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Chapter1

# Chapter1 - Chapter 1 The Real Numbers PWhite Discussion...

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Chapter 1: The Real Numbers PWhite Discussion Some Preliminaries The Axiom of Completeness Consequences of Completeness Cantor’s Theorem Epilogue Chapter 1: The Real Numbers Peter W. White Initial development by Keith E. Emmert Department of Mathematics Tarleton State University Fall 2011 / Real Anaylsis I

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Chapter 1: The Real Numbers PWhite Discussion Some Preliminaries The Axiom of Completeness Consequences of Completeness Cantor’s Theorem Epilogue Overview Discussion: The Irrationality of 2 Some Preliminaries The Axiom of Completeness Consequences of Completeness Cantor’s Theorem Epilogue
Chapter 1: The Real Numbers PWhite Discussion Some Preliminaries The Axiom of Completeness Consequences of Completeness Cantor’s Theorem Epilogue The Big Pythagorean Oopsie I The Pythagoreans worshiped integers and fractions of integers (rational numbers). I They believed that everything could be connected to an integer or rational number. I Around 500 BC, the Pythagoreans discovered the irrationality of 2. I The wheels came off the bus. I Real Analysis began and is often driven by the study of things that caused the “wheels to come off the bus.”

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Chapter 1: The Real Numbers PWhite Discussion Some Preliminaries The Axiom of Completeness Consequences of Completeness Cantor’s Theorem Epilogue Basic Deﬁnitions Deﬁnition 1 I The natural numbers are deﬁned by N = { 1 , 2 , 3 , . . . } . I The integers are deﬁned by Z = { . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , . . . } . I The rational numbers are deﬁned by Q = ± p q | p , q Z , q 6 = 0 ² . Theorem 2 There is no rational number whose square is 2. Proof:
Chapter 1: The Real Numbers PWhite Discussion Some Preliminaries The Axiom of Completeness Consequences of Completeness Cantor’s Theorem Epilogue Questions and Observations I Thus 2 6∈ Q . Where does 2 “belong?” I We deﬁne irrational numbers as those that are not rational. I Fact: n p with p prime is irrational. I Note that n p is a root of f ( x ) = x n - p . I Are there other irrational numbers that are not algebraic roots of polynomials with rational coefﬁcients? I The real numbers could be deﬁned as Q with it’s “holes” ﬁlled in. I More about these things later.

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Chapter 1: The Real Numbers PWhite Discussion Some Preliminaries The Axiom of Completeness Consequences of Completeness Cantor’s Theorem Epilogue Overview Discussion: The Irrationality of 2 Some Preliminaries The Axiom of Completeness Consequences of Completeness Cantor’s Theorem Epilogue
Chapter 1: The Real Numbers PWhite Discussion Some Preliminaries The Axiom of Completeness Consequences of Completeness Cantor’s Theorem Epilogue Sets Deﬁnition 3 I A set is a collection of any objects, called elements . I

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Chapter1 - Chapter 1 The Real Numbers PWhite Discussion...

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