A rigorous deductive approach to elementary Euclidean geometry
... teaching elementary Euclidean geometry at the secondary education levels. Euclidean geometry is a privileged area of mathematics, since it allows from an early stage to practice rigorous reasonings and to exercise vision and intuition. Our concern is that the successive reforms of curricula in the l ...
... teaching elementary Euclidean geometry at the secondary education levels. Euclidean geometry is a privileged area of mathematics, since it allows from an early stage to practice rigorous reasonings and to exercise vision and intuition. Our concern is that the successive reforms of curricula in the l ...
08. Non-Euclidean Geometry 1. Euclidean Geometry
... • There are indefinitely many lines through a given point that are parallel to any given straight line. • The sum of angles of a triangle < 2 right angles. ...
... • There are indefinitely many lines through a given point that are parallel to any given straight line. • The sum of angles of a triangle < 2 right angles. ...
Taxicab Geometry
... If two points line in a plane, then the line containing these points lies in the same plane. ...
... If two points line in a plane, then the line containing these points lies in the same plane. ...
what`s coordinate geometry - Study Hall Educational Foundation
... trigonometry. It is also applied in scanners which make use of coordinate geometry to reproduce the exact image of the selected picture in the computer. It manipulates the points of each information in the original documents and reproduces them in soft copy. ...
... trigonometry. It is also applied in scanners which make use of coordinate geometry to reproduce the exact image of the selected picture in the computer. It manipulates the points of each information in the original documents and reproduces them in soft copy. ...
2 - Trent University
... triangles are exactly the same shape, but not necessarily the same size. We will mainly be concerned with triangles when dealing with congruence and similarity, but the definitions can be extended in obvious ways to polygons with more sides, and to two-dimensional shapes in general. 1. Show that con ...
... triangles are exactly the same shape, but not necessarily the same size. We will mainly be concerned with triangles when dealing with congruence and similarity, but the definitions can be extended in obvious ways to polygons with more sides, and to two-dimensional shapes in general. 1. Show that con ...
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On Euclidean and Non-Euclidean Geometry by Hukum Singh DESM
... 3-dimensional Euclidean space was studied by Gauss. German mathematician B.Riemann (1826-1866) discovered n-dimensional geometry which is now known as Riemannian geometry. Riemann also studied on spherical geometry and showed that every line passing through a point R not on the line PQ meets the lin ...
... 3-dimensional Euclidean space was studied by Gauss. German mathematician B.Riemann (1826-1866) discovered n-dimensional geometry which is now known as Riemannian geometry. Riemann also studied on spherical geometry and showed that every line passing through a point R not on the line PQ meets the lin ...
Practice Exam 5: Topology
... that if ω, η ∈ Λ∗ (W ) then A∗ (ω ∧ η) = A∗ (ω) ∧ A∗ (η). 3. Produce a C ∞ -compatible atlas of coordinate charts for S n , the set of vectors of length one in Rn+1 to show that it is a smooth n-manifold. Justify your answer. 4. Does there exist a submersion f : T 2 → R2 ? By T 2 we mean S 1 × S 1 g ...
... that if ω, η ∈ Λ∗ (W ) then A∗ (ω ∧ η) = A∗ (ω) ∧ A∗ (η). 3. Produce a C ∞ -compatible atlas of coordinate charts for S n , the set of vectors of length one in Rn+1 to show that it is a smooth n-manifold. Justify your answer. 4. Does there exist a submersion f : T 2 → R2 ? By T 2 we mean S 1 × S 1 g ...
2.7.1 Euclidean Parallel Postulate
... equivalent? The exercises ask you to prove one direction on a few of the statements and to find a counterexample in the Poincaré Half-plane. Exercises 2.65. Show the Poincaré Half-plane does not satisfy the Euclidean Parallel Postulate. (a) Use dynamic geometry software to construct an example. (b) ...
... equivalent? The exercises ask you to prove one direction on a few of the statements and to find a counterexample in the Poincaré Half-plane. Exercises 2.65. Show the Poincaré Half-plane does not satisfy the Euclidean Parallel Postulate. (a) Use dynamic geometry software to construct an example. (b) ...
Local and Global Scores in Selective Editing
... Euclidean score perform well with a large number of key items, it appears to perform at least as well as the maximum score for small numbers of ...
... Euclidean score perform well with a large number of key items, it appears to perform at least as well as the maximum score for small numbers of ...
Solutions to Homework 1, Quantum Mechanics
... Are they linearly independent? Support your answer with details. Answ: The three vectors given are not linearly independent, since |1i − 2 |2i = |3i. ~ = 3î + 4ĵ and B ~ = 2î − 6ĵ in the 2-dimensional space of 4) Consider the two vectors A the x-y plane. Do they form a suitable set of basis vect ...
... Are they linearly independent? Support your answer with details. Answ: The three vectors given are not linearly independent, since |1i − 2 |2i = |3i. ~ = 3î + 4ĵ and B ~ = 2î − 6ĵ in the 2-dimensional space of 4) Consider the two vectors A the x-y plane. Do they form a suitable set of basis vect ...
Some Basic Topological Concepts
... X need not be either, open or closed. Moreover, as it was the case for open sets, the same set C might be closed in one space X but not in another. An important relationship between the concepts of open and closed is the following Property: Let U ⊆ X. Then U is open in X if and only if X \ U is clos ...
... X need not be either, open or closed. Moreover, as it was the case for open sets, the same set C might be closed in one space X but not in another. An important relationship between the concepts of open and closed is the following Property: Let U ⊆ X. Then U is open in X if and only if X \ U is clos ...
Solutions - Stony Brook Mathematics
... (definition of midpoint) and PX = PX, we can apply SSS to see that 4CPX = 4APX. Thus angles ∠CPX and ∠APX are congruent. Since they are also supplementary angles, we have shown that they must be right angles. ...
... (definition of midpoint) and PX = PX, we can apply SSS to see that 4CPX = 4APX. Thus angles ∠CPX and ∠APX are congruent. Since they are also supplementary angles, we have shown that they must be right angles. ...
Contents Euclidean n
... With n > 3 we lose our usual geometric view of Euclidean n-space. All is not lost, however. We can often extend the ideas present in the cases where ...
... With n > 3 we lose our usual geometric view of Euclidean n-space. All is not lost, however. We can often extend the ideas present in the cases where ...
329homework7 - WordPress.com
... geometry, the fourth angle would be a right angle and follow the definition of a quadrilateral as we know it. In non-Euclidean geometries, the angle can be acute. My perception of these quadrilaterals is different now that we have analyzed other types of geometries. The existence of these figures is ...
... geometry, the fourth angle would be a right angle and follow the definition of a quadrilateral as we know it. In non-Euclidean geometries, the angle can be acute. My perception of these quadrilaterals is different now that we have analyzed other types of geometries. The existence of these figures is ...
Topology/Geometry Jan 2014
... 1. Answer each question on a separate page. Turn in a page for each problem even if you cannot do the problem. 2. Label each answer sheet with the problem number. 3. Put your number, not your name, in the upper right hand corner of each page. If you have not received a number, please choose one (123 ...
... 1. Answer each question on a separate page. Turn in a page for each problem even if you cannot do the problem. 2. Label each answer sheet with the problem number. 3. Put your number, not your name, in the upper right hand corner of each page. If you have not received a number, please choose one (123 ...
Topology 640, Midterm exam
... (b) Show that if A1 and A2 are open dense subsets of X then A1 ∩ A2 is also a dense subset of X. (c) Show by example that the statement from part (b) above fails if A1 and A2 are not required to be open sets. 2. Show that the product of two metrizable topologies is again metrizable.1 3. Let (X, TX ) ...
... (b) Show that if A1 and A2 are open dense subsets of X then A1 ∩ A2 is also a dense subset of X. (c) Show by example that the statement from part (b) above fails if A1 and A2 are not required to be open sets. 2. Show that the product of two metrizable topologies is again metrizable.1 3. Let (X, TX ) ...
zero and infinity in the non euclidean geometry
... geometry and Euclidean Geometry is that instead of describing a plane as a flat surface a plane is a sphere. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small tr ...
... geometry and Euclidean Geometry is that instead of describing a plane as a flat surface a plane is a sphere. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small tr ...
Lecture 5 Notes
... of the elements of B. If X has a countable basis at each point x 2 X, we call X firstcountable. Definition 1.3. A topological space X is called second-countable if there exists a countable basis for its topology. All second-countable spaces turn out to be first-countable as well if you fiddle with t ...
... of the elements of B. If X has a countable basis at each point x 2 X, we call X firstcountable. Definition 1.3. A topological space X is called second-countable if there exists a countable basis for its topology. All second-countable spaces turn out to be first-countable as well if you fiddle with t ...
COURSE TITLE – UNIT X
... #28—Identifies and evaluates tangent, sine, and cosine ratios for an acute angle of a right triangle; uses a table, calculator, or computer to find the ratio for a given angle or find the angle for a given ratio #29—Uses the tangent, sine, and cosine ratios for right triangles to solve application p ...
... #28—Identifies and evaluates tangent, sine, and cosine ratios for an acute angle of a right triangle; uses a table, calculator, or computer to find the ratio for a given angle or find the angle for a given ratio #29—Uses the tangent, sine, and cosine ratios for right triangles to solve application p ...
SCALAR PRODUCTS, NORMS AND METRIC SPACES 1
... the real numbers. The main example for MATH 411 is V = Rn . Also, keep in mind that “0” is a many splendored symbol, with meaning depending on context. It could for example mean the number zero, or the zero vector in a vector space. Definition 1.1. A scalar product is a function which associates to ...
... the real numbers. The main example for MATH 411 is V = Rn . Also, keep in mind that “0” is a many splendored symbol, with meaning depending on context. It could for example mean the number zero, or the zero vector in a vector space. Definition 1.1. A scalar product is a function which associates to ...
Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term ""Euclidean"" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions.Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are also used to define rational numbers. When algebra and mathematical analysis became developed enough, this relation reversed and now it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. It means that points of the space are specified with collections of real numbers, and geometric shapes are defined as equations and inequalities. This approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions.From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the real coordinate space (Rn) of the same dimension. In one dimension, this is the real line; in two dimensions, it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or more real number coordinates. Mathematicians denote the n-dimensional Euclidean space by En if they wish to emphasize its Euclidean nature, but Rn is used as well since the latter is assumed to have the standard Euclidean structure, and these two structures are not always distinguished. Euclidean spaces have finite dimension.