Mathematics: From the Birth of Numbers

Jan Gullberg, Peter Hilton9780393040029, 0-393-04002-X

This gently guided, profusely illustrated Grand Tour of the world of mathematics takes the reader on a long and fascinating journey – from the dual invention of numbers and language, through the primary realms of arithmetic, algebra, geometry, trigonometry, and calculus, to the final destination of differential equations, with excursions into symbolic logic, set theory, topology, fractals, probability, and assorted other mathematical byways. Mathematics: From the Birth of Numbers is unique among popular books on mathematics in combining an engaging, easy-to-read history of the subject with a comprehensive mathematical survey text. Intended, in the author’s words, “for the benefit of those who never studied the subject, those who think they have forgotten what they once learned, and those with a sincere desire for more knowledge,” it links mathematics to the humanities, linguistics, the natural sciences, and technology.

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Table of contents :
MATHEMATICS- FROM THE BIRTH OF NUMBERS……Page 1
Contents in Brief……Page 6
Preface……Page 7
Contents……Page 9
Foreword: “Mathematics in Our Culture” by Peter Hilton……Page 16
Information for the Reader……Page 22
01| Numbers and Language……Page 23
1.0 The Origins of Reckoning……Page 25
1.1 Numbers and Numerals……Page 27
1.2 Number Names……Page 29
1.3 Etymology of English Number Names……Page 48
1.4 Numbers vs. Infinity……Page 51
02| Systems of Numeration……Page 53
2.0 Forms of Notation……Page 54
2.1 Additive Notation……Page 56
2.2 Multiplicative Notation……Page 66
2.3 Positional Notation……Page 68
2.4 Decimal Position System……Page 70
2.5 Sexagesimal Numeration……Page 78
2.6 Vigesimal Numeration……Page 80
2.7 Duodecimal Numeration……Page 83
2.8 Binary, Octal, Hexadecimal……Page 84
2.9 Special Forms of Notation……Page 88
03| Types of Numbers……Page 91
3.0 An Expanding Universe of Numbers……Page 92
3.1 Rational Numbers……Page 94
3.2 Prime Numbers……Page 99
3.3 Perfect and Amicable Numbers……Page 104
3.4 Irrational Numbers……Page 106
3.5 Imaginary and Complex Numbers……Page 109
3.6 The Quest for PI……Page 111
04| Cornerstones of Mathematics……Page 119
4.0 Beginnings……Page 120
4.1 Symbols Galore……Page 123
4.2 Fundamental Operations……Page 135
4.3 Laws of Arithmetic and Algebra……Page 154
4.4 Powers and Roots……Page 156
4.5 Logarithms……Page 172
4.6 Mathematical Proof……Page 179
4.7 Reliability of Digits and Calculations……Page 183
4.8 Simple Calculating Devices……Page 190
05| Combinatorics……Page 205
5.0 Historical Notes……Page 206
5.1 Multiplication Principle……Page 208
5.2 Permutations……Page 210
5.3 Combinations……Page 218
5.4 Samples with Replacement……Page 221
5.5 Graph Theory……Page 223
5.6 Magic Squares and Their Kin……Page 227
06| Symbolic Logic……Page 237
6.0 Historical Notes……Page 238
6.1 Pitfalls……Page 241
6.2 Propositions……Page 242
6.3 Tautologies……Page 247
6.4 Syllogisms and Proofs……Page 249
6.5 Logic Circuits……Page 251
07| Set Theory……Page 253
7.0 Introduction……Page 254
7.1 Sets and Their Contents……Page 255
7.2 Venn Diagrams……Page 264
7.3 Algebra of Sets……Page 271
7.4 Boolean Algebra……Page 274
7.5 Transfinite Numbers……Page 279
08| Introduction to Sequences and Series……Page 285
8.1 Terminology……Page 286
8.2 Finite Sequences and Series……Page 288
8.3 Infinite Series……Page 292
8.4 The Tower of Hanoi……Page 307
8.5 The Fibonacci and Related Sequences……Page 308
8.6 Figurate Numbers……Page 311
09| Theory of Equations……Page 317
9.01 History……Page 319
9.02 Groundwork……Page 324
9.1 Linear Equations……Page 329
9.2 Equations with Absolute Values……Page 330
9.3 Quadratic Equations……Page 331
9.4 Inequalities……Page 334
9.5 Root, Exponential, and Logarithmic Equations……Page 335
9.6 Cubic Equations……Page 338
9.7 Quartic Equations……Page 342
9.8 Systems of Equations……Page 347
9.9 Diophantine Equations……Page 352
10| Introduction to Functions……Page 357
10.1 Groundwork……Page 358
10.2 Elementary Functions……Page 370
10.3 Continuity and Limits……Page 377
11| Overture to the Geometries……Page 385
11.0 History……Page 386
11.1 Geometric Abstraction……Page 392
11.2 Perspective and Projection……Page 394
11.3 Form and Shape……Page 398
11.4 Survey of Geometries……Page 399
11.5 Topology……Page 400
11.6 Euclidean and Non-Euclidean Geometries……Page 403
12| Elementary Geometry……Page 405
12.1 Geometric Elements and Figures……Page 408
12.2 Units of Measurement……Page 431
12.3 Euclidean Construction……Page 435
12.4 Theorems and Formulas……Page 447
12.4.1 Plane Geometry……Page 448
12.4.2 Solid Geometry……Page 468
13| Trigonometry……Page 479
13.0 Scope and History……Page 480
13.1 Fundamental Trigonometric Functions……Page 492
13.2 Inverse Trigonometric Functions……Page 499
13.3 Solving Triangles……Page 500
13.4 Graphs, Domains, Ranges……Page 524
13.5 Trigonometric Identities……Page 529
13.6 Trigonometric Equations……Page 538
13.7 Limits……Page 550
14| Hyperbolic Functions……Page 554
14.0 Introduction……Page 555
14.1 Fundamental Hyperbolic Functions……Page 557
14.2 Inverse Hyperbolic Functions……Page 560
14.3 Identities……Page 562
15| Analytic Geometry……Page 567
15.0 Scope and History……Page 568
15.1 Rectilinear Figures……Page 569
15.2 Conic Sections……Page 579
15.3 Shifting Orthogonal Coordinates……Page 592
15.4 Polar Coordinate Systems……Page 596
15.5 Parametric Equations……Page 604
16| Vector Analysis……Page 617
16.0 Scope and History……Page 618
16.1 Basic Vector Algebra……Page 620
16.2 Scalar and Vector Components……Page 625
16.3 Multiplication of Vectors……Page 632
17| Fractals……Page 643
17.0 What Are Fractals?……Page 644
17.1 The Snowflake Curve……Page 645
17.2 Anti-Snowflake and Anti-Square Curves……Page 647
17.3 The Cantor Set……Page 648
17.4 Sierpinski Triangle, Carpet, and Sponge……Page 649
17.5 The Mandelbrot Set……Page 651
17.6 The Dimension Concept……Page 653
18| Matrices and Determinants……Page 654
18.0 Scope and History……Page 655
18.1 Matrices – Presentation……Page 657
18.2 Matrices – Rules of Operation……Page 659
18.3 Determinants……Page 663
18.4 Special Matrices……Page 671
18.5 Cofactors and the Inverse of a Matrix……Page 676
18.6 Solving Systems of Linear Equations……Page 678
19| Embarking on Calculus……Page 688
19.1 What Is Calculus?……Page 690
19.2 History……Page 691
20| Introduction to Differential Calculus……Page 700
20.1 Derivatives and Differentials……Page 702
20.2 Differentiating Algebraic Functions……Page 706
20.3 Differentiating Transcendental Functions……Page 712
20.4 Special Techniques of Differentiation……Page 722
20.5 Partial Differentiation……Page 727
20.6 Mean-Value Theorems……Page 733
21| Introduction to Integral Calculus……Page 737
21.1 Basic Concepts……Page 738
21.2 Methods of Integration……Page 739
21.3 The Definite Integral……Page 763
21.4 Multiple Integrals……Page 769
21.5 Improper or Unrestricted Integrals……Page 772
22| Power Series……Page 781
22.1 Convergence……Page 782
22.2 Taylor’s and Maclaurin’s Series……Page 783
22.3 Expanding Transcendental Functions……Page 787
22.4 Binomial Expansion……Page 791
22.5 The Riemann Zeta Function and Hypothesis……Page 794
23| Indeterminate Limits……Page 795
23.1 A Retrospect……Page 797
23.2 L’Hospital’s Rule……Page 798
24| Complex Numbers Revisited……Page 803
24.1 Introduction……Page 804
24.2 Sums and Differences……Page 805
24.3 Products and Quotients……Page 806
24.4 Powers……Page 807
24.5 Roots……Page 809
24.6 Logarithms……Page 810
25| Extrema and Critical Points……Page 813
25.1 One Independent Variable……Page 814
25.2 More than One Independent Variable……Page 831
25.3 Functions with Restrictions……Page 838
26| Arc Length……Page 843
26.1 Basic Principle……Page 845
26.2 The Catenary……Page 847
26.3 Arc Length in Parametric Form……Page 848
26.4 Arc Length in a Polar Coordinate System……Page 851
27| Centroids……Page 853
27.1 Mass Point Systems……Page 854
27.2 Plane Figures and Laminas……Page 858
27.3 Center of Mass of Solids of Revolution……Page 863
28| Area……Page 867
28.1 Plane Surfaces……Page 868
28.2 Surface of Revolution……Page 876
28.3 Work……Page 883
29| Volume……Page 887
29.1 Disk Method……Page 889
29.2 Shell Method……Page 893
29.3 Solids Generated by Area Bounded by Two Curves……Page 896
29.4 Translation of Axes……Page 898
29.5 Guldin’s Second Rule……Page 900
29.6 Solids with Known Cross-Section Areas……Page 901
29.7 Transforming Double Integrals from Orthogonal to Polar Coordinates……Page 903
30| Motion……Page 907
30.1 Laws of Kepler and Newton……Page 909
30.2 Differentiating Distance and Velocity……Page 912
30.3 Integrating Acceleration and Velocity……Page 916
30.4 Velocity Vectors and Acceleration Vectors……Page 917
30.5 Space-Time; Mass and Energy……Page 920
31| Harmonic Analysis……Page 923
31.0 Historical Notes……Page 925
31.1 Fourier Series……Page 926
31.2 Expanding Discontinuous Functions……Page 931
31.3 Expanding Even or Odd Functions……Page 934
32| Methods of Approximation……Page 939
32.1 Negligible Terms……Page 941
32.2 Interpolation……Page 942
32.3 Graphic and Iterative Methods……Page 947
32.4 Numerical Integration……Page 963
33| Probability Theory……Page 977
33.0 Introduction……Page 978
33.1 History……Page 979
33.2 The Basics……Page 982
33.3 The Probability Density Function……Page 987
33.4 Central Tendency……Page 990
33.5 Dispersion……Page 992
33.6 Normal Distribution……Page 995
34| Differential Equations……Page 1005
34.1 Fundamental Concepts……Page 1007
34.2 First-Order Ordinary Differential Equations……Page 1010
34.3 Formulating Differential Equations……Page 1016
34.4 Second-Order Ordinary Differential Equations……Page 1031
Bibliography……Page 1056
Works Cited……Page 1063
Name Index……Page 1068
Subject Index……Page 1075
Symbols in Common Use……Page 1107