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Elements of Mathematics: A Problem-Centered Approach to History and Foundations 2021 ed. [Hardback]

  • Formāts: Hardback, 527 pages, height x width: 235x155 mm, weight: 1073 g, 3 Illustrations, color; 30 Illustrations, black and white; XV, 527 p. 33 illus., 3 illus. in color., 1 Hardback
  • Sērija : Readings in Mathematics
  • Izdošanas datums: 24-Sep-2021
  • Izdevniecība: Springer Nature Switzerland AG
  • ISBN-10: 3030750507
  • ISBN-13: 9783030750503
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Elements of Mathematics: A Problem-Centered Approach to History and Foundations 2021 ed.Palielināt
  • Formāts: Hardback, 527 pages, height x width: 235x155 mm, weight: 1073 g, 3 Illustrations, color; 30 Illustrations, black and white; XV, 527 p. 33 illus., 3 illus. in color., 1 Hardback
  • Sērija : Readings in Mathematics
  • Izdošanas datums: 24-Sep-2021
  • Izdevniecība: Springer Nature Switzerland AG
  • ISBN-10: 3030750507
  • ISBN-13: 9783030750503
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783030750503.html
Keywords:
This textbook offers a rigorous presentation of mathematics before the advent of calculus. Fundamental concepts in algebra, geometry, and number theory are developed from the foundations of set theory along an elementary, inquiry-driven path. Thought-provoking examples and challenging problems inspired by mathematical contests motivate the theory, while frequent historical asides reveal the story of how the ideas were originally developed.





Beginning with a thorough treatment of the natural numbers via Peanos axioms, the opening chapters focus on establishing the natural, integral, rational, and real number systems. Plane geometry is introduced via Birkhoffs axioms of metric geometry, and chapters on polynomials traverse arithmetical operations, roots, and factoring multivariate expressions. An elementary classification of conics is given, followed by an in-depth study of rational expressions. Exponential, logarithmic, and trigonometric functions complete the picture, driven by inequalities that compare them with polynomial and rational functions. Axioms and limits underpin the treatment throughout, offering not only powerful tools, but insights into non-trivial connections between topics.





Elements of Mathematics is ideal for students seeking a deep and engaging mathematical challenge based on elementary tools. Whether enhancing the early undergraduate curriculum for high achievers, or constructing a reflective senior capstone, instructors will find ample material for enquiring mathematics majors. No formal prerequisites are assumed beyond high school algebra, making the book ideal for mathematics circles and competition preparation. Readers who are more advanced in their mathematical studies will appreciate the interleaving of ideas and illuminating historical details.

Recenzijas

Elements of mathematics is a curious book. The most challenging aspect of this volume to assess is its purpose. (Jeff Johannes, Mathematical Reviews, October, 2022) Transparency of explanation and gradually built material are outstanding features of the textbook. In addition, solutions to some problems are designed using more than one approach, making it adaptable to various students' backgrounds. The book makes itself accessible to a vast population of students. The book can enhance the undergraduate curriculum or serve as a reflective resource for graduate mathematics students. (Andrzej Sokolowski, MAA Reviews, March 20, 2022) A historical concern is present throughout,with pieces of information on the history of concepts and theorems. (Victor V. Pambuccian, zbMATH 1479.00002, 2022)

0 Preliminaries: Sets, Relations, Maps
1(36)
0.1 Sets
2(6)
0.2 Relations
8(3)
0.3 Maps and Real Functions
11(6)
0.4 Cardinality
17(8)
0.5 The Zermelo--Fraenkel Axiomatic Set Theory*
25(12)
1 Natural, Integral, and Rational Numbers
37(36)
1.1 Natural Numbers
37(15)
1.2 Integers
52(7)
1.3 The Division Algorithm for Integers
59(7)
1.4 Rational Numbers
66(7)
2 Real Numbers
73(62)
2.1 Real Numbers via Dedekind Cuts
74(21)
2.2 Infinite Decimals as Real Numbers
95(9)
2.3 Real Numbers via Cauchy Sequences
104(22)
2.4 Dirichlet Approximation and Equidistribution*
126(9)
3 Rational and Real Exponentiation
135(52)
3.1 Arithmetic Properties of the Limit
135(17)
3.2 Roots, Rational and Real Exponents
152(22)
3.3 Logarithms
174(5)
3.4 The Stolz--Cesaro Theorems
179(8)
4 Limits of Real Functions
187(20)
4.1 Limit Inferior and Limit Superior
187(6)
4.2 Continuity
193(5)
4.3 Differentiability
198(9)
5 Real Analytic Plane Geometry
207(56)
5.1 The Birkhoff Metric Geometry
208(4)
5.2 The Cartesian Model of the Birkhoff Plane
212(4)
5.3 The Cartesian Distance
216(4)
5.4 The Triangle Inequality
220(5)
5.5 Lines and Circles
225(9)
5.6 Arc Length on the Unit Circle
234(11)
5.7 The Birkhoff Angle Measure
245(7)
5.8 The Principle of Shortest Distance*
252(5)
5.9 Π According to Archimedes*
257(6)
6 Polynomial Expressions
263(56)
6.1 Polynomials
264(6)
6.2 Arithmetic Operations on Polynomials
270(4)
6.3 The Binomial Formula
274(10)
6.4 Factoring Polynomials
284(5)
6.5 The Division Algorithm for Polynomials
289(11)
6.6 Symmetric Polynomials
300(13)
6.7 The Cauchy--Schwarz Inequality
313(6)
7 Polynomial Functions
319(32)
7.1 Polynomials as Functions
319(7)
7.2 Roots of Cubic Polynomials
326(6)
7.3 Roots of Quartic and Quintic Polynomials
332(3)
7.4 Polynomials with Rational Coefficients
335(4)
7.5 Factoring Multivariate Polynomials
339(8)
7.6 The Greatest Common Factor
347(4)
8 Conies
351(28)
8.1 The General Conic
351(7)
8.2 Parabolas
358(6)
8.3 Ellipses
364(6)
8.4 Hyperbolas
370(9)
9 Rational and Algebraic Expressions and Functions
379(44)
9.1 Rational Expressions and Rational Functions
380(5)
9.2 The Partial Fraction Decomposition
385(10)
9.3 Asymptotes of Rational Functions
395(5)
9.4 Algebraic Expressions and Functions, Rationalization
400(4)
9.5 Harmonic, Geometric, Arithmetic, Quadratic Means
404(13)
9.6 The Greatest Integer Function
417(6)
10 Exponential and Logarithmic Functions
423(46)
10.1 The Natural Exponential Function According to Newton
423(12)
10.2 The Bernoulli Numbers*
435(4)
10.3 The Natural Logarithm
439(15)
10.4 The General Exponential and Logarithmic Functions
454(3)
10.5 The Natural Exponential Function According to Euler
457(12)
11 Trigonometry
469(50)
11.1 The Unit Circle S vs. the Real Line 1
469(2)
11.2 The Sine and Cosine Functions
471(6)
11.3 Principal Identities for Sine and Cosine
477(17)
11.4 Trigonometric Rational Functions
494(6)
11.5 Trigonometric Limits
500(7)
11.6 Cosine and Sine Series According to Newton
507(3)
11.7 The Basel Problem of Euler*
510(5)
11.8 Ptolemy's Theorem
515(4)
Further Reading 519(2)
Index 521
Gabor Toth is Distinguished Professor of Mathematics at Rutgers University, Camden. His research interests include convex geometry and differential geometry, in particular, harmonic maps and minimal immersions. Beyond mathematics, he teaches Ancient Egyptian Grammar and the history of precolonial Africa. He regularly trains gifted high school students for mathematical contests in Princeton. His previous books include Measures of Symmetry for Convex Sets and Stability, Glimpses of Algebra and Geometry, and Finite Möbius Groups, Spherical Minimal Immersions, and Moduli.