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Linear Geometric Constructions Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers Friday, July 8 th , 2011. Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric

  1. Linear Geometric Constructions Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers Friday, July 8 th , 2011. Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 1 / 24

  2. Introduction What is a Geometric Construction? Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 2 / 24

  3. Introduction What is a Geometric Construction? Types of Geometric Constructions Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 2 / 24

  4. Introduction What is a Geometric Construction? Types of Geometric Constructions Mathematicians Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 2 / 24

  5. Definitions and Rules for Basic Construcitons Tools Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 3 / 24

  6. Definitions and Rules for Basic Construcitons Tools Compass Straightedge Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 3 / 24

  7. Definitions and Rules for Basic Construcitons Tools Compass Straightedge Postulates for Basic Constructions Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 3 / 24

  8. Definitions and Rules for Basic Construcitons Tools Compass Straightedge Postulates for Basic Constructions Assume we can construct two points (the origin and (1,0)) Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 3 / 24

  9. Definitions and Rules for Basic Construcitons Tools Compass Straightedge Postulates for Basic Constructions Assume we can construct two points (the origin and (1,0)) Constructions of lines and circles Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 3 / 24

  10. Definitions and Rules for Basic Construcitons Tools Compass Straightedge Postulates for Basic Constructions Assume we can construct two points (the origin and (1,0)) Constructions of lines and circles Use of the intersections of those lines and cirlces to constuct new points Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 3 / 24

  11. Basic constructions Perpendicular Lines Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 4 / 24

  12. Basic constructions Perpendicular Lines Parallel Lines Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 4 / 24

  13. Basic constructions Perpendicular Lines Parallel Lines Squareroots Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 4 / 24

  14. Basic constructions Perpendicular Lines Parallel Lines Squareroots Bisecting an angle Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 4 / 24

  15. Examples of Basic Constructions Theorem Given a line L and a point P on the line, we can draw a line perpendicular to L that passes through P. Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 5 / 24

  16. Examples of Basic Constructions Theorem Given a line L and a point P on the line, we can draw a line perpendicular to L that passes through P. Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 6 / 24

  17. Examples of Basic Constructions Theorem Given a line L and a point P on the line, we can draw a line perpendicular to L that passes through P. Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 7 / 24

  18. Examples of Basic Constructions Theorem Given a constructible number a, we can construct √ a. Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 8 / 24

  19. Examples of Basic Constructions Theorem Given an angle BAC, we can bisect the angle Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 9 / 24

  20. Examples of Basic Constructions Theorem Given an angle BAC, we can bisect the angle Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 10 / 24

  21. Fields What is a field? Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 11 / 24

  22. Fields What is a field? Operations Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 11 / 24

  23. Fields What is a field? Operations Addition Subtraction Multiplication Division Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 11 / 24

  24. Fields What is a field? Operations Addition Subtraction Multiplication Division Properties Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 11 / 24

  25. Fields What is a field? Operations Addition Subtraction Multiplication Division Properties Associative Communitive Distributive Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 11 / 24

  26. Fields What is a field? Operations Addition Subtraction Multiplication Division Properties Associative Communitive Distributive Identities Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 11 / 24

  27. Fields What is a field? Operations Addition Subtraction Multiplication Division Properties Associative Communitive Distributive Identities Additive identity Multiplicative identity Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 11 / 24

  28. Fields What is a field? Operations Addition Subtraction Multiplication Division Properties Associative Communitive Distributive Identities Additive identity Multiplicative identity Inverses Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 11 / 24

  29. Fields What is a field? Operations Addition Subtraction Multiplication Division Properties Associative Communitive Distributive Identities Additive identity Multiplicative identity Inverses Additive inverse Multiplicative inverse Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 11 / 24

  30. Examples of Fields Q , the rational numbers Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 12 / 24

  31. Examples of Fields Q , the rational numbers R , the real numbers Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 12 / 24

  32. Examples of Fields Q , the rational numbers R , the real numbers C , the complex numbers Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 12 / 24

  33. Examples of Fields Q , the rational numbers R , the real numbers C , the complex numbers E , the constructible numbers Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 12 / 24

  34. Two-tower over Q Theorem Theorem The number α ∈ R is constructible with straight edge and compass ( α ∈ constructiblenumbers ) if and only there is a sequence of field extensions Q = F 0 ⊂ F 1 ⊂ F 2 ⊂ ...... ⊂ F n so that [ F i : F i − 1 ] = 2 or 1 for i = 1 , ......., n (i.e. F i = F i − 1 ( √ β i )), β i ∈ F i − 1 and α ∈ F n The Theorem states ( Q ) is constructible with only a straight edge and compass. � p is constructible with only a straight edge and compass q Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 13 / 24

  35. Limitations of Basic Constructions What constructions are we not able to do with simply a straightedge and compass? Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 14 / 24

  36. Limitations of Basic Constructions What constructions are we not able to do with simply a straightedge and compass? Trisecting an angle Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions 14 / 24

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