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1. Introduction
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Math 112
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Math 112
Index
Numbers
R. Mayer
Spring 2006
Acknowledgements
Introduction
Remembrance of Things Past
The Goal of the Course
Some General Remarks
1. Notation, Undefined Concepts and Examples
1.1 Sets
1.2 Propositions
1.3 Equality
1.4 More Sets
1.5 Functions
1.6
Russell's Paradox
2. Fields
2.1 Binary Operations
2.2 Some Examples
2.3 The Field Axioms
2.4 Some Consequences of the Field Axioms.
2.5 Subtraction and Division
2.6 Ordered Fields
2.7 Absolute Value
3. Induction and Integers
3.1 Natural Numbers and Induction
3.2 Integers and Rationals.
3.3 Recursive Definitions.
3.4 Summation.
3.5 Maximum Function
4. The Complexification of a Field.
4.1 Construction of
.
4.2 Complex Conjugate.
5. Real Numbers
5.1 Sequences and Search Sequences
5.2 Completeness
5.3 Existence of Roots
6. The Complex Numbers
6.1 Absolute Value and Complex Conjugate
6.2 Geometrical Representation
6.3 Roots of Complex Numbers
6.4 Square Roots
6.5 Complex Functions
7. Complex Sequences
7.1 Some Examples.
7.2 Convergence
7.3 Null Sequences
7.4 Sums and Products of Null Sequences
7.5 Theorems About Convergent Sequences
7.6 Geometric Series
7.7 The Translation Theorem
7.8 Bounded Monotonic Sequences
8. Continuity
8.1 Compositions with Sequences
8.2 Continuity
8.3 Limits
9. Properties of Continuous Functions
9.1 Extreme Values
9.2 Intermediate Value Theorem
10. The Derivative
10.1 Derivatives of Complex Functions
10.2 Differentiable Functions on
10.3 Trigonometric Functions
11. Infinite Series
11.1 Infinite Series
11.2 Convergence Tests
11.3 Alternating Series
11.4 Absolute Convergence
12. Power Series
12.1 Definition and Examples
12.2 Radius of Convergence
12.3 Differentiation of Power Series
12.4 The Exponential Function
12.5 Logarithms
12.6 Trigonometric Functions
12.7 Special Values of Trigonometric Functions
12.8 Proof of the Differentiation Theorem
12.9 Some XVIII-th Century Calculations
Bibliography
A. Hints and Answers
B. Associativity and Distributivity of Operations in
Index
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Russell's Paradox
.