ImpossibleConstructions

ImpossibleConstructions - Math 461 F Impossible...

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Math 461 F Spring 2011 Impossible Constructions Drew Armstrong In class we proved that 2 is not a rational number (the edge and diagonal of a square are incommensurable). This “crisis of incommensurables” forced the Greeks to base their mathematics on the concept of “length” instead of “number”. In particular, the Euclidean system was based on constructions with straightedge-and-compass, which are restricted to drawing the line containing two given points, drawing the circle with a given center and radius, drawing the intersection points of lines and circles. (Why did they do this? Well, you can’t prove anything if you don’t have some rules.) However, this system quickly ran into trouble. Classical mathematics was unable to solve the following problems, and it was not for lack of trying: (1) to draw a square with area equal to a given circle, (2) to draw the edge of a cube with volume double that of a given cube, (3) to draw a line that trisects the angle between two given lines, (4) to draw the regular heptagon. Many people suspected that these problems were impossible, but no one could prove it until Fermat and Descartes ( La G´ eom´ etrie , 1637) finally healed the rift between geometry and algebra by introducing coordinate geometry
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This note was uploaded on 01/08/2012 for the course MATH 561 taught by Professor Armstrong during the Spring '11 term at University of Miami.

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ImpossibleConstructions - Math 461 F Impossible...

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