Model Answer_NIMAVI

Algebraic Structure

Algebraic Structure Note

Number Theory

Graph Theory MAVI

Introduction Euclidean geometry and it's Elements

Geometry is developed from the time of human civilization. The tradition of the development is basically rooted in three ancient civilizations. These civilizations are (a) Babylonian, (b) Egypt and (c) Greek. There are number of reasons to evident these civilization as root of geometry development. Among them (a) trade route of Babylonian, (b) Nile river of Egypt and (c) “Geo + Metron” version of geometry in Greek are vital.

Along with three ancient civilization, there were number of mathematicians who contributed for the development of Geometry. Among them, following contributions are significantly counted.

1. Demonstrative approach of Thales (600 BC)
2. Pythagoras theorem of Pythagoras (500 BC)
3. Elements of geometry of Hippocrates-(400 BC)
4. Academy of Plato (400 BC):
5. Axiomatization of Eudoxus (300 BC)

Comprising the early development of Geometry dated back to 300BC, a Greek mathematician Euclid had systematically organized (compiled) Elements in 300BC. This Elements is known as Geometry. This is also known as Euclidean Geometry. In this way, formal birth of geometry get by Euclid in 300BC.

By that time, Euclid defined the geometry as “Geo+Metron” or “measuring earthly things”. In his speculation, the geometry was based on flat surface. Therefore, it is also called plane geometry. However, these days, the meaning of Geometry has been widen. These days, Geometry can be defined in different perspectives. Among them, the four common perspectives are mentioned below.

1. One of the mother structure of Mathematics
2. Study of two properties
3. Study of five invariants
4. An axiomatic System

Elements

Around 300 BC, a Greek mathematician Euclid compiled volume of 13 books. These books were a systemic organization of geometrical knowledge developed by that time. The volume of these 13 books is called Elements. The Elements consists of following major components.

1. Five common notions
2. Five postulates
3. 118 definitions
4. 465 propositions

The five common notions are

1. Things equal to same thing are equal.
2. If equals are added to equals, the result is equal.
3. If equals are subtracted from equals, the result is equal.
4. Things coincide to each other are equal.
5. The whole is greater than the part.

The five postulates are

1. A straight line can be drawn between two points.
2. A straight line can be extend continuously in a straight line.
3. A circle can be describe with a point and a distance.
4. All right angles are equal.
5. If a straight line is falling on two straight lines, make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.

In addition to this, the details of Elements are given below.

SN	C	P	D	T	C=common notions, P=Postulates, D=Definitions, T=Propositions (Theorems)
1	5	5	23	48	Basic Geometry
2			2	14	Geometric algebra
3			11	37	Circles
4			7	16	Construction
5			18	25	Theory of abstract proportions
6			4	33	Similarity
7			22	39	Fundamental of number theory
8				27	Continued proportions in number theory
9				36	Advances in number theory
10			3	115	Incommensurable number
11			28	39	Solid geometry
12				18	Mensuration
13				18	Regular solids

Incommensurable number:

Two numbers are called Incommensurable number if their ratio is an irrational number.

a

a

Regular solid

A polygon whose all interior angles are equal, all sides are equal, is called regular polygon. For example, square is a regular polygon.

There are infinite number of regular polygon that exist.

A solid whose all sides (edges) are equal, all faces are equal, is called regular solid. For example, cube is a regular solid.

There are only five regular solids that exist in the universe. This solid were introduced by Plato in 400BC. Therefore, it is also called Platonic solids. These solids are given as below.

SN	Name	V	E	F	V-E+F	Dual
1	Tetrahedron	4	6	4	2	Self
2	Cube	8	12	6	2	Octahedron
3	Octahedron	6	12	8	2	Cube
4	Icosahedron	12	30	20	2	Dodecahedron
5	Dodecahedron	20	30	12	2	Icosahedron

SMSG Model of Euclidean Geometry

SMSG model for Euclidean Geometry (EG) is an axiomatic system developed by a team of American mathematicians (School Mathematics Study Group-SMSG) in mid-19^th century.  The main characteristics of this model was

• ·         Introducing pedagogical thought (pedagogical orientation) into EG
• ·         Making (Bringing) EG as a discipline of school education

SMSG model for Euclidean Geometry is a re-engineering of Euclidean Geometry to make it more efficient combining the features of H and B (Fusion of H and B to make EG more efficient).
 In SMSG model for Euclidean Geometry, there are three undefined terms and 22 postulates.

• ·  Point
• ·   Line
• ·    Plane

Definitions

In SMSG model, there are number of definitions. Among them, one example is given below.

Triangle: a plane figure enclosed by three lines.

Axioms (Postulates)

In SMSG model, the axioms (postulates) are classified into eight groups. These groups are given below.

Group              I-Incidence-1

II-Distance-2-4 (3)

III-Space relations-5-8 (4)

IV-Separations-9-10 (2)

++++++++++++++++++

V-Angle measure-11-14 (4)

VI-Congruence-15 (1)

VII-Parallelism-16 (1)

VIII-Area and volume -17-22 (6)

Postulates-I: Incidence

1.      (Line Uniqueness) Given any two distinct points there is exactly one line on both.  

Postulates-II: Distance

2.      (Distance Postulate) To every pair of distinct points there corresponds a unique positive number. This number is called the distance between them.  

3.       (Ruler Postulate) The points of a line can be placed in a correspondence with the real numbers such that:

4.      To every point of the line there corresponds exactly one real number.

•  To every real number there corresponds exactly one point of the line.
• The distance between two distinct points is the absolute value of the difference of the corresponding real numbers.
•  (Ruler Placement Postulate) Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive. 

Postulates-III: Space Relations

5.      (Existence of Points) Every plane contains at least three non-collinear points. Space contains at least four non-coplanar points.

6.      (Points on a Line Lie in a Plane) If two points lie in a plane, then the line containing these points lies in the same plane.  

7.      (Plane Uniqueness) Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane.

8.       (Plane Intersection) If two planes intersect, then intersection is a line.  

Postulates-IV: Separations

9.      (Plane Separation Postulate) Given a line and a plane containing it, the points of the plane that do not lie on the line form two sets such that: each of the sets is convex; if P is in one set and Q is in the other, then segment PQ intersects the line.

10.  (Space Separation Postulate) The points of space that do not lie in a plane such that, if P is in one side and Q is in the other, then segment PQ intersects the plane.

Postulates-V: Angle construction (measure)

11.  (Angle Measurement Postulate) To every angle there corresponds a real number between 0° and 180°.  

12.   (Angle Construction Postulate) Let, AB be a ray on the edge of the half-plane H. For every r between 0 and 180, there is exactly one ray AP with P in H such that ∠PAB = r.

13.  (Angle Addition Postulate) If D is a point in the interior of ∠BAC, then ∠BAC = ∠BAD + ∠DAC.  

14.  (Supplement Postulate) If two angles form a linear pair, then they are supplementary.  

Postulates-VIII: Area and Volume

17 .(Area of Polygonal Region) To every polygonal region there corresponds a unique positive real number called the area.

18.   (Area of Congruent Triangles) If two triangles are congruent, then the triangular regions have the same area.  

19    (Summation of Areas of Regions) Suppose that the region R is the union of two regions R₁ and R₂. If R₁ and R₂ intersect at most in a finite number of segments and points, then the area of R is the sum of the areas of R₁ and R₂.  

20.      (Area of a Rectangle) The area of a rectangle is the product of the length of its base and the length of its altitude.  

21.      (Volume of Rectangular Parallelpiped) The volume of a rectangular parallelpiped is equal to the product of the length of its altitude and the area of its base.  

22.      (Cavalieri's Principle) Given two solids and a plane. If for every plane that intersects the solids and is parallel to the given plane, and the two intersections determine regions that have the same area, then the two solids have the same volume.

Theorem