Calculus (Book): Difference between revisions

Calculus (Book) (edit)

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 | <nowiki>*</nowiki>I 1.4 || Exercises || 8
 |-
-| I 1.5 || Rational numbers || 8
+| I 1.5 || A critical analysis of the Archimedes' method || 8
 |-
-| I 1.6 || Multiplicative inverses || 10
+| I 1.6 || The approach to calculus to be used in this book || 10
 |-
-! colspan="3" | Chapter 2: Linear Equations
+! colspan="3" | Part 2: Some Basic Concepts of the Theory of Sets
 |-
-| 1 || Equations in two unknowns || 53
+| I 2.1 || Introduction to set theory || 11
 |-
-| 2 || Equations in three unknowns || 57
+| I 2.2 || Notations for designating sets || 12
 |-
-! colspan="3" | Chapter 3: Real Numbers
+| I 2.3 || Subsets || 12
 |-
-| 1 || Addition and multiplication || 61
+| I 2.4 || Unions, intersections, complements || 13
 |-
-| 2 || Real numbers: positivity || 64
+| I 2.5 || Exercises || 15
 |-
-| 3 || Powers and roots || 70
+! colspan="3" | Part 3: A set of Axioms for the Real-Number System
 |-
-| 4 || Inequalities || 75
+| I 3.1 || Introduction || 17
 |-
-! colspan="3" | Chapter 4: Quadratic Equations
+| I 3.2 || The field axioms || 17
+|-
+| <nowiki>*</nowiki>I 3.3 || Exercises || 19
+|-
+| I 3.4 || The order axioms || 19
+|-
+| <nowiki>*</nowiki>I 3.5 || Exercises || 21
+|-
+| I 3.6 || Integers and rational numbers || 21
+|-
+| I 3.7 || Geometric interpretation of real numbers as points on a line || 22
+|-
+| I 3.8 || Upper bound of a set, maximum element, least upper bound (supremum) || 23
+|-
+| I 3.9 || The least-Upper-bound axiom (completeness axiom) || 25
+|-
+| I 3.10 || The Archimedean property of the real-number system || 25
+|-
+| I 3.11 || Fundamental properties of the supremum and infimum || 26
+|-
+| <nowiki>*</nowiki>I 3.12 || Exercises || 28
+|-
+| <nowiki>*</nowiki>I 3.13 || Existence of square roots of nonnegative real numbers || 29
+|-
+| <nowiki>*</nowiki>I 3.14 || Roots of higher order. Rational powers || 30
+|-
+| <nowiki>*</nowiki>I 3.15 || Representation of real numbers by decimals || 30
+|-
+! colspan="3" | Part 4: Mathematical Induction, Summation Notation, and Related Topics
+|-
+| I 4.1 || An example of a proof by mathematical induction || 32
+|-
+| I 4.2 || The principle of mathematical induction || 34
+|-
+| <nowiki>*</nowiki>I 4.3 || The well-ordering principle || 34
+|-
+| I 4.4 || Exercises || 35
+|-
+| <nowiki>*</nowiki>I 4.5 || Proof of the well-ordering principle || 37
+|-
+| I 4.6 || The summation notation || 37
+|-
+| I 4.7 || Exercises || 39
+|-
+| I 4.8 || Absolute values and the triangle inequality || 41
+|-
+| I 4.9 || Exercises || 43
+|-
+| <nowiki>*</nowiki>I 4.10 || Miscellaneous exercises involving induction || 44
 |-
 ! colspan="3" | Interlude: On Logic and Mathematical Expressions