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| Courant R., Robbins H. — What Is Mathematics?: An Elementary Approach to Ideas and Methods |
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| Ïðåäìåòíûé óêàçàòåëü |
Hyperbolic geometry 214—224
Hyperbolic paraboloid 286
Hyperbolic points 226
Hyperboloid 212—214
Hyperreal numbers 519—521
Hypocycloid 154
Ideal elements in projective geometry 180—185
Image point (of mapping) 141
imaginary numbers see "Complex numbers"
incidence 169
Incommensurable segments 58—61
Independent variable 275
Indirect proof 86—87
Induction, empirical 10
Induction, mathematical 9—20
Inequalities 3—4 15—16 57 58 94 322 361—366 501
Infinite continued fractions 301—303
Infinite decimals 61—63
Infinite products 300 481—481 482
Infinite series 63—66 472—477
Infinitude of primes 22 26—27 481
infinity 56 77—88
Infinity, elements at (in projective geometry) 180—185 493
Infinity, mathematical analysis of 77—88
Infinity, point at (in inversion geometry) 142
Integer, principle of the smallest 18—19
Integers and continuum hypothesis 493
Integers and definition of dimension 499
Integers and fractal sets 501
Integers and Hausdorff dimension 500 501
Integers and nonstandard analysis 520
Integers, negative 55
Integers, positive 1—9
integral 399—414 464—465 504—510
Interest, compound 457
Intersection of sets 110 494—495
interval 57
Intervals, nested 68
Intuitionism 86—87 215
Invariance 165—167
Invariance of angles under inversion 158—159
Invariance of cross-ratio 173—174
Inverse functions 278—281
Inverse operations 3
Inverse points 141
Inverse points, construction of 144—145
Inversion geometry 140—146 158—164
Inversors 155—158
Irrational numbers, as infinite decimals 63
Irrational numbers, defined by nested intervals 68—71
Irrational numbers, defined by ruts 71—72
Irrational numbers, defined by sequences 72
Isoperimetric problems 373—373 376
Iteration, limits by 326—327
Jones polynomial 501 503 505
Jordan curve theorem 244—246 267—269
Jump discontinuity 284
Klein's bottle 262
Klein's model 219—222
Knots 255—256 501—505
Least squares, method of 365—366
Leibniz and nonstandard analysis 519 521—522
Leibniz' formula for  441
Length of a curve 466—469
Level lines 286—287
Light rays, extremum property of 330—332
Light triangles 352—353
Limits 289—321
Limits by continuous approach 303—312
Limits by iteration 326—327
Limits of geometrical series 65—66
Limits of infinite decimals 63—66
Limits of sequences 289—303
Limits, examples on 322—327
Line at infinity 182
Line conic 207
Lines, concurrent 170
Lines, contour 286—287
Lines, Coplanar 176
Lines, pencil of 203
Linkages 155—158 501—505
Liouville's theorem 104—107
Log n!, order of magnitude of 471—472
Logarithm, natural 28 443—446 450—453 469—470 500
Logic, mathematical 87—88 112—114
Logical product 110 494
Logical sum 110 494
Magnitude, orders of 469—472
Mandelbrot set 499—500 501
Map, regular 264
Map-coloring problem 246—248 264—267
Mapping 141
Mascheroni constructions 147—152
Mathematical induction 9—20
Mathematical logic 87—88 112—114
Maxima and minima 329—397 426—427 433
Mean, arithmetical 361—365
Mean, geometrical 361—365
Means, inequality connecting 361—365
Mechanical instruments, constructions with 152—155
Mechanics, problem in 505—507
Metamathematics 88
Metric geometry 169
Minimax, points of 343—345
Modulo d 32
Modulus of complex number 93
Moebius strip 259—262
Monotone function 280
Monotone sequence 295—297
Morse relations 345
Motion, equations of 460—461
Motion, ergodic 353—354
Motion, rigid 141
Multiplicity of roots of algebraic equation 102
n-dimensional geometry 227—234
Natural numbers 1—20 520
negative numbers 54—55
Newtonian dynamics 460—461 506
Non-denumerability of continuum 81—83
Non-Euclidean geometry 213—227
Nonstandard analysis 518—523
NP-complete 512
Number fields 127—134
Number system 51—107 501
Numbers, algebraic 103—104
Numbers, cardinal 83—86 493
Numbers, complex 88—103
Numbers, composite 22
Numbers, constructlble 127—134
Numbers, Fermat 25 119
Numbers, natural 1—20 520
numbers, negative 54—55
Numbers, prime 21—31
Numbers, prime, Pythagorean 40—42
Numbers, prime, rational 52—58 520
Numbers, prime, real 68—72
Numbers, prime, transcendental 103—104
One-sided surfaces 259—264
orders of magnitude 469—472
Pappus' theorem 188
Paradoxes of the infinite 87
Paradoxes of Zeno 305—306
Parallel postulate 218
Parallelism and infinity 180—185 493
Pascal's theorem 188 191 209—212
Pascal's triangle 17
Peaucellier's inversor 155—167
Pencil of lines 203
Pentagon, construction of regular 100 122—123
Perspective 169
| PI 140 299—300 303 441—442
Plane at infinity 184—185
Plateau's problem 386 514 515—516 518
Point conic 204
Points, at infinity 180—185
Points, collinear 170
Points, range of 207
Polyhedra, Euler characteristic of 236—240 258—259 262
Polyhedra, genus of 256—258 262
Polyhedra, in n dimensions 227—234
Polyhedra, one-sided 259—262
Polyhedra, regular 236—240
Polyhedra, simple 236
Polynomial time 509—612
Polynomials and computability 488
Polynomials and formula for primes 487—488
Polynomials and knots 501 503 505
Polynomials, Alexander 503 505
Polynomials, HOMFLY 505
Polynomials, Jones 501 503 505
Polynomials, variables of 487—488
positional notation 4
Postulates 214
Prime number theorem 27—30 482—486 487—490
PRIMES 21—31 481 482—486 487—490
Primitive functions 438
probability 114—116
Product, infinite 300 481—482
Product, logical 110
Progressions, Arithmetical 12—13 26—27 487—488
Projective correspondence 178—261
Projective geometry 165—214
Projective transformation 167—170
Proof and sense of "theorem" 438
Proof via nonstandard analysis 523
Proof: constructive, indirect, and existential 86—87
Pythagorean numbers 40—42 492
Quadrants 73
Quadratic equation 91—92 302
Quadratic residues 38
Quadric surfaces 212—214
Quadrilateral, complete 179—180
Radian measure 277—278
Radioactive disintegration 455—457
Range of points 207
Rational numbers 52—58 520
Rational numbers, density of 58
Rational numbers, denumerability of 79—80
Rational numbers, operations with 53—54
Rational quantities, geometrical construction of 120—122
Real numbers 58—72
Real numbers, continuum of 68 493
Real numbers, operations with 70—71 519
Reflection in a circle 140—146
Reflection in a system of circles 163—164
Reflection in one or more lines 330—332
Reflection in triangles 352—353
Reflection, general extremum problems in 353—354
Reflection, repeated 162—164
Regular polygons, construction of 119 122—125 495
Regular polyhedra 236—240
Regular polyhedra, n dimensional 227—234
Relativity 227 229
Residues, quadratic 38—40
Riemannian geometry 224 227 489—490
Rigid motion 141
Roots of unity 98—100
Schwarz's triangle problem 346—354 377
Second derivative 426 435
Segment 57 73
Sense (of angles) 159
Septimal system 5—6 7—8
Sequences 289—303
Sequences, bounded 295
Sequences, convergent, divergent, and oscillating 294
Sequences, monotone 295—297
Sequences, theorem on 315—316
Series, infinite 472—477
Set 78
Set, Cantorian 494 501
Set, compact 316
Set, complement of 111
Set, empty 18 494
Set, fractal 501
Set, Mandelbrot 499—500 501
Sets, algebra of 108—116 494—495
Sets, equivalence of 78
Sierpirtski gasket 501
Sieves 25 489 490
Simple dosed curve 244
Simple polyhedron 236
Simply connected 243
slope 415 490
Smallest integer, principle of 18—19
Soap film experiments 385—397 513—518
Solvability of problems 118
Square root, geometrical construction of 122
Squaring the circle 140 147
Stationary points 3414146
Steiner constructions 151—152 196—198
Steiner's problem 354—361 377—379 391 507—513 515
Straight line, equation of 75
Straightedge constructions 151—152 196—198
Street network problem see "Steiner's problem"
Subfield 138
Subscripts 5
Subset 109
Subset, proper 78
Sum of first n cubes 15
Sum of first n squares 14
Sum, logical 110 494
Surfaces, minimal 513—518
Surfaces, one-sided 259—264
Surfaces, quadric 212—214
Synthetic geometry 165
Tangent 415
Tangent properties of ellipse and hyperbola 333—336
Taniyama Conjecture 492—493
Taylor series 476—477
Theory of numbers 21—51 481 482—486 491
Topological classification of surfaces 256—261 502—503
Topological transformation 241
topology 235—271
Topology and critical points 345
Torus 248
Torus, three-dimensional 262—264
Transcendence of pi 104 140
Transcendental numbers 103—104 104—107
Transformations, equations of 288—289
Transformations, geometrical 140—141 165—167
Transformations, projective 167—170
Transformations, topological 241
Triangles and Steiner's problem 507—513
Triangles, extremum properties of 330 332—333 346—353 354—359
Trigonometric functions, definition of 277
Trisection of angle 117 137—138
Union (of sets) 110 494—495
Unique factorization 23 46—48
Unit circle 93 492
Unity, roots of 98—100
Unsolvability of Greek problems 134—140
Unsolvabitity, proofs of 120—140
Variable 273—277
Variable, complex 478
Variable, dependent 275
Variable, general notion of 273
Variable, independent 275
Variations, calculus of 379—385
Velocity 423—425
Vibrations 458—459
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