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Intro Math What Is Mathematics? An Elementary Approach to Ideas and Methods by Courant

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  1. Jan 22, 2013 #1

    micromass

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    Author: What Is Mathematics? An Elementary Approach to Ideas and Methods
    Title: Richard Courant
    Amazon Link: https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192

    Table of Contents:
    Code (Text):

    [LIST]
    [*] Preface
    [*] How to use the book
    [*] What is mathematics?
    [*] The natural numbers
    [LIST]
    [*] Introduction
    [*] Calculations with Integers
    [LIST]
    [*] Laws of Arithmetic
    [*] The Representation of Integers
    [*] Computation in Systems Other than the Decimal
    [/LIST]
    [*] The Infinitude of the Number System. Mathematical Induction
    [LIST]
    [*] The Principle of Mathematical Induction
    [*] The Arithmetical Progression
    [*] The Geometrical Progression
    [*] The Sum of the First n Squares
    [*] An Important Inequality
    [*] The Binomial Theorem
    [*] Further Remarks on Mathematical Induction
    [/LIST]
    [/LIST]
    [*] The Theory of Numbers
    [LIST]
    [*] Introduction
    [*] The Prime Numbers
    [LIST]
    [*] Fundamental Facts
    [*] The Distribution of the Primes
    [*] Formulas Producing Primes
    [*] Primes in Arithmetical Progressions
    [*] The Prime Number Theorem
    [*] Two Unsolved Problems Concering Prime Numbers
    [/LIST]
    [*] Congruences
    [LIST]
    [*] General Concepts
    [*] Fermat's Theorem
    [*] Quadratic Residues
    [/LIST]
    [*] Pythagorean Numbers and Fermat's Last Theorem
    [*] The Euclidean Algorithm
    [LIST]
    [*] General Theory
    [*] Application to the Fundamental Theorem of Arithmetic
    [*] Euler's [itex]\varphi[/itex] Function. Fermat's Theorem Again
    [*] Continued Fractions. Diophantine Equations
    [/LIST]
    [/LIST]
    [*] The Number System of Mathematics
    [LIST]
    [*] Introduction
    [*] The Rational Numbers
    [LIST]
    [*] Rational Numbers as Device for Measuring
    [*] Intrinsic Need for the Rational Numbers. Principal of Generalization
    [*] Geometric Interpretation of Rational Numbers
    [/LIST]
    [*] Incommensurable Segments, Irrational Numbers, and the Concept of Limit
    [LIST]
    [*] Introduction
    [*] Decimal Fractions. Infinite Decimals
    [*] Limits. Infinite Geometrical Series
    [*] Rational Numbers and Periodic Decimals
    [*] General Definition of Irrational Numbers by Nested Intervals
    [*] Alternative Methods of Defining Irrational Numbers. Dedeking Cuts
    [/LIST]
    [*] Remarks on Analytic Geometry
    [LIST]
    [*] The Basic Principle
    [*] Equations of Lines and Curves
    [/LIST]
    [*] The Mathematical Analysis of Infinity
    [LIST]
    [*] Fundamental Concepts
    [*] The Denumerability of the Rational Numbers and the Non-Denumerability of the Continuum
    [*] Cantor's "Cardinal Numbers"
    [*] The Indirect Method of Proof
    [*] The Paradoxed of the Infinite
    [*] The Foundations of Mathematics
    [/LIST]
    [*] Complex Numbers
    [LIST]
    [*] The Origin of Complex Numbers
    [*] The Geometrical Interpretation of Complex Numbers
    [*] De Moivre's Formula and the Roots of Unity
    [*] The Fundamental Theorem of Algebra
    [/LIST]
    [*] Algebraic and Transcendental Numbers
    [LIST]
    [*] Definitions and Existence
    [*] Liouville's Theorem and the Construction of Transcendental Numbers
    [/LIST]
    [/LIST]
    [*] The Algebra of Sets
    [LIST]
    [*] General Theory
    [*] Application to Mathematical Logic
    [*] An Applications to the Theory of Probability
    [/LIST]
    [*] Geometrical Constructions. The Algebra of Number Fields
    [LIST]
    [*] Introduction
    [*] Impossibility Proofs and Algebra
    [LIST]
    [*] Fundamental Geometrical Constructions
    [LIST]
    [*] Constructions of Fields and Square Root Extraction
    [*] Regular Polygons
    [*] Apollonius' Problem
    [/LIST]
    [*] Constructible Numbers and Number Fields
    [LIST]
    [*] General Theory
    [*] All Constructible Numbers are Algebraic
    [/LIST]
    [*] The Unsolvability of the Three Greek Problems
    [LIST]
    [*] Doubling the Cube
    [*] A Theorem on Cubic Equations
    [*] Trisecting the Angle
    [*] The Regular Heptagon
    [*] Remarks on the Problem of the Squaring the Circle
    [/LIST]
    [/LIST]
    [*] Various Methods for Performing Constructions
    [LIST]
    [*] Geometrical Transformations. Inversion
    [LIST]
    [*] General Remarks
    [*] Properties of Inversion
    [*] Geometrical Construction of Inverse Points
    [*] How to Bisect a Segment and Find the Center of a Circle with the Compass Alone
    [/LIST]
    [*] Constructions with Other Tools. Mascheroni Constructions with Compass Alone
    [LIST]
    [*] A Classical Construction for Doubling the Cube
    [*] Restriction to the Use of the Compass Alone
    [*] Drawing with Mechanical Instruments. Mechanical Curves. Cycloids
    [*] Linkages. Peaucellier's and Hart's Inversors
    [/LIST]
    [*] More about Inversions and its Applications
    [LIST]
    [*] Invariance of Angles. Families of Circles
    [*] Application to the Problem of Apollonius
    [*] Repeated Reflections
    [/LIST]
    [/LIST]
    [/LIST]
    [*] Projective Geometry. Axiomatics. Non-Euclidean Geometries
    [LIST]
    [*] Introduction
    [LIST]
    [*] Classification of Geometrical Properties. Invariance under Transformations
    [*] Projective Transformations
    [/LIST]
    [*] Fundamental Concepts
    [LIST]
    [*] The Group of Projective Transformations
    [*] Desargues's Theorem
    [/LIST]
    [*] Cross-Ratio
    [LIST]
    [*] Definition and Proof of Invariance
    [*] Application to the Complete Quadrilateral
    [/LIST]
    [*] Parallelism and Infinity
    [LIST]
    [*] Points at Infinity as "Ideal Points"
    [*] Ideal Elements and Projection
    [*] Cross-Ratio with Elements at Infinity
    [/LIST]
    [*] Applications
    [LIST]
    [*] Preliminary Remarks
    [*] Proof of Desargues's Theorem in the Plane
    [*] Pascal's Theorem
    [*] Brianchon's Theorem
    [*] Remark on Duality
    [/LIST]
    [*] Analytic Representation
    [LIST]
    [*] Introductory Remarks
    [*] Homogeneous Coordinates. The Algebraic Basis of Duality
    [/LIST]
    [*] Problems on Constructions with Straightedge Alone
    [*] Conics and Quadric Surfaces
    [LIST]
    [*] Elementary Metric Geometry of Conics
    [*] Projective Properties of Conics
    [*] Conics as Line Curves
    [*] Pascal's and Brianchon's General Theorems for Conics
    [*] The Hyperboloid
    [/LIST]
    [*] Axiomatics and Non-Euclidean Geometry
    [LIST]
    [*] The Axiomatic Method
    [*] Hyperbolic Non-Euclidean Geometry
    [*] Geometry and Reality
    [*] Poincaré's Model
    [*] Elliptic or Riemannian Geometry
    [/LIST]
    [*] Appendix: Geometry in more than Three Dimensions
    [LIST]
    [*] Introduction
    [*] Analytic Approach
    [*] Geometrical or Combinatorial Approach
    [/LIST]
    [/LIST]
    [*] Topology
    [LIST]
    [*] Introduction
    [*] Euler's Formula for Polyhedra
    [*] Topological Properties of Figures
    [LIST]
    [*] Topological Properties
    [*] Connectivity
    [/LIST]
    [*] Other Examples of Topological Theorems
    [LIST]
    [*] The Jordan Curve Theorem
    [*] The Four Color Problem
    [*] The Concept of Dimension
    [*] A Fixed Point Theorem
    [*] Knots
    [/LIST]
    [*] The Topological Classification of Surfaces
    [LIST]
    [*] The Genus of a Surface
    [*] The Euler Characteristic of a Surface
    [*] One-Sided Surface
    [/LIST]
    [*] Appendix
    [LIST]
    [*] The Five Color Theorem
    [*] The Jordan Curve Theorem for Polygons
    [*] The Fundamental Theorem of Algebra
    [/LIST]
    [/LIST]
    [*] Functions and Limits
    [LIST]
    [*] Introduction
    [*] Variable and Function
    [LIST]
    [*] Definitions and Examples
    [*] Radian Measure of Angles
    [*] The Graph of a Function. Inverse Functions
    [*] Compound Functions
    [*] Continutity
    [*] Fundtions of Several Variables
    [*] Functions and Transformations
    [/LIST]
    [*] Limits
    [LIST]
    [*] The Limit of a Sequence [itex]a_n[/itex]
    [*] Monotone Sequences
    [*] Euler's Number e
    [*] The Number [itex]\pi[/itex]
    [*] Continued Fractions
    [/LIST]
    [*] Limits by Continues Approach
    [LIST]
    [*] Introduction. General Definition
    [*] Remarks on the Limit Concept
    [*] The Limit of sin(x)/x
    [*] Limits as [itex]x\rightarrow \infty[/itex]
    [/LIST]
    [*] Precide Definition of Continuity
    [*] Two Fundamental Theorems on Continuous Functions
    [LIST]
    [*] Bolzano's Theorem
    [*] Proof of Bolzano's Theorem
    [*] Weierstrass' Theorem on Extreme Values
    [*] A Theorem on Sequences. Compact Sets
    [/LIST]
    [*] Some Applications of Bolzano's Theorem
    [LIST]
    [*] Geometrical Applications
    [*] Applications to a Problem in Mechanics
    [/LIST]
    [/LIST]
    [*] More Examples on Limits and Continuity
    [LIST]
    [*] Examples of Limits
    [LIST]
    [*] General Remarks
    [*] The Limit of [itex]q^n[/itex]
    [*] The limit of [itex]\sqrt[n]{p}[/itex]
    [*] Discontinuous Functions as Limits of Continuous Functions
    [*] Limits by Iteration
    [/LIST]
    [*] Example on Continuity
    [/LIST]
    [*] Maxima and Minima
    [LIST]
    [*] Introduction
    [*] Problems in Elementary Geometry
    [LIST]
    [*] Maximum Area of a Triangle with Two Sides Given
    [*] Heron's Theorem. Extremum Property of Light Rays
    [*] Applications to Problems on Triangles
    [*] Tangent Properties of Ellipse and Hyperbola. Corresponding Extremum Properties
    [*] Extreme Distance to a Given Curve
    [/LIST]
    [*] A General Principal Underlying Extreme Value Problems
    [LIST]
    [*] The Principle
    [*] Examples
    [/LIST]
    [*] Stationary Points and the Differential Calculus
    [LIST]
    [*] Extrema and Stationary Points
    [*] Maxima and Minima of Functions of Several Variables. Saddle Points
    [*] Minimax Points and Topology
    [*] The Distance from a Point to a Surface
    [/LIST]
    [*] Schwarz's Triangle Problem
    [LIST]
    [*] Schwarz's Proof
    [*] Another Proof
    [*] Obtuse Triangles
    [*] Triangles Formed by Light Rays
    [*] Remarks Concering Problems of Reflection and Ergodic Motion
    [/LIST]
    [*] Steiner's Problem
    [LIST]
    [*] Problem and Solution
    [*] Analysis of the Alternatives
    [*] A Complementary Problem
    [*] Remarks and Exercises
    [*] Generalization to the Street Network Problem
    [/LIST]
    [*] Extrema and Inequalities
    [LIST]
    [*] The Arithmetical and Geometrical Mean of Two Positive Quantities
    [*] Generalization to n Variables
    [*] The Method of Least Squares
    [/LIST]
    [*] The Existence of an Extremum. Dirichlet's Principle
    [LIST]
    [*] General Remarks
    [*] Examples
    [*] Elementary Extremum Problems
    [*] Difficulties in Higher Cases
    [/LIST]
    [*] The Isoperimetric Problem
    [*] Extremum Problems with Boundary Conditions. Connection Between Steiner's Problem and the Isoperimetric Problem
    [*] The Calculus of Variations
    [LIST]
    [*] Introduction
    [*] The Calculus of Variations. Fermat's Principle in Optics
    [*] Bernouilli's Treatment of the Brachistochrone Problem
    [*] Geodesic on a Sphere. Geodesics and Maxi-Minima
    [/LIST]
    [*] Experimental Solutions of Minimum Problems. Soap Film Experiments
    [LIST]
    [*] Introduction
    [*] Soap film Experiments
    [*] New Experiments on Plateau's Problem
    [*] Experimental Solutions of Other Mathematical Problems
    [/LIST]
    [/LIST]
    [*] The Calculus
    [LIST]
    [*] The Integral
    [LIST]
    [*] Area as Limit
    [*] The Integral
    [*] General Remarks on the Integral Concept. General Definition
    [*] Examples of Integration. Integration of [itex]x^r[/itex]
    [*] Rules for the "Integral Calculus"
    [/LIST]
    [*] The Derivative
    [LIST]
    [*] The Derivative as a Slope
    [*] The Derivative as a Limit
    [*] Examples
    [*] Derivatives of Trigonometrical Functions
    [*] Differentiation and Continuity
    [*] Derivative and Velocity. Second Derivative and Acceleration
    [*] Geometrical Meaning of the Second Derivative
    [*] Maxima and Minima
    [/LIST]
    [*] The Technique of Differentiation
    [*] Leibniz' Notation and the "Infinitely Small"
    [*] The Fundamental Theorem of the Calculus
    [LIST]
    [*] The Fundamental Theorem
    [*] First Applications. Integration of [itex]x^r[/itex], cos(x), sin(x), Arctan(x)
    [*] Leibniz' Formula for [itex]\pi[/itex]
    [/LIST]
    [*] The Exponential Function and the Logarithm
    [LiST]
    [*] Definition and Properties of the Logarithm Euler's Number e
    [*] The Exponential Function
    [*] Formula's for Differentiation of [itex]e^r[/itex] [tex]a^x[/itex], [itex]x^s[/itex]
    [*] Explicit Expression for e, [itex]e^r[/itex] and log(x) as Limits
    [*] Infinite Series for the Logarithm. Numerical Calculation
    [/LIST]
    [*] Differential Equations
    [LIST]
    [*] Definition
    [*] The Differential Equation of the Exponential Function. Radioactive Disintegration. Law of Growth. Compound Interest
    [*] Other Examples. Simplest Vibrations
    [*] Newton's Law of Dynamics
    [/LIST]
    [/LIST]
    [*] Supplement
    [LIST]
    [*] Matters of Principle
    [LIST]
    [*] Differentiability
    [*] The Integral
    [*] Other Applications of the Concept of Integral. Work. Length
    [/LIST]
    [*] Order of Magnitude
    [LIST]
    [*] The Exponential Function and Powers of x
    [*] Order of Maginute of log(n!)
    [/LIST]
    [*] Infinite Series and Infinite Products
    [LIST]
    [*] Infinite Series of Functions
    [*] Euler's Formula, [itex] cos(x)+isin(x) = e^{ix}[/itex]
    [*] The Harmonic Series and the Zeta Function. Euler's Product for the Sine
    [/LIST]
    [*] The Prime Number Theorem Obtained by Statistical Methods
    [/LIST]
    [*] Recent Developments
    [LIST]
    [*] A Formula for Primes
    [*] The Goldbach Conjecture and Twin Primes
    [*] Fermat's Last Theorem
    [*] The Continuum Hypothesis
    [*] Set-Theoretic Notation
    [*] The Four Color Theorem
    [*] Hausdorff Dimension and Fractals
    [*] Knots
    [*] A Problem in Mechanics
    [*] Steiner's Problem
    [*] soap Films and Minimal Surfaces
    [*] Nonstandard Analysis
    [/LIST]
    [*] Appendix: Supplementary Remarks, Problems, and Exercises
    [LIST]
    [*] Arithmetic and Algebra
    [*] Analytic Geometry
    [*] Geometrical Constructions
    [*] Projective and Non-Euclidean Geometry
    [*] Topology
    [*] Functions, Limits, and Continuity
    [*] Maxima and Minima
    [*] The Calculus
    [*] Technique of Integration
    [/LIST]
    [*] Suggestions for Further Reading
    [*] Suggestions for Additional Reading
    [*] Index
    [/LIST]
     
     
    Last edited by a moderator: May 6, 2017
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  3. Jan 23, 2013 #2

    MarneMath

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    Education Advisor

    I read this book back when I was in high school, and I credit it for teaching me that mathematics is much more than solving for x. I didn't understand much of the book nor was I able to solve any of the problems, but I think being exposed to real math by a person who obviously knew a lot about it was enough to get me interested in learning how these connections were made. Definitely a great read.
     
  4. Feb 14, 2013 #3
    Re: What Is Mathematics? An Elementary Approach to Ideas and Methods b

    LOVE this book. It's answered a lot of the questions I've had since I've started studying pure math. My only complaint is that there are no solutions to the problems.
     
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