The nature and power of mathematics

Cover......Page __sk_0000.djvu
Copyright......Page __sk_0002.djvu
Contents......Page __sk_0003.djvu
Preface......Page __sk_0007.djvu
1.1 Pi and Irrational Numbers: An Introduction to Greek Mathematics......Page __sk_0013.djvu
The Number Pi......Page __sk_0016.djvu
Irrational Numbers......Page __sk_0020.djvu
Repeating Decimals......Page __sk_0022.djvu
Focus: Archimedes......Page __sk_0025.djvu
Exercises......Page __sk_0027.djvu
1.2 Euclidean Geometry......Page __sk_0029.djvu
Logic......Page __sk_0031.djvu
Euclid\'s Definitions and Postulates......Page __sk_0033.djvu
The First Thirty-Two Propositions......Page __sk_0037.djvu
The Pythagorean Theorem......Page __sk_0044.djvu
Aside: Poincare Conjecture......Page __sk_0047.djvu
Exercises......Page __sk_0049.djvu
Regular Polyhedra......Page __sk_0052.djvu
Euler\'s Formula......Page __sk_0057.djvu
Conic Sections......Page __sk_0061.djvu
Pre-Keplerian Theories of Planetary Motion......Page __sk_0065.djvu
Kepler and the Regular Polyhedra......Page __sk_0068.djvu
Kepler\'s Laws......Page __sk_0071.djvu
Focus: Kepler......Page __sk_0073.djvu
Exercises......Page __sk_0077.djvu
2.1 Formal Axiom Systems......Page __sk_0079.djvu
Consistency......Page __sk_0082.djvu
Independence......Page __sk_0085.djvu
Equivalent Axioms......Page __sk_0086.djvu
Hilbert\'s Foundations of Geometry......Page __sk_0087.djvu
Aside: Russell\'s Paradox and Set Theory......Page __sk_0090.djvu
Exercises......Page __sk_0092.djvu
Playfair\'s Postulate......Page __sk_0094.djvu
Angle Sums of Triangles......Page __sk_0096.djvu
Saccheri\'s Approach......Page __sk_0099.djvu
Gauss\'s Work......Page __sk_0102.djvu
Focus: Gauss......Page __sk_0104.djvu
Exercises......Page __sk_0107.djvu
2.3 Hyperbolic Geometry......Page __sk_0110.djvu
Bolyai and Lobachevsky......Page __sk_0111.djvu
Some Theorems of Hyperbolic Geometry......Page __sk_0116.djvu
Properties of Parallels and Divergent Lines......Page __sk_0118.djvu
Area and Defect......Page __sk_0123.djvu
Aside: Kant and Geometry......Page __sk_0127.djvu
Exercises......Page __sk_0128.djvu
2.4 Spherical Geometry......Page __sk_0132.djvu
Flatland......Page __sk_0134.djvu
Postulates......Page __sk_0138.djvu
Propositions......Page __sk_0142.djvu
Area and Excess......Page __sk_0144.djvu
Consistency......Page __sk_0146.djvu
Aside: A Flat Model of Spherical Geometry......Page __sk_0147.djvu
Exercises......Page __sk_0150.djvu
2.5 Models of Hyperbolic Geometry......Page __sk_0153.djvu
Beltrami\'s Pseudo sphere......Page __sk_0154.djvu
The Klein-Beltrami Disk Model......Page __sk_0157.djvu
The Poincare Disk Model......Page __sk_0158.djvu
Aside: Hyperbolic Paper......Page __sk_0165.djvu
Exercises......Page __sk_0168.djvu
Higher-Dimensional Spaces......Page __sk_0170.djvu
Riemann\'s Lecture......Page __sk_0176.djvu
Special Theory of Relativity......Page __sk_0177.djvu
General Theory of Relativity......Page __sk_0180.djvu
The Large-Scale Geometry of the Universe......Page __sk_0184.djvu
Focus: Albert Einstein......Page __sk_0187.djvu
Exercises......Page __sk_0190.djvu
3.1 Prime Numbers......Page __sk_0192.djvu
Unique Factorization......Page __sk_0193.djvu
Divisibility and Primes......Page __sk_0194.djvu
The Infinitude of the Primes......Page __sk_0197.djvu
The Prime Number Theorem......Page __sk_0200.djvu
Mersenne Primes and Perfect Numbers......Page __sk_0204.djvu
Aside: Fermat\'s Last Theorem......Page __sk_0209.djvu
Exercises......Page __sk_0214.djvu
Finding the GCD......Page __sk_0218.djvu
Writing the GCD as a Combination of the Numbers......Page __sk_0221.djvu
Linear Diophantine Equations......Page __sk_0224.djvu
Aside: Proof of the Fundamental Theorem of Arithmetic......Page __sk_0227.djvu
Exercises......Page __sk_0229.djvu
3.3 Congruence Arithmetic......Page __sk_0230.djvu
Addition and Multiplication......Page __sk_0233.djvu
Exponentiation......Page __sk_0235.djvu
Congruence Equations......Page __sk_0239.djvu
Aside: Bases of Number Systems......Page __sk_0242.djvu
Exercises......Page __sk_0246.djvu
3.4 The Little Fermat Theorem......Page __sk_0248.djvu
Two Proofs......Page __sk_0250.djvu
Applications......Page __sk_0252.djvu
Pseudoprimes......Page __sk_0253.djvu
Aside: Primality Testing......Page __sk_0254.djvu
Exercises......Page __sk_0256.djvu
Substitution Ciphers......Page __sk_0259.djvu
The Enigma Ciphers......Page __sk_0266.djvu
Focus: Alan Turing and His \"Machine\"......Page __sk_0269.djvu
Exercises......Page __sk_0275.djvu
The RSA Method......Page __sk_0277.djvu
Signatures......Page __sk_0283.djvu
Security......Page __sk_0284.djvu
The Knapsack Method......Page __sk_0287.djvu
Aside: Computational Complexity......Page __sk_0290.djvu
Exercises......Page __sk_0292.djvu
The Koch Curve......Page __sk_0294.djvu
Similarity Dimension......Page __sk_0297.djvu
Box Dimension and the Cantor Set......Page __sk_0300.djvu
Fractals in Nature......Page __sk_0305.djvu
Aside: Fractal Ferns and Image Compression......Page __sk_0310.djvu
Exercises......Page __sk_0316.djvu
Population Growth......Page __sk_0324.djvu
Computer Experiments......Page __sk_0325.djvu
Fixed Points......Page __sk_0329.djvu
Chaos......Page __sk_0335.djvu
Bifurcation Diagrams......Page __sk_0336.djvu
A Similar Family of Functions......Page __sk_0339.djvu
Aside: Lorenz\'s Weather Forecast......Page __sk_0340.djvu
Exercises......Page __sk_0344.djvu
Complex Numbers......Page __sk_0349.djvu
The Mandelbrot Set......Page __sk_0351.djvu
Julia Sets......Page __sk_0369.djvu
Newton\'s Method......Page __sk_0377.djvu
Focus: Isaac Newton......Page __sk_0380.djvu
Exercises......Page __sk_0384.djvu
Epilogue......Page __sk_0391.djvu
Bibliography......Page __sk_0393.djvu
Credits......Page __sk_0397.djvu
Index......Page __sk_0399.djvu