Bøger i Dolciani Mathematical Expositions serien

A Cornucopia of Quadrilaterals collects and organizes hundreds of beautiful and surprising results about four-sided figures-for example, that the midpoints of the sides of any quadrilateral are the vertices of a parallelogram, or that in a convex quadrilateral (not a parallelogram) the line through the midpoints of the diagonals (the Newton line) is equidistant from opposite vertices, or that, if your quadrilateral has an inscribed circle, its center lies on the Newton line. There are results dating back to Euclid: the side-lengths of a pentagon, a hexagon, and a decagon inscribed in a circle can be assembled into a right triangle (the proof uses a quadrilateral and circumscribing circle); and results dating to Erdos: from any point in a triangle the sum of the distances to the vertices is at least twice as large as the sum of the distances to the sides. The book is suitable for serious study, but it equally rewards the reader who dips in randomly. It contains hundreds of challenging four-sided problems. Instructors of number theory, combinatorics, analysis, and geometry will find examples and problems to enrich their courses. The authors have carefully and skillfully organized the presentation into a variety of themes so the chapters flow seamlessly in a coherent narrative journey through the landscape of quadrilaterals. The authors' exposition is beautifully clear and compelling and is accessible to anyone with a high school background in geometry.

Explores recurrent phenomena, including astronimcal transits and conjuctions, lifecycles of cicadas, and eclipses, by way of continued fraction expansions. The deeper purpose is to find patterns, solve puzzles, and discover some appealing number theory.

Challenge: Can you find all the integers $a$, $b$, $c$ satisfying $2a^{2} + 3b^{2} = 5c^{2}$? Looks simple, and there are in fact a number of easy solutions. But most of them turn out to be anything but obvious! A major advance in number theory means this book can give an easy answer to this and countless similar questions.

The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there is a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman.

Offers an introduction to algebraic number theory for readers with a moderate knowledge of elementary number theory and some familiarity with the terminology of abstract algebra. With its exceptionally clear prose, hundreds of exercises, and historical motivation, this is a textbook for a second undergraduate course in number theory.

A collection of 90 short elementary gems from algebra, geometry, combinatorics, and Number Theory.

Represents the fruits of 15 years of work in geometry by prize-winning authors Tom Apostol and Mamikon Mnatsakanian. Using new and intuitively rich methods, they give beautifully illustrated proofs of results, the majority of which are new, and frequently develop extensions of familiar theorems.

A beautifully illustrated collection of striking and original results in geometry. Recommended for students and teachers of geometry and calculus.

An exploration of the mathematics of twenty geometric diagrams that play a crucial role in visualizing mathematical proofs.

An accessible introduction to plane algebraic curves that also serves as a natural entry point to algebraic geometry.

A concise but accessible guide to functional analysis that begins with the basics before moving on to more advanced topics.

A concise guide to support an undergraduate real analysis course.

A collection of over 250 multiple-choice problems to challenge and delight everyone from school students to professional mathematicians.

A guide to the hidden byways of algebra, calculus and geometry, for the seasoned mathematical traveller.

Introduces the richness and variety of inequalities in mathematics using illustration and visualisation.

Describes techniques for solving problems in maxima and minima other than the methods of calculus.