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UnivHypGeom20: Pure and applied geometry--understanding the continuum

The distinction between pure and applied geometry is closely related to the difference between rational numbers and decimal numbers. Especially when we treat decimal numbers in an approximate way: specifying rather an interval or range rather than a particular value. This gives us a way of explaining the distinction between a line meeting a circle exactly or only roughly. This video addresses a very big confusion in mathematics: the idea that `real numbers' are a proper model for the `continuum'. THEY ARE NOT!! The true foundation for mathematics rests in the rational numbers and concrete constructions made from them. So we point out some of the logical deficiencies in the usual chat about the square root of 2, or pi, or e. And show the way towards a much more sensible approach to one of the most important problems in mathematics: how to understand the hierarchy of continuums.CONTENT SUMMARY: pg 1: @00:11 Circles, lines, rational numbers, real numberspg 2: @04:00 Errett Bishop quote; Pure Geometry and Applied Geometry comparedpg 3: @05:58 Pure Geometry|rational numbers :: Applied Geometry|decimal numbers; rational number? frameworkpg 4: @07:32 Decimal numberspg 5: @11:40 infinite decimals; pg 6: @22:31 Applied mathematicians; rough decimalpg 7: @26:06 example; look at pixelspg 8: @30:58 rough or exact? solutions of a polynomial curve, Fermat curvepg 9: @32:52 unit circlepg 10: @34:53 Continuum Problem: To understand the hierarchy of continuums (THANKS to EmptySpaceEnterprise)
Length: 38:44


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