Introduction to GEOMETRY

1. Introduction to GEOMETRY By S.Karthik Rangasai

2. Geometry Means:- • Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metron “measurement“) is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. • A mathematician who works in the field of geometry is called a geometer. • Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales (6th Century BC). • By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. • Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. • The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. • Both geometry and astronomy were considered in the classical world to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.

3. Structure of Geometry – Axiomatic Approach The structure of GEOMETRY will therefore spotlight the following essential stages namely :- • Undefined and Defined Terms • Axioms • Theorems • Further definitions and deductions Body of Prepositions (or) Theorems Undefined / Defined terms and statements Application of Theorems and Logic Nature Riders (or) Deductions Application of Riders and Logic

4. Difference between Axiom and Theorem • The Axioms forms from the 1st and 2nd stage of the Structure of Geometry, but Theorems forms from the 3rd stage of the Structure of Geometry. • Using Undefined Terms, defined Terms and Axioms we can form some new relations. These Relations are called THEOREMS. • Axioms is a statement which has no question of proving it, but Theorem is the a statement which has to be proved using an Axiom. Body of Prepositions (or) Theorems Undefined / Defined terms and statements Application of Theorems and Logic Application of Riders and Logic Nature Riders (or) Deduction

5. Undefined and Defined Terms • Undefined Terms:- Terms like “Point”, “Line”, “Plane” etc. which cannot be defined are called UNDEFINED TERMS. • Defined Terms:- Terms like “Line Segment”, “Triangle”, “Angle” which can be defined are called DEFINED TERMS.

6. AXIOMS • Axioms are those statements which are self evident truths (or) basic facts formed as a result of observation and institution. • The AXIOMS are the foundation stones on which the structure of GEOMETRY is structured. These Axioms arise in different situations. • Let us know some situations:-

7. 1) Firstly we observe:- • There is one and only one line containing any two distinct points in a plane. Therefore, This Axiom shows the inter-relationship between the Basic Concepts namely ‘Points’, ‘Line’. 2) Similarly, the Axioms:- • “There is one and only one plane containing 3 non – collinear points”. • “The line containing any two points in a plane lies wholly in that plane” show the inter-relationship between the basic concepts ‘Points’ and ‘Plane’. From these examples, we understand that, sometimes, the axioms arise to show the inter-relationship between basic concepts.

8. 2) Secondly, we observe:- • The definition of A ‘Line Segment’, which is given as the set of points A, B and all the points between A and B, we see that the undefined terms ‘set’ and ‘points’ are connected by the terms ‘Between’ which has not been previously defined. We take it up at this stage, through the following Axiom ----- “If A, B, C are distinct points in line, then we say that B is in between A and C if only AB + BC = AC”. From this example, we understand that, sometimes, the axioms arise, also to define new terms connecting undefined (or) defined concepts.

9. 3)Thirdly, we observe:- • The 2 edges of a table meet in a corner, it suggests the axiom “Two lines intersect in one and only one point”. From this example, we also understand that, sometimes, the axioms arise, as abstractions from concrete situations.

10. But at the same time, we must take the following precautions when we formulate these Axioms. • The axioms should not contradict each other. They must be consistent. • We must create only necessary and sufficient number of axioms for the development of the structure. That is, we should not create more Axioms than what is required. • The Axioms created should be independent. That is, no Axiom should be created based on another Axiom. So, consistency, independence and completeness are the 3 important characteristics required for the Axioms

11. THEOREMS • Theorem is a usually a conditional statement. • When the conditions are satisfied, then only the Theorem will be proved to be true. • For ex:- “ If in a Triangle 2 sides are equal, then the angles opposite to them are equal ”. It consists of 2 parts. The 1st part is the conditional part starting with ‘if’ and the 2nd part is the conclusion part starting with ‘then’. • The 1st part is called ‘Hypothesis’ and 2nd part is called the ‘ Conclusion’. • The Hypothesis indicates the conditions and the Conclusion part indicates what to be proved. • Every Theorem In Geometry is formulated in this way.

12. Various Stages in the Structures of a Theorem • General Education:- Proposition of the Theorem. • Figure:- A figure may be drawn relevant to what is described in general enunciation and it is to be named. • Hypothesis:- The given conditions of the Theorem are particularly mentioned with respect to the figure. • Conclusion:- In this, what is required to be proved is particularly mentioned using the names in the figure, if necessary. The above 2, 3, 4 stages are together called as “Particular Enunciation”. • Construction:- In some Theorems, the figures as per general enunciation may not be sufficient to prove the Theorem. Then, we may be required to add some more details like lines, angles etc. Such additional details drawn is called construction. Of course, this construction part may not be necessary for every Theorem. • Proof:- This is an important stage in establishing a Theorem. Here we establish what is required to be proved as mentioned in the conclusion part by logical steps. This chain of steps starts with what is given in the hypothesis. Each step must be supported only by an Axiom or by a definition, (or) by Theorems previously proved, but not by any observation (or) by intuition.

13. Methods of Proof • There are 2 kinds of proving the Methods of Proof. • They are the • The Direct Method of Proving:- In this method of proving, we begin with what is given in the Hypothesis of the Theorem and arrive the Conclusion by means of a chain of logical steps. But it is important to follow here, that each step must be justified by a reason. • The Indirect Method of Proving:- Sometimes there is difficulty in developing a direct proof. Then we adopt the method of Indirect proof. In this method instead of starting the proof with Hypothesis as in Direct proof, we start with the assumption contrary to conclusion and write steps. As a result of this, at one stage or the other we arrive at a Mathematical Absurdity. Logical thinking enables us to know that the reason for the absurdity is our wrong assumption made contrary to the Conclusion. From this we establish that the original Conclusion should be True. This method of proof is also called as “Reductio ad Absurdum”.

14. Direct Method of Proof • Example:- “If x is odd, x2 is odd.” • This implication can be proved by Direct Method in the following way.

15. Indirect Method of proof • Example:- “If the alternate angles are equal when 2 straight lines are cut by a transversal, then the 2 lines are parallel.” • Hypothesis:- Straight lines l and m are cut by a transversal p and ∠a = ∠b (Alternate Angles) • Conclusion:- l // m • Proof:- Let us assume that l and m not parallel. • Then they must meet at a point on one side of the transversal when they are produced. Let that point be P. • Then PAB is a triangle and ∠a is the exterior angle formed by producing PA, and ∠b is the interior opposite angle. • ∠a > ∠b (The exterior angle of a triangle is greater than the interior angle) • But ∠a = ∠b (Hypothesis) When we think of the reason for this absurdity, it is clear that it is due to the wrong assuming that l and m not parallel. Therefore we conclude that l and m must be parallel. Just as we have the above 2 methods of proving true statements of Mathematics, in the same way we have methods of disproving false statements also. A l a p P b m B

16. Disproof by Counter Example • Though we give any number of examples to prove a true statement, we cannot establish the truth of it with the help of those examples. That is to say, the science of Mathematics will not accept of the Theorems by citing particular examples. • But to disprove a false statement it will suffice to give just 1 example proving the falseness of the statement. This example is called the “Counter Example”. From this, we can establish that the statement is false. This method is called “Disproof by Counter Example”. • Example:- “All prime numbers are odd” Let this statement be required to be proved as false. For this we list out a few first prime numbers. 2, 3, 5, 7, 11, 13 . . . . . . . . . . . • When we observe these numbers, we see that all are not odd numbers. There is 1 even number in them. That number is ‘2’. So, the number ‘2’ is an example contradicting the given statement. This is called the counter example. With one example we conclude that the given statement is false.

17. India’s Contribution to GEOMETRY • Our Country has contributed many things indirectly to this field of Geometry. Like Axioms and Theorems our country has not been systematically proved. It is been initially learnt as a technical art like Yagnas and Yagas. This gradually developed as science later. Our country has developed Geometry in correlation with Astronomy. • It is ‘Kalpa’ a part of ‘Vedas’, which has later on become the geometrical called ‘The laws of Kalpa’ in Yagnas and Yagas. • There are many famous Historians of Mathematicians who have worked in the of Geometry like “Boudhayana”, “Aapastambha”, “Katyayana”, “Aryabhatta”, “Brahma Gupta”, “Bhaskara”, etc.

18. The Pythagoras theorem was originally founded by the famous ‘Boudhayana’ in 600 B.C. • The Right angle Triangle theorem founded by Euclid was originally founded by the famous ‘Bhaskaracharya II’ in 1150 A.D. • Between 300 B.C and 300 A.D the book “Stanaanga Sutramu” had the finest works of Geometry. • Aryabhatta I – 499 A.D, Brahma Gupta - 628 A.D, Mahaveeracharyudu - 850 A.D, Sridharacharyudu - 900 A.D, Narayana Panditudu - 1356 A.D, Munneswarudu - 1603 A.D are great Mathematicians who worked in the field of Geometry.