Edward J. Barbeau Livres

Mistakes in mathematical reasoning, taken from a variety of sources, are exposed and analysed in this entertaining book.

Pell's equation is part of a central area of algebraic number theory that treats quadratic forms and the structure of the rings of integers in algebraic number fields. It is an ideal topic to lead college students and talented high school students to a better appreciation of the power of mathematical technique. To appreciate this focused exercise book, a high school background in mathematics is all that is needed, and teachers and others interested in mathematics who do not have a background in advanced mathematics may find that it is a suitable vehicle for keeping up an independent interest in the subject.

The book extends the high school curriculum and provides a backdrop for later study in calculus, modern algebra, numerical analysis, and complex variable theory. Exercises introduce many techniques and topics in the theory of equations, such as evolution and factorization of polynomials, solution of equations, interpolation, approximation, and congruences. The theory is not treated formally, but rather illustrated through examples. Over 300 problems drawn from journals, contests, and examinations test understanding, ingenuity, and skill. Each chapter ends with a list of hints; there are answers to many of the exercises and solutions to all of the problems. In addition, 69 "explorations" invite the reader to investigate research problems and related topics.

The book features a collection of problems from the first 15 annual undergraduate mathematics competitions at the University of Toronto, spanning topics such as calculus, linear algebra, and number theory. Problems are organized chronologically, with solutions provided in subsequent chapters categorized by subject. Additional appendices offer background material and recognize students who excelled in the competitions, making it a valuable resource for both learners and educators in mathematics.

This text records the problems given for the first 15 annual undergraduate mathematics competitions, held in March each year since 2001 at the University of Toronto. Problems cover areas of single-variable differential and integral calculus, linear algebra, advanced algebra, analytic geometry, combinatorics, basic group theory, and number theory. The problems of the competitions are given in chronological order as presented to the students. The solutions appear in subsequent chapters according to subject matter. Appendices recall some background material and list the names of students who did well. The University of Toronto Undergraduate Competition was founded to provide additional competition experience for undergraduates preparing for the Putnam competition, and is particularly useful for the freshman or sophomore undergraduate. Lecturers, instructors, and coaches for mathematics competitions will find this presentation useful. Many of the problems are of intermediate difficulty and relate to the first two years of the undergraduate curriculum. The problems presented may be particularly useful for regular class assignments. Moreover, this text contains problems that lie outside the regular syllabus and may interest students who are eager to learn beyond the classroom.