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Beaumont R.A., Pierce R.S. — The Algebraic Foundations of Mathematics
Beaumont R.A., Pierce R.S. — The Algebraic Foundations of Mathematics

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Название: The Algebraic Foundations of Mathematics

Авторы: Beaumont R.A., Pierce R.S.

Аннотация:

This book is an offspring of two beliefs which the authors have held for many years: it is worthwhile for the average person to understand what mathematics is all about; it is impossible to learn much about mathematics without doing mathematics. The first of these convictions seems to be accepted by most educated people. The second opinion is less widely held. Mathematicians teaching in liberal arts colleges and universities are often under pressure from their colleagues in the humanities and social sciences to offer short courses which will painlessly explain mathematics to students with varying backgrounds who are seeking a broad, liberal education. The extent to which such courses do not exist is a credit to the good sense of professional mathematicians. Mathematics is a big and difficult subject. It embraces a rigid method of reasoning, a concise form of expression, and a variety of new concepts and viewpoints which are quite different from those encountered in everyday life. There is no such thing as "descriptive" mathematics. In order to find answers to the questions "What is mathematics?" and "What do mathematicians do?", it is necessary to learn something of the logic, the language, and the philosophy of mathematics. This cannot be done by listening to a few entertaining lectures, but only by active contact with the content of real mathematics. It is the authors' hope that this book will provide the means for this necessary contact.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1963

Количество страниц: 486

Добавлена в каталог: 12.04.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abscissa      299
Absolute value      132 292
Addition in a ring      107
Addition of complex numbers      287 301
Addition of integers      101
Addition of matrices      430
Addition of natural numbers      89
Addition of polynomials      314
Addition of rational numbers      219
Addition of real numbers      241
Algebraic number      385
Amplitude      304
Archimedes      226
Archimedes’ Principle      233
Argument      304
Associate      331
Associative law for addition      90
Associative law for matrix multiplication      437
Associative law for multiplication      93
Associative law for set operations      37
Base      138
Basis of induction      57 68
Binary operation for rings      107 114
Binary operation for sets      30
Binary system      140
Binomial coefficients      61 64
Binomial theorem      63
Bombelli, Rafael      286
Bounds for roots      374
C in a ring      121
C of multiplication      93 96
Cancellation law of addition      90 96
Cantor, Georg      18
Cantor’s theorem      279
Cardan, Girolamo      309
Cardan’s formulas      364
Cardinal number of a finite set      18 86
Cardinal number of a set      20 83
Cardinality      18 20 166
Cartesian coordinates      299
Casting out nines      179
Characteristic of an integral, domain      210
Chinese remainder theorem      188
Coefficient      313
Column matrix      429
Common divisor      144 150
Common multiple      151
Commutative law for addition      90
Commutative law for multiplication      93
Commutative ring      108
Comparison Test      269
Complement of a set      31
Complete factorization      336
Complete ordered integral, domain      251
Completing the square      296
Complex conjugate      291
complex numbers      2 287
Complex plane      300
Composite number      153
Congruence      176
Congruence classes      182
Congruent modulo m      176 215
conjugate      291
Consistent systems of, equations      411
Constant polynomial      317
Constant term      317
Contrapositive      7
Convergent sequence      258
Convergent Series      264
CONVERSE      7
Coordinate line      230
Coordinate plane      299
Coordinate system      230 238
Course of values induction      68
Cubic equation      361
Decimal fraction      226
Decimal representation of real numbers      277
Decimal representation, of, rational numbers      282
Dedekind cuts      237
Dedekind, Richard      224
Definition      7
Degree of a polynomial      321 397
Degree of an algebraic number      386
DeMoivre’s Theorem      305
Density property      233
Denumerable      23
Derivative      338
Descartes, Rene      299
Determinant      412 429
Diagonal method      280
Difference of natural numbers      96
Difference of sets      36
Dimensions of a matrix      429
Diophantine equation      169
Direct proof      9
Discriminant      368
Disjoint sets      44
Disjunctive normal form, theorem      40
Distance      299
Distributive law for natural, numbers      93
Distributive law for set operations      39
Divergent series      264
Division algorithm for integers      135
Division algorithm for polynomials      322
Division in an integral domain      122
Division of polynomials      326
Divisor of zero      122
Domain of polynomials      317 394
Duodecimal system      139
E of a system of equations      416
Echelon form of a matrix      447
Element of a matrix      429
Element of a set      11 17
Elementary symmetric, polynomial      406
Elementary transformation, matrix      451
Elementary transformations of a system of equations      413
Elementary transformations, of, a matrix      446
Empty set      13 17
Equal, matrices      430
Equal, polynomials      313
Equal, sets      11
Equivalence class      216
Equivalence of sets      20
Equivalence of statements      6
Equivalence relation      214
Equivalent systems of, equations      412
Euclid      148 160
Euclidean algorithm for polynomials      330
Euclidean algorithm, for integers      148
Euler, Leonhard      189
Euler’s theorem      191
EXPONENT      194 206
Factor      122 326
Factor Theorem      347
Fermat conjecture      172
Fermat numbers      162
Fermat, Pierre      162
Fermat’s theorem      191
Fibonacci sequence      79 151 152
Field      204
Finite ordinal numbers      84
Finite sets      18 20 86
Fundamental Theorem of Arithmetic      155
Fundamental theorem of decimal representation      272
Fundamental theorem of symmetric polynomials      407 466
Fundamental theorem, of, algebra      355 474
General associative law      115 117
General commutative law      115 117
General distributive law      118
Godel numbering      165
Graph      369
Greatest common divisor for integers      146 148 150 157
Greatest common divisor for polynomials      327 332
Greatest element      130
Greatest integer function      278
Greatest lower bound      249
Homogeneous system of, equations      424
Identity      4
Identity element      93 121
Identity matrix      443
Imaginary part of a complex, number      291
Implication      4
Incongruent solutions, modulo m      183
Inconsistent system of, equations      411
Indeterminate      317
Index of summation      115
Index set      37
Indirect proof      9
Induction hypothesis      57 68
Induction step      57 68
Inductive definitions      79
Inequality      129
Infinite decimal sequence      228
Infinite sequence      258
Infinite series      264
Infinite set      18 20
Integers      2 100
Integral domain      121
Intersection of sets      30 36
Inverse of a square matrix      443
Inverse of an element      205
Inverse of an implication      7
Irrational number      224
Irreducible polynomial      333 357 358
Isomorphic rings      111
Isomorphism      111
largest element      130
Latin square      185
Law of substitution      8
Leading coefficient      321
Least common multiple of polynomials      332
Least common multiple, of, integers      151 157
Least element      130
Least upper bound      249
LIMIT      260
Linear congruence      181
Lower bound      248
Lower bound for roots      374
Mathematical induction      53
Matrix      428
Matrix of coefficients      434
Maximum element      130
Measure of a set      41 45
Mersenne number      162
Method of infinite descent      174
Method of infinite descent, m-fold root      348
Minimum element      130
Minimum polynomial      386
Modulus      176 292
Monic associate      328
Monic polynomial      328
Monic polynomial, mth root      256
Multiple factor      345
Multiple root      349
Multiplication in a ring      107
Multiplication of complex, numbers      287 305
Multiplication of integers      104
Multiplication of matrices      433
Multiplication of natural numbers      91 92
Multiplication of polynomials      315
Multiplication of rational numbers      219
Multiplication of real numbers      244
Multiplicity      348
Natural numbers      2 82 85
Negation in a ring      107
Negation of complex numbers      287
Negation of integers      101
Negation of polynomials      315
Negation of rational numbers      219
Negation of real numbers      243
Negative elements      127
negative numbers      100 244
Negative of a matrix      432
Negative of a polynomial      315
NIM      140
Nonnegative elements      128
Nonnegative real numbers      244
Nonsingular matrix      443
Nonsingular matrix, n-place decimal approximation      273
Nonsingular matrix, n-place decimal fraction      270
Nonsingular matrix, n-rowed square matrix      439
Nonsingular matrix, nth roots of unity      308
Number of divisors      156
One-to-one correspondence      19
Order modulo m      194 352
Ordered field      207
Ordered integral domain      126
Ordered pair      24
Ordering of integers      125
Ordering of natural numbers      95
Ordering of rational numbers      219
Ordering of real numbers      240
Ordinate      299
Origin      230 298
Orthogonal latin square      185
Pairwise disjoint collection of, sets      44
Parallelogram rule      302
Partial sum      264
Partially ordered set      248
Partition      218
Pascal triangle      61
Peano’s axioms      87
Perfect number      163
Permutation      20
Polar representation      303
Polynomial      313
Polynomial in several indeterminates      394
Positive elements      127
Positive integers      128
Positive real numbers      244
Power set      26
Prime characteristic      210
Prime number      68 153 159
Prime number theorem      161
Prime pair      161
Primitive root      195 352
Principle of mathematical, induction      57 76
Probability measure      43
Product of matrices      433
Product of natural numbers      91 92
Product of polynomials      315
Product of sets      25
Product sign      118
Proofs      8
Proper divisor of zero      122
Proper subset      15
Pythagoras’ theorem      224 234
Quadratic equation      296
Quartic equation      365
Quotient      124 136 325
Range of a variable      3
Rational numbers      2 200 218
Real numbers      2 224 238
Real part of a complex number      291
recursive definitions      79
Reduced cubic equation      362
Reducible polynomial      333
Reflexive law      215
Relation      213
Relatively prime integers      147
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