Real Analysis builds the theory behind calculus directly from the basic concepts of real numbers, limits, and open and closed sets in $\mathbb{R}^n$. It gives the three characterizations of continuity: via epsilon-delta, sequences, and open sets. It gives the three characterizations of compactness: as ``closed and bounded,'' via sequences, and via open covers. Topics include Fourier series, the Gamma function, metric spaces, and Ascoli's Theorem. The text not only provides efficient proofs, but also shows the student how to come up with them. The excellent exercises come with select solutions in the back. Here is a real analysis text that is short enough for the student to read and understand and complete enough to be the primary text for a serious undergraduate course. Frank Morgan is the author of five books and over one hundred articles on mathematics. He is an inaugural recipient of the Mathematical Association of America's national Haimo award for excellence in teaching. With this book, Morgan has finally brought his famous direct style to an undergraduate real analysis text.

Real Analysis and Applications starts with a streamlined, but complete approach to real analysis. It finishes with a wide variety of applications in Fourier series and the calculus of variations, including minimal surfaces, physics, economics, Riemannian geometry, and general relativity. The basic theory includes all the standard topics: limits of sequences, topology, compactness, the Cantor set and fractals, calculus with the Riemann integral, a chapter on the Lebesgue theory, sequences of functions, infinite series, and the exponential and Gamma functions. The applications conclude with a computation of the relativistic precession of Mercury's orbit, which Einstein called "convincing proof of the correctness of the theory [of General Relativity]." The text not only provides clear, logical proofs, but also shows the student how to come up with them. The excellent exercises come with select solutions in the back. Here is a text which makes it possible to do the full theory and significant applications in one semester. Frank Morgan is the author of six books and over one hundred articles on mathematics. He is an inaugural recipient of the Mathematical Association of America's national Haimo award for excellence in teaching. With this applied version of his Real Analysis text, Morgan brings his famous direct style to the growing numbers of potential mathematics majors who want to see applications right along with the theory.

A recreational-and-problem-solving math book, CHRISTmaths: A Creative Problem Solving Math Book attempts to bring together the joy (or spirit) of Christmas and the spirit (or joy) of mathematics. Looking at topics linking Mathematics and Christmas—what the queen of the sciences and the king of the public holidays have in common—CHRISTmaths will not only appeal to a Christmas or Christian audience, but also to any problem solvers who enjoy mathematics recreationally. CHRISTmaths should appeal to • creative problem solvers who are bored by drill-and-kill math titles, and who desire to get an intellectual kick out of solving non-routine questions; • mathletes who long for some creative mathematical problem solving to tickle their mathematical bones. CHRISTmaths hopes to give readers the opportunity to experience the Ah, Aha! and Ha Ha of Mathematics. Contents Preface Biodata of 25 B.C. and A.D. Are You Christmas-Literate? The 12 Puzzles of Christmas Santa’s Itinerary 12 Daffynitions of CHRISTMAS A CHRISTMAS Spell Guesstimation on Christmas Day 7 Beautiful Xmas Series 12 Challenges @ Christmastime A Mathematician’s Musings on Xmas Day Mathematical Graphiti I Xmas Philamath 12 Myths about Christ and Christmas Mathematical Graphiti II Mathematical Graphiti III 25 No-Frills Christmas Crackers Did You Know…. The Mathematics of Christmas 25 Mathematical Quickies & Trickies Was Pythagoras a pre-Christian Christian? A Formula for Christmas Day Q&A about Christmas Clausophobia and the Rest Mathematical Graphiti III Mathematical Graphiti IV Number of Zeros in 1 × 2 × 3 ×⋯× 24 × 25 25 Math Things You Can Do on Christmas 1 × 2 × 3 ×⋯× (n − 1) × n ends in 25 zeros Taking Up Your Cross Mathematicians Christened Number of Digits in 2525 Christmas Tangrams CHRISTMAS By Numbers What day Is Christmas in 2025? The Mathematical Fathers The Answer Is Not 25 Christmas Countdown A Christmas Potpourri CHRISTMAS Alphametics Mathematical Graphiti IV Celebrate Father Christmas Week 25 Illegal Things You May Want to Do on Xmas The Twelve Days of Christmas A Green Christmas Answers/Hints/Solutions Bibliography & References Type of e-book: Nonfiction, problem solving, recreational, Singapore math, trick questions Audiences: Suitable for Grades 5-10

The primary aim of this book is to provide teachers of mathematics with all the tools they would need to conduct most effective mathematics instruction. The book guides teachers through the all-important planning process, which includes short and long-term planning as well as constructing most effective lessons, with an emphasis on motivation, classroom management, emphasizing problem-solving techniques, assessment, enriching instruction for students at all levels, and introducing relevant extracurricular mathematics activities. Technology applications are woven throughout the text.A unique feature of this book is the second half, which provides 125 highly motivating enrichment units for all levels of secondary school mathematics. Many years of proven success makes this book essential for both pre-service and in-service mathematics teachers.

For a thousand years, infinity has proven to be a difficult and illuminating challenge for mathematicians and theologians. It certainly is the strangest idea that humans have ever thought. Where did it come from and what is it telling us about our Universe? Can there actually be infinities? Is matter infinitely divisible into ever-smaller pieces? But infinity is also the place where things happen that don't. All manner of strange paradoxes and fantasies characterize an infinite universe. If our Universe is infinite then an infinite number of exact copies of you are, at this very moment, reading an identical sentence on an identical planet somewhere else in the Universe. Now Infinity is the darling of cutting edge research, the measuring stick used by physicists, cosmologists, and mathematicians to determine the accuracy of their theories. From the paradox of Zeno’s arrow to string theory, Cambridge professor John Barrow takes us on a grand tour of this most elusive of ideas and describes with clarifying subtlety how this subject has shaped, and continues to shape, our very sense of the world in which we live. The Infinite Book is a thoroughly entertaining and completely accessible account of the biggest subject of them all–infinity.

Mathematical ideas with aesthetic appeal for any mathematically minded person.

The Early Mathematics of Leonhard Euler gives an article-by-article description of Leonhard Euler's early mathematical works; the 50 or so mathematical articles he wrote before he left St. Petersburg in 1741 to join the Academy of Frederick the Great in Berlin. These early pieces contain some of Euler's greatest work, the Konigsberg bridge problem, his solution to the Basel problem, and his first proof of the Euler-Fermat theorem. It also presents important results that we seldom realize are due to Euler; that mixed partial derivatives are (usually) equal, our f(x) f(x) notation, and the integrating factor in differential equations. The books shows how contributions in diverse fields are related, how number theory relates to series, which, in turn, relate to elliptic integrals and then to differential equations. There are dozens of such strands in this beautiful web of mathematics. At the same time, we see Euler grow in power and sophistication, from a young student when at 18 he published his first work on differential equations (a paper with a serious flaw) to the most celebrated mathematician and scientist of his time. It is a portrait of the world's most exciting mathematics between 1725 and 1741, rich in technical detail, woven with connections within Euler's work and with the work of other mathematicians in other times and places, laced with historical context.