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

[micromass]

Author: What Is Mathematics? An Elementary Approach to Ideas and Methods
Title: Richard Courant
Amazon Link: https://www.amazon.com/dp/0195105192/?tag=pfamazon01-20

Table of Contents:

[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]

 

[MarneMath]

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.

 

[QuantumP7]

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.