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Elements of Mathematics = A Problem-Centered Approach to History and Foundations /

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Elements of Mathematics/ by Gabor Toth.
其他題名:	A Problem-Centered Approach to History and Foundations /
作者:	Toth, Gabor.
面頁冊數:	XV, 527 p. 33 illus., 3 illus. in color.online resource. :
Contained By:	Springer Nature eBook
標題:	History of Mathematical Sciences. -
電子資源:	https://doi.org/10.1007/978-3-030-75051-0
ISBN:	9783030750510

Elements of Mathematics = A Problem-Centered Approach to History and Foundations /
Toth, Gabor.

Elements of MathematicsA Problem-Centered Approach to History and Foundations /[electronic resource] :by Gabor Toth. - 1st ed. 2021. - XV, 527 p. 33 illus., 3 illus. in color.online resource. - Readings in Mathematics. - Readings in Mathematics.

0. Preliminaries: Sets, Relations, Maps -- 1. Natural, Integral and Rational Numbers -- 2. Real Numbers -- 3. Rational and Real Exponentiation -- 4. Limits of Real Functions -- 5. Real Analytic Plane Geometry -- 6. Polynomial Expressions -- 7. Polynomial Functions -- 8. Conics -- 9. Rational and Algebraic Expressions and Functions -- 10. Exponential and Logarithmic Functions -- 11. Trigonometry -- Further Reading -- Index.

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 Peano’s axioms, the opening chapters focus on establishing the natural, integral, rational, and real number systems. Plane geometry is introduced via Birkhoff’s 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.

ISBN: 9783030750510

Standard No.: 10.1007/978-3-030-75051-0doiSubjects--Topical Terms:

785417
History of Mathematical Sciences.

LC Class. No.: QA331.5

Dewey Class. No.: 515.8

Elements of Mathematics = A Problem-Centered Approach to History and Foundations /
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