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High SchoolMathematicsExtensions

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Contents

HIGH SCHOOL MATHEMATICS EXTENSIONS..............................................................................................................1CONTENTS.................................................................................................................................................................2AUTHORS...................................................................................................................................................................4PRIMES AND MODULAR ARITHMETIC .........................................................................................................................5

Primes ..................................................................................................................................................................5Introduction..........................................................................................................................................................5Factorisation.........................................................................................................................................................62, 5 and 3..............................................................................................................................................................8Theorem - Divisibility by 3..................................................................................................................................8Finding primes .....................................................................................................................................................9Infinitely many primes .......................................................................................................................................10Modular Arithmetic............................................................................................................................................12To go further ......................................................................................................................................................33Acknowledgement..............................................................................................................................................33Reference ...........................................................................................................................................................34Feedback ............................................................................................................................................................34Problem Set ........................................................................................................................................................34Square root of minus 1 .......................................................................................................................................37Project -- The Square Root of -1 ........................................................................................................................37

SOLUTIONS TO EXERCISES.......................................................................................................................................39HSE Primes|Primes and Modular Arithmetic.....................................................................................................39Diophantine equation Exercises .........................................................................................................................45Problem set solutions .........................................................................................................................................50

DEFINITIONS............................................................................................................................................................54LOGIC......................................................................................................................................................................54

Introduction........................................................................................................................................................54Boolean algebra..................................................................................................................................................54Propositions........................................................................................................................................................69Logic Puzzles .....................................................................................................................................................72Problem Set ........................................................................................................................................................74Feedback ............................................................................................................................................................75

MATHEMATICAL PROOFS.........................................................................................................................................76Introduction........................................................................................................................................................76Mathematical induction......................................................................................................................................76Proof by contradiction........................................................................................................................................79Reading higher mathematics ..............................................................................................................................81Axioms and Inference ........................................................................................................................................83Problem Set ........................................................................................................................................................86Feedback ............................................................................................................................................................87

EXERCISES...............................................................................................................................................................88Mathematical proofs...........................................................................................................................................88Problem set.........................................................................................................................................................92Mathematical Proofs Problem Set......................................................................................................................92

INFINITY AND INFINITE PROCESSES..........................................................................................................................95Introduction........................................................................................................................................................95Infinite Sets ........................................................................................................................................................95Limits Infinity got rid of ..................................................................................................................................102Infinite series....................................................................................................................................................105

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Feedback ..........................................................................................................................................................106Exercises ..........................................................................................................................................................107Infinity and infinite processes..........................................................................................................................107Problem set.......................................................................................................................................................109

COUNTING AND GENERATING FUNCTIONS.............................................................................................................110Some Counting Problems.................................................................................................................................110Generating functions ........................................................................................................................................110Further Counting ..............................................................................................................................................116Problem Set ......................................................................................................................................................121Feedback ..........................................................................................................................................................125Exercises ..........................................................................................................................................................126Counting and Generating functions..................................................................