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This introductory textbook takes a problem-solving approach to number theory, situating each concept within the framework of an example or a problem for solving. Starting with the essentials, the text covers divisibility, unique factorization, modular arithmetic and the Chinese Remainder Theorem, Diophantine equations, binomial coefficients, Fermat and Mersenne primes and other special numbers, and special sequences. Included are sections on mathematical induction and the pigeonhole principle, as well as a discussion of other number systems. By emphasizing examples and applications the...

* Problem-solving tactics and practical test-taking techniques provide in-depth enrichment and preparation for various math competitions

* Comprehensive introduction to trigonometric functions, their relations and functional properties, and their applications in the Euclidean plane and solid geometry

* A cogent problem-solving resource for advanced high school students, undergraduates, and mathematics teachers engaged in competition training

Rooted in a pedagogically successful problem-solving approach to linear algebra, this work fills a gap in the literature that is sharply divided between, on the one end, elementary texts with only limited exercises and examples, and, at the other extreme, books too advanced in prerequisites and too specialized in focus to appeal to a wide audience. Instead, "Essential Linear Algebra" clearly develops the theoretical foundations of vector spaces, linear equations, matrix algebra, eigenvectors, and orthogonality, while simultaneously emphasizing applications and connections to fields such as...

Rooted in a pedagogically successful problem-solving approach to linear algebra, this work fills a gap in the literature that is sharply divided between, on the one end, elementary texts with only limited exercises and examples, and, at the other extreme, books too advanced in prerequisites and too specialized in focus to appeal to a wide audience. Instead, "Essential Linear Algebra" clearly develops the theoretical foundations of vector spaces, linear equations, matrix algebra, eigenvectors, and orthogonality, while simultaneously emphasizing applications and connections to fields such as...

This text treats the classical theory of quadratic diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area. The presentation features two basic methods to investigate and motivate the study of quadratic diophantine equations: the theories of continued fractions and quadratic fields. It also discusses Pell's equation and its generalizations, and presents some important quadratic diophantine equations and applications. The inclusion of examples makes this book useful for both research and classroom settings.

* Learn how complex numbers may be used to solve algebraic equations, as well as their geometric interpretation

* Theoretical aspects are augmented with rich exercises and problems at various levels of difficulty

* A special feature is a selection of outstanding Olympiad problems solved by employing the methods presented

* May serve as an engaging supplemental text for an introductory undergrad course on complex numbers or number theory

This text treats the classical theory of quadratic diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area. The presentation features two basic methods to investigate and motivate the study of quadratic diophantine equations: the theories of continued fractions and quadratic fields. It also discusses Pell's equation and its generalizations, and presents some important quadratic diophantine equations and applications. The inclusion of examples makes this book useful for both research and classroom settings.

Questions of maxima and minima have great practical significance, with applications to physics, engineering, and economics; they have also given rise to theoretical advances, notably in calculus and optimization. Indeed, while most texts view the study of extrema within the context of calculus, this carefully constructed problem book takes a uniquely intuitive approach to the subject: it presents hundreds of extreme value problems, examples, and solutions primarily through Euclidean geometry.

Key features and topics:

* Comprehensive selection of problems, including...