEOMETRY MATH 532

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This 2 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 532 at University of South Carolina - Columbia taught by Ma Filaseta in Fall. Since its upload, it has received 9 views. For similar materials see /class/229525/math-532-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15

MATH 5327361 LECTURE 1 1 Hand out and go over syllabus 2 Class photos 3 No homework today 4 Logic in the Last Century 0 Are there statements that can be made in mathematics which are true but which we cannot prove Remark 1 The answer cannot be Yes since if there exist such statements we would not know they are true so we would not know they exist Remark 2 Remark 1 is wrong Such statements are known to exist but that does not mean that we know what they are o Is mathematics consistent Is it possible that some day a proof will exist that 11 3 Remark 1 If mathematics were not consistent we wouldn t have this class Therefore mathematics must be consistent Remark 2 Remark 1 is questionable No proof exists that mathematics is consistent It is known however that if mathematics is consistent then we cannot prove it is consistent 5 Back to Euclid s Time i Constructions Find the perpendicular bisector of a line segment Duplicate an angle Divide a segment into 3 equal pieces Given a line Z and a point P not on Z construct a line Z parallel to Z that passes through P Find the center of a given circle Given a line E and two points A and B on one side of the line nd 0 on E so that AOD BOE Given a circle 0 and a point P outside 0 construct a line Z tangent to O that passes through P Given a circle 0 and two points P and Q outside 0 construct the circles that are tangent to O and pass through P and Q ii Even More Basic Constructions Given two points construct a line through them What if the points are far apart and the straightedge and compass are small by comparison Draw a circle with a given radius and center iii Euclid s Axioms or Postulates A straight line can be drawn through any 2 points D A straight line can be extended in either direction as long as we wish U Given any point P and any distance r we can draw a cirlce of radius r centered at All right angles are equal 1148 If a line Z intersects 2 lines 1 and 2 and makes 2 interior angles on the same side of Z each less then a right angle then the lines 1 and 2 intersect on that side of Z Comment From these axioms Euclid developed theorems or propositions 6 Between Euclid and a Century Ago 0 What if the axioms were different Consider something like 1 There exist 3 non collinear points 2 Given two points there exists a line passing through them 3 Any two distinct lines intersect in exactly two points Do these axioms make sense Are they simply describing something that s not true and therefore nonsense 0 Describe great circles on a sphere and explain why the axioms above all hold in this context

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