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# s_notes01 - I— l MATHEMATICAL OBJECTS MATHEMATICAL WORLD...

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Unformatted text preview: I— l MATHEMATICAL OBJECTS MATHEMATICAL WORLD VS. PHYSICAL UNIVERSE Math world: numbers, functions, operations, geometries, inﬁnities, limits. Existence of math objects. Creation of math objects: —from sets —by deﬁnition from existing math objects “Ideal” and perfect nature of math objects Study and exploration of math objects and their properties by deduction (“proof”). Assumptions and theorems (= facts) Remarkable agreement of community of scientists and mathematicians as to correctness of mathematical investigations and results 1-2 2-DIMENSIONAL EUCLIDEAN GEOMETRY (E2) and 3—DIMENSIONAL EUCLIDEAN GEOMETRY (E3) How we visualize them and make copies if them Form of the deﬁnition for E2 (one of several approaches): -inﬁnite set of basic “atomic” objects called oints -special sets of points to be called ﬁnes Form of the deﬁnition for E3: -inﬁnite set of basic “atomic” objects called points -special sets of points to be called lines -special sets of points to be called planes Deﬁnitions: A given set S of points in E2 is said to be collinear if there is some line L in E2 such that every point in S is also in L. A given set S of points in E3 is said to be collinear if there is some line L in E3 such that every point in S is also in L. A given set S in E3 is said to be coplanar if there is some plane M such that every point in S is also in M. 1-3 INCIDENCE RELATIONS (Part of deﬁnition of E2 and E3) For E’: Given two distinct points, there is a unique (one and only one) line which contains both points. For any two points P and Q, there is a measure which gives the distance d(P,Q) between P and Q. For E3: Given two distinct points, there is a unique (one and only one) line which contains both points. Given two distinct points in a plane M, the unique line determined'by those points must lie entirely in (be a subset of) M. Given three distinct non-collinear points, there is a unique plane which contains all three points. (Equivalent assertion: Given two intersecting lines, there is a unique plane which contains both lines.) Given two planes which intersect (have a common point), then their intersection (the set of all common points) must be a line. For any two points P and Q, there is a measure which gives the distance d(P,Q) between P and Q. Every plane in E3 is a copy of E2. 1-4 PARALLELISM (Deﬁnitions) For E2: Two lines are said to be parallel if they do not intersect. For E3: Two lines are said to be parallel if they do not intersect but do lie in a common plane. Two lines are said to be _s_l_gg:_v_v if they do not intersect and do not lie in a common plane. If each of two given distinct lines is parallel to the same third line, then those two given lines must be parallel. CONGRUENCE The idea of congmence is fundamental for E2 and E3. A mapping on E2 (or E3) is a function f which carries every point P to some corresponding point f(P). A mapping is said to be an isometg if for every pair of points P and Q, d(f(P),f(Q)) = d(P,Q). A set of points S is said to be congr_uent to a set of points 8’ if there is an isometry which carries S exactly onto 8'. An angle consists of two rays (half-lines) from a common vertex point. An angle is said to be a right angle if it is congruent to the “supplementary” angle formed by reversing the direction of one of its rays. ‘ I - 5 PERPENDICULARIT Y For E2: Two lines are said to be pegp endicular, if they intersect at a right angle. For E3: Two lines are said to be peppendicular, if they intersect at a right angle. A line and a plane are said to be pegpendicular, if they intersect at a single point P and if the given line is perpendicular to every line lying in the given plane and going through P. PROJECTIONS For E’: Fact: For a given ﬁle L and any point P, there is a unique line L' such that L’ contains P and is perpendicular to L. The intersection point of L’ with L is called the projection of point P on L. For any set S of points and any line L, the set of all projections on L of points in S is called the projection of S on L. For E3: Fact: For a given plane M and any point P, there is a unique line going through P and perpendicular to M. The intersection point this line with M is called the projection of the point P on the plane M. For any set S of points, the set of all projections on M of points in S is called the projection of S on M. I - 6 PROJECTIONS (continued) F act: For a given line L and any point P, there is a unique plane going through P and perpendicular to L. The intersection point of L with this plane is called the projection of the point P on the line L. For any set S of points, the set of all projections on L of points in S is called the projection of S on L. MISCELLANEOUS DEFINITIONS (to be found in Chapter 1 of the text) parallelogram dihedral angle polygon n-gon quadrilateral planar polygon convex polygon regular polygon polyhedron n-hedron convex polyhedron regular polyhedron octahedron dodecahedron icosahedron parallelepiped prism I - 7 DEFINITIONS continued pyramid sphere cylinder cone Also: simple mensuration formulas (area and volume) ...
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s_notes01 - I— l MATHEMATICAL OBJECTS MATHEMATICAL WORLD...

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