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An introduction for readers with some high school mathematics to both the higher and the more fundamental developments of the basic themes of elementary mathematics. Chapters begin with a series of elementary problems, cleverly concealing more advanced mathematical ideas. These are then made explicit and further developments explored, thereby deepending and broadening the readers' understanding of mathematics. The text arose from a course taught for several years at St. Petersburg University, and nearly every chapter ends with an interesting commentary on the relevance of its subject matter to the actual classroom setting. However, it may be recommended to a much wider readership; even the professional mathematician will derive much pleasureable instruction from it.
Designed for students preparing to engage in their first struggles to understand and write proofs and to read mathematics independently, this is well suited as a supplementary text in courses on introductory real analysis, advanced calculus, abstract algebra, or topology. The book teaches in detail how to construct examples and non-examples to help understand a new theorem or definition; it shows how to discover the outline of a proof in the form of the theorem and how logical structures determine the forms that proofs may take. Throughout, the text asks the reader to pause and work on an example or a problem before continuing, and encourages the student to engage the topic at hand and to learn from failed attempts at solving problems. The book may also be used as the main text for a "transitions" course bridging the gap between calculus and higher mathematics. The whole concludes with a set of "Laboratories" in which students can practice the skills learned in the earlier chapters on set theory and function theory.
The new edition of Pearson Edexcel GCSE (9-1) Mathematics Higher Student Book 1 develops reasoning, fluency and problem-solving to boost students’ confidence and give them the best preparation for GCSE study. Purposefully updated based on feedback from thousands of teachers and students, as well as academic research and impact studies Bolsters preparation for GCSE with new questions that reflect the latest exams and a format that seamlessly aligns with our GCSE Maths courses Shown to help GCSE students master maths with confidence with a UK-specific approach that draws upon global best practices and cutting-edge research Tried-and-tested differentiation with a unique unit structure and improved pacing to support every student’s progress Extra skills-building support, problem-solving, and meaningful practice to consolidate learning and deepen understanding New additions to boost progression and post-GCSE study such as ‘Future skills questions’ and ‘Working towards A level’ features
A Bridge to Higher Mathematics is more than simply another book to aid the transition to advanced mathematics. The authors intend to assist students in developing a deeper understanding of mathematics and mathematical thought. The only way to understand mathematics is by doing mathematics. The reader will learn the language of axioms and theorems and will write convincing and cogent proofs using quantifiers. Students will solve many puzzles and encounter some mysteries and challenging problems. The emphasis is on proof. To progress towards mathematical maturity, it is necessary to be trained in two aspects: the ability to read and understand a proof and the ability to write a proof. The journey begins with elements of logic and techniques of proof, then with elementary set theory, relations and functions. Peano axioms for positive integers and for natural numbers follow, in particular mathematical and other forms of induction. Next is the construction of integers including some elementary number theory. The notions of finite and infinite sets, cardinality of counting techniques and combinatorics illustrate more techniques of proof. For more advanced readers, the text concludes with sets of rational numbers, the set of reals and the set of complex numbers. Topics, like Zorn’s lemma and the axiom of choice are included. More challenging problems are marked with a star. All these materials are optional, depending on the instructor and the goals of the course.
Please note this title is suitable for any student studying: Exam Board: AQA Level: GCSE Subject: Mathematics First teaching: September 2015 First exams: June 2017 AQA GCSE Maths, Higher Student Book has been approved by AQA and specially written by a team of maths experts for the Higher tier of AQA's 2015 GCSE specification. Designed to fully support the new style of assessment, the book adopts a clear style to focus on delivering exam success via the careful development of fluency and problem solving practice. Powered by MyMaths the book links directly to the ever popular web site offering students a further source of appropriate support.
A new series of bespoke, full-coverage resources developed for the 2015 GCSE Mathematics qualifications. Endorsed for the OCR J560 GCSE Mathematics Higher tier specification for first teaching from 2015, this Student Book provides full coverage of the new GCSE Mathematics qualification. With a strong focus on developing problem-solving skills, reasoning and fluency, it helps students understand concepts, apply techniques, solve problems, reason, interpret and communicate mathematically. Written by experienced teachers, it also includes a solid breadth and depth of quality questions set in a variety of contexts. GCSE Mathematics Online - an enhanced digital resource incorporating progression tracking - is also available, as well as Problem-solving Books, Homework Books and a free Teacher's Resource.
This book is written for students who have taken calculus and want to learn what "real mathematics" is.
Written by an experienced Soviet teacher, Candidate of Physico-mathematical sciences, I. F. Suvorov, this text represents a concise course in higher mathematics, with problems and exercises arranged in the same sequence as the theoretical material. It is designed for intermediate and higher educational institutions with short courses of mathematics and also for self-instruction.
This book has been specifically edited for the two-tier GCSE specification by the highly successful author, David Rayner. The book is targeted at the Higher tier GCSE, and it comprises units organised clearly by topic. Each unit offers: BL Summary of objectives at the start so it is clear what students need to know BL Clear explanations with examples showing the key techniques BL Plenty of practice with clearly differentiated questions pitched at an appropriate level BL Summaries and past exam questions to help students gain responsibility for their learning It forms part of a series of two student books at GCSE, in which the other book caters for the Foundation tier.
A new series of bespoke, full-coverage resources developed for the 2015 GCSE Mathematics qualifications. Written for the AQA GCSE Mathematics Foundation tier specification for first teaching from 2015, our Homework Book is an ideal companion to the AQA Foundation tier Student Book and can be used as a standalone resource. With exercises that correspond to each section of the Student Book, it offers a wealth of additional questions for practice and consolidation. Our Homework Books contain a breadth and depth of questions covering a variety of skills, including problem-solving and mathematical reasoning, as well as extensive drill questions. Answers to all questions are available free on the Cambridge University Press UK Schools website.