Books like How to think like a mathematician : a companion to undergraduate mathematics by Kevin Houston

Concrete Mathematics introduces the mathematical foundations that underpin advanced computing and algorithm analysis, offering essential techniques for solving complex problems and uncovering patterns in data.

Mathematics hold a bad reputation, but they are a necessity. Their bad reputation can be attributed to how poorly we are being taught mathematics. American teachers have realized how counterproductive classical methodologies are, in which problems are presented with an air of mystery so as not to not detract the "wisdom" of those who disclose them for us, and have gracefully dealt with this problem, making mathematics look pleasant and familiar. This allows many technicians, intellectuals and parents who do not want to fall behind with regard to their children, because they have forgotten some concepts, dare again to reconsider math. But ... How? Older systems seem arid and people eagerly seek new easy and effective methods. Hence the success, both in Europe and in America, of works like this one we are offering our readers. **How has it come to this simplicity?** ... based on the idea that in mathematics the only quality that must be possessed is to understand that everything makes sense. Once this premise is established, the author starts from known elements to deduce consequences instead of making statements, all using everyday language. The first chapter is worked out with a pencil and paper. Within a few (exciting, of course)minutes the reader will be ready for practical exercises. To avoid a misstep, this book provides the entire resolution process of each excercise and its result as well. It is a true assimilation method, similar to those already used to learn languages, drawing or radio. Without producing heaviness, the mechanism of the exponential and the binary system, by numbering systems, is shown. From exponentials to radicals, which progress is made from pure algebra towards first equations, logarithms, trigonometry and first integrals. Functions of geometry, hyperbolas, parabolas, etc., will allow the reader to easily solve equations using graphs. **Do you know how to add?** This is the only essential knowledge: mathematics are simple arithmetic, but for this statement to be true it must get rid of all that is opposed to its understanding. In saying this we do not mean its natural difficulties, because they are resolved skillfully by the author. Formulas should not be memorized: the brain must not be turned into a passive registry; it must understand the "whys and wherefores." This book explains how to achieve a right formula so that, then you yourself can find those submitted within the text. **Its practical purpose** The book has been given a deliberately oriented practice and that minimum of essential theory included tends to faciltate methods that may be used any time anywhere.

This is a text in basic mathematics with multiple uses for either high school or college level courses. Readers will get a firm foundation in basic principles of mathematics which are necessary to know in order to go ahead in calculus, linear algebra or other topics. The subject matter is clearly covered and the author develops concepts so the reader can see how one subject matter can relate and grow into another.

Introduction to the Problems in Analysis outlines an elementary, one semester course which exposes students to both the process of rigor, and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Does the Cantor set contain any irrational numbers? Can the set of points where a function is discontinuous be arbitrary? Can the rational numbers be written as a countable intersection of open sets? Is an infinitely differentiable function necessarily the limit of its Taylor series? Giving these topics center stage, the motivation for a rigorous approach is justified by the fact that they are inaccessible without it.

This arsenal of tips and techniques eases new students into undergraduate mathematics, unlocking the world of definitions, theorems, and proofs.

Problem-Solving Strategies is a unique collection of competition problems from over twenty major national and international mathematical competitions for high school students. The discussion of problem-solving strategies is extensive. It is written for trainers and participants of contests of all levels up to the highest level: IMO, Tournament of the Towns, and the noncalculus parts of the Putnam competition. It will appeal to high school teachers conducting a mathematics club who need a range of simple to complex problems and to those instructors wishing to pose a "problem of the week," "problem of the month," and "research problem of the year" to their students, thus bringing a creative atmosphere into their classrooms with continuous discussions of mathematical problems. This volume is a must-have for instructors wishing to enrich their teaching with some interesting nonroutine problems and for individuals who are just interested in solving difficult and challenging problems.