MGA4U0 UNIT 1 – AN INTRODUCTION TO PROOF THROUGH EUCLIDEAN GEOMETRY MGA4U0 UNIT 1 – AN INTRODUCTION TO PROOF THROUGH EUCLIDEAN GEOMETRY..............................................1 WHAT IS A PROOF? – INDUCTIVE AND DEDUCTIVE REASONING .......................................................................................4 NOLFI’S INTUITIVE DEFINITION OF “PROOF” .........................................................................................................................................4 PYTHAGOREAN THEOREM EXAMPLE .....................................................................................................................................................4 Proof: ...............................................................................................................................................................................................4 RESEARCH EXERCISES ...........................................................................................................................................................................4 THE MEANING OF Π – AN EXAMPLE OF DEDUCTIVE REASONING ..........................................................................................................5 Examples ..........................................................................................................................................................................................5 TIPS ON BECOMING A POWERFUL “PROVER”..........................................................................................................................................6 EXAMPLE OF INDUCTIVE REASONING (THE “CONJECTURE, TEST THE CONJECTURE, TRY TO PROVE” PROCESS) ..................................7 Conjecture........................................................................................................................................................................................7 Test the Conjecture...........................................................................................................................................................................7 EXAMPLE OF DEDUCTIVE REASONING – DEDUCTIVE PROOF OF THE CONJECTURE................................................................................7 Proof of Above Conjecture:..............................................................................................................................................................7 HOW TO DISTINGUISH BETWEEN A DEDUCTIVE ARGUMENT AND AN INDUCTIVE ARGUMENT ...............................................................7 Deductive Reasoning........................................................................................................................................................................7 Inductive Reasoning .........................................................................................................................................................................7 Examples of Deductive Reasoning from Everyday Life....................................................................................................................7 Examples of Inductive Reasoning from Everyday Life.....................................................................................................................7 Homework ........................................................................................................................................................................................7 AXIOMATIC SYSTEMS OF REASONING ........................................................................................................................................8 FOUNDATION OF AXIOMATIC SYSTEM OF REASONING – PRIMITIVE (UNDEFINED) TERMS AND AXIOMS ...............................................8 DEFINITIONS AND PROPOSITIONS IN AXIOMATIC SYSTEM OF REASONING.............................................................................................9 SUMMARY..............................................................................................................................................................................................9 EXERCISES .............................................................................................................................................................................................9 AN EXAMPLE OF DEDUCTIVE REASONING IN SCIENCE – EINSTEIN’S SPECIAL THEORY OF RELATIVITY (ENRICHMENT MATERIAL) ...10 Postulates (Axioms) of Special Relativity.......................................................................................................................................10 SOME IMPORTANT RESULTS DEDUCED FROM THE POSTULATES OF SPECIAL RELATIVITY ...................................................................10 A GRAND “THEOREM” THAT OVERTURNS ALL CURRENTLY ACCEPTED MATH?......................................................11 “THEOREM” .........................................................................................................................................................................................11 “Proof” ..........................................................................................................................................................................................11 PROOF IN EUCLIDEAN GEOMETRY.............................................................................................................................................12 WHAT IS EUCLIDEAN GEOMETRY? ......................................................................................................................................................12 GENERAL APPROACH TO LEARNING THEOREMS..................................................................................................................................13 GEOMETRY THEOREMS YOU MIGHT HAVE (OR SHOULD HAVE) LEARNED BEFORE TAKING THIS COURSE.............................................13 PROOF OF ASTT ..................................................................................................................................................................................15 PROOF OF ITT......................................................................................................................................................................................15 ALTERNATIVE PROOF OF ITT...............................................................................................................................................................15 PROOF OF OAT....................................................................................................................................................................................15 Homework ......................................................................................................................................................................................15 PROOF IN CARTESIAN GEOMETRY.............................................................................................................................................16 WHAT IS CARTESIAN GEOMETRY?.......................................................................................................................................................16 PROOF IN CARTESIAN GEOMETRY........................................................................................................................................................16 EXAMPLE 1 ..........................................................................................................................................................................................16 Proof...............................................................................................................................................................................................16 EXAMPLE 2 – A COROLLARY OF THE PYTHAGOREAN THEOREM..........................................................................................................16 Proof...............................................................................................................................................................................................16 EXAMPLE 3 ..........................................................................................................................................................................................16 Proof...............................................................................................................................................................................................16 HOMEWORK.........................................................................................................................................................................................17 CRITICAL PREREQUISITE KNOWLEDGE ................................................................................................................................................17 THE INTIMATE CONNECTION BETWEEN ALGEBRA AND GEOMETRY ...........................................................................18 INTRODUCTION ....................................................................................................................................................................................18 AN EXERCISE IN THE LANGUAGE OF MATHEMATICS ...........................................................................................................................18 Copyright ©, Nick E. Nolfi MGA4U0 An Introduction to Deductive Geometry and the Nature of Proof IPEG-1