| Description |
1 online resource (317 p.) |
| Contents |
Cover -- Half Title -- Title Page -- Copyright Page -- Contents -- Preface and Acknowledgments -- 1. Introduction: The Purpose of This Book -- 1.1. A Very Brief Historical Context -- 1.2. The Axiomatic Method -- 1.3. The Place of Number Systems within Mathematics -- 1.4. Mathematical Writing, Notation, and Terminology -- 1.5. Logic and Methods of Proof -- 2. Sets and Relations -- 2.1. Sets -- 2.1.1. Quantifiers -- 2.1.2. Subsets and Equality -- 2.1.3. Union, Intersection, and Complement -- 2.1.4. Ordered Sets -- 2.2. Relations between Sets -- 2.2.1. Relations in General -- 2.2.2. Functions |
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2.3. Relations on a Set -- 2.3.1. Equivalence Relations -- 2.3.2. Order Relations -- 2.3.3. Transitivity and Proofs -- 2.4. Binary Operations and Algebraic Structures -- 3. Natural Numbers, N -- 3.1. Peano's Axioms -- 3.2. Addition of Natural Numbers -- 3.3. Multiplication of Natural Numbers -- 3.4. Exponentiation (Powers) of Natural Numbers -- 3.5. Order in the Natural Numbers -- 3.6. Bounded Sets in N -- 3.7. Cardinality, Finite and Infinite Sets -- 3.7.1. Some Useful Notations -- 3.7.2. Finite Sets, Their Subsets and Injections -- 3.7.3. Finiteness and Boundedness of Sets |
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3.7.4. Infinite Sets -- 3.8. Subtraction: The Inverse of Addition -- 4. Integers, Z -- 4.1. Definition of the Integers -- 4.2. Arithmetic on Z -- 4.3. Algebraic Structure of Z -- 4.3.1. An Abelian Group -- 4.3.2. A Commutative Ring -- 4.4. Order in Z -- 4.4.1. How to Solve Inequalities -- 4.5. Finite, Infinite, and Bounded Sets in Z -- 5. Foundations of Number Theory -- 5.1. Integer Division -- 5.2. Expressing Integers in Any Base -- 5.3. Prime Numbers and Prime Factorisation -- 5.3.1. Prime Numbers and Prime Factorisation in N -- 5.3.2. Primes in Z and Other Number Systems -- 5.4. Congruence |
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5.5. Modular Arithmetic -- 5.6. Zd as an Algebraic Structure -- 6. Rational Numbers, Q -- 6.1. Definition of the Rationals -- 6.2. Addition and Multiplication on Q -- 6.3. Countability of Q -- 6.4. Exponentiation and Its Inverse(s) on Q -- 6.4.1. Integer Powers -- 6.4.2. Roots and Fractional Powers -- 6.4.3. Logarithms -- 6.5. Order in Q -- 6.6. Bounded Sets in Q -- 6.7. Expressing Rational Numbers in Any Base -- 6.7.1. Terminating Base-b Representations -- 6.7.2. Repeating Base-b Representations -- 6.7.3. Fractions from Repeating Base-b Representations -- 6.8. Sequences and Series |
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7. Real Numbers, R -- 7.1. The Requirements for Our Next Number System -- 7.2. Dedekind Cuts -- 7.3. Order and Bounded Sets in R -- 7.4. Addition in R -- 7.5. Multiplication in R -- 7.6. Exponentiation in R -- 7.7. Expressing Real Numbers in Any Base -- 7.8. Cardinality of R -- 7.9. Algebraic and Transcendental Numbers -- 8. Quadratic Extensions I: General Concepts and Extensions of Z and Q -- 8.1. General Concepts of Quadratic Extensions -- 8.2. Introduction to Quadratic Rings: Extensions of Z -- 8.3. Units in Z[p k] -- 8.4. Primes in Z[p k] -- 8.4.1. Basic Theorems about Primes |
| Notes |
Description based upon print version of record |
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8.4.2. Associates Classes and Conjugates of Primes |
| Form |
Electronic book
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| ISBN |
9780429602245 |
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0429602243 |
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