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

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In summary, "What Is Mathematics? An Elementary Approach to Ideas and Methods" by Richard Courant is a comprehensive guide to the fundamental concepts and methods of mathematics. The book covers topics such as natural numbers, laws of arithmetic, theory of numbers, algebra, complex numbers, geometry, topology, and calculus. It also delves into advanced topics such as projective geometry, non-Euclidean geometries, and functions and limits. The book provides a thorough understanding of mathematics and its applications through clear explanations and examples.

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Author: What Is Mathematics? An Elementary Approach to Ideas and Methods
Title: Richard Courant

Code:
[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
[/LIST]
[*] Pythagorean Numbers and Fermat's Last Theorem
[*] The Euclidean Algorithm
[LIST]
[*] General Theory
[*] Application to the Fundamental Theorem of Arithmetic
[*] Euler's $\varphi$ 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
[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
[*] 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 $a_n$
[*] Monotone Sequences
[*] Euler's Number e
[*] The Number $\pi$
[*] Continued Fractions
[/LIST]
[*] Limits by Continues Approach
[LIST]
[*] Introduction. General Definition
[*] Remarks on the Limit Concept
[*] The Limit of sin(x)/x
[*] Limits as $x\rightarrow \infty$
[/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 $q^n$
[*] The limit of $\sqrt[n]{p}$
[*] 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 $x^r$
[*] 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 $x^r$, cos(x), sin(x), Arctan(x)
[*] Leibniz' Formula for $\pi$
[/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 $e^r$ [tex]a^x[/itex], $x^s$
[*] Explicit Expression for e, $e^r$ 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, $cos(x)+isin(x) = e^{ix}$
[*] 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]
[*] Index
[/LIST]

Last edited by a moderator:
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.

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.

## 1. What topics are covered in "What Is Mathematics?"

The book covers a wide range of topics including basic arithmetic, algebra, geometry, trigonometry, calculus, and introductory topics in modern mathematics such as set theory and logic.

## 2. Is "What Is Mathematics?" suitable for beginners?

Yes, the book is designed as an elementary approach to mathematics and is suitable for beginners with little to no background in math.

## 3. Are there any real-world applications discussed in "What Is Mathematics?"

Yes, the book includes numerous examples and applications of mathematical concepts in various fields such as physics, engineering, and economics.

## 4. Does "What Is Mathematics?" include exercises for practice?

Yes, the book includes numerous exercises and problems for readers to practice and apply the concepts learned in each chapter.

## 5. Can "What Is Mathematics?" be used as a textbook for a math course?

Yes, the book can be used as a textbook for introductory math courses or as a supplement for self-study. It provides a solid foundation for further studies in mathematics.

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