Introductory Mathematics: Algebra and Analysis (Springer Undergraduate Mathematics Series)
by Geoffrey C. Smith
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Paperback (02 February, 2000)
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Reviews (3)

An excellent introduction to analysis and algebra
If you want to understand mathematics after high school you must read this book first before plunging into university level mathematics. It will make your journey in mathematics much easier.

A lucid, laugh-a-minute delight
Here is an author who brings a fresh vitality to basic undergraduate mathematics with his witty, discursive treatment of topics ranging from the nature of a proof to a kit for the construction of the real numbers.

Smith adopts a tone of camaraderie which lures the reader along a trailof mathematical discovery, all the while remaining true to the standards ofexactitude which must characterise mathematics at this level. Nevercondescending, he blithely raises issues which other authors would sweepunder the carpet in a beginners' textbook, treating them with hisinimitable forthright professionalism.The author's exuberant interest andkeen mind pervade the book, making it a perfect delight to read.

Iwould recommend that all young undergraduates/school-leavers are given thisbook before more traditional dusty texts blind them to the sheer pleasureof mathematics.

A first-rate and funny introduction to algebra and analysis
As a returning novice to university mathematics, (it is some15 years since the last time I last took an introductory maths course) I found this book to be both refreshing and highly original.The material and layout isdifferent to most textbooks.It is probably a book for people who want tograsp the idea of mathematics rather than just pass an exam. However, asthe author notes in the preface, it is (other things being equal) both a'gentle and relaxed introduction'.The mathematics is pure and theemphasis is on the idea rather than on how to solve particular problems inthe life sciences or engineering.Topics covered include; Sets, functionsand relations; Proofs; Complex numbers; Vectors and matrices; Group theory;Sequences and series; Real numbers; and Mathematical analysis.It is anexcellent book for those interested in learning and understandingmathematics.The colloquial tone helps alot and it is a style thatdeserves to become more common in the future.The book also offers aninteresting glimpse of the mathematical mind.The author has a remarkablesense of humour and the book is hilarious at times. ... Read more

Isbn: 3540761780
Sales Rank: 852532
Subjects:  1. Algebra    2. Algebra - General    3. General    4. Mathematical Analysis    5. Mathematics    6. Science/Mathematics    7. Mathematics / Mathematical Analysis   

Reading, Writing, and Proving : A Closer Look at Mathematics (Undergraduate Texts in Mathematics)
by Ulrich Daepp, Pamela Gorkin
Average Customer Review:
Hardcover (07 August, 2003)
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Agood text for an introduction to proofs course
I have used the Daepp, Gorkin text twice, for an introduction to proofs type of course. This course is usually taken by Math and Computer Science majors after Calculus and either with or after a course in Linear Algebra. This type of course was not in existence when I was a student, in the 70's. In those days, there was some proofs in Calculus (certainly some delta-epsilon type arguments were given) and Linear Algebra was much more proof oriented. Hence, most math majors picked up the ability to read and learn abstract mathematics during the first two years. These days, Linear Algebra has become a course in row-reducing matrices and very little abstraction takes place. Hence there is a real need for a course (and texts) to pave the way for courses in Analysis and Abstract Algebra. The Daepp, Gorkin text compares favorably to all similar texts I have looked at and it is priced reasonably. I passed on another text that I liked because it was $125, which is ridiculous for studenst who are not wealthy. On the plus side, this text covers all the material you would need in such a course and, in fact, there are several avenues open to the instructor of a one semester course. Besides the usual material on sets and mappings, there are chapters on cardinality issues, intoductory analysis ideas and slightly more advanced topics in number theory. The chapters are short and "digestable." There are some possible independent research topics at the end of the text. On the negative side, the examples given in the text are mostly all drawn from the standard number systems. This makes it harder to motivate basic concepts of sets and mappings. Why not give some examples from sets of mappings (e.g. the composite of two odd functions is odd, does a better job of teaching about composition than just composing two standard function), 2 by 2 matrices, and some examples from Calculus (e.g. the derivative, viewed as a function from degree n polynomials to degree n-1 polynomials is a non one-to-one, onto map)? The Division Algorithm is not mentioned until page 315 and it is not proved and there is no discussion on the Fundamental Theorem of Arithmetic. In fact, the number theory is a bit strange. While there are proofs of Fermat and Euler's Theorems, they omit more elementary number theory needed to get to these results. The Binomial Theorem is left as an exercise and no applications are given. As an application, I show my classes tha the sequence (1+1/n)^n is increasing and bounded above. Also, the exercises are a little strange. Too many are very easy and there are not enough basic practice types or challenge types. I would eliminate the logic chapters in these texts. They really don't help with the math and their elimination would create more room for math topics.
I would recommend this book, but warn the reader and/or teacher, that some supplementing would be needed. ... Read more

Isbn: 0387008349
Sales Rank: 552626
Subjects:  1. General    2. Logic    3. Mathematical Analysis    4. Mathematics    5. Number Theory    6. Science/Mathematics    7. Study and teaching (Higher)    8. Technical writing    9. United States    10. Mathematics / Logic   

An Accompaniment to Higher Mathematics (Undergraduate Texts in Mathematics)
by George R. Exner
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Paperback (22 June, 1999)
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Reviews (1)

Good book on proofs
I really appreciate "An accompaniement to higher mathematics", because it presents a great amount of stuff concerning the technique of proof. The first chapter present why examples are so importants by showing how to test definitions, how to construct good and bad examples and how to test the validity of a theorem by the mean of extreme example. The material in this chapter is basic and easy to follow. The second chapter is about the infirmal language and some technics of proof. First, it presents the logic behind the proofs. Also, this chapter presents technics such as induction, proof by case, differents forms of proof based on implication. The third chapter is about the use of quantifier in the proof. It shows how to use and when to use quantifier. Also how to find the structure of a proof. Why I find this book interresting ? First the book is full of exercises of different kind (set, function, analysis, ...and the forth chapter contain laboratories that give you again plenty of exercises), It is written clearly, the author give a lot of advice about proofs, I find the book very suitable to undergraduted, I find the style of writting of the book very motivating. This book is definitely a good one but it is not perfect. I found some of the exercises too easy (about 30 % of the exercises), I don't think that this book is suitable for graduate student but it may help in the way you work proofs and problems. Also this book give me some help in analysis course. I recommend this book for anyone who want to learn the basic and more about proofs. ... Read more

Isbn: 0387946179
Sales Rank: 559095
Subjects:  1. Logic    2. Mathematical And Symbolic Logic    3. Mathematics    4. Philosophy Of Mathematics    5. Proof theory    6. Science/Mathematics    7. Mathematics / Mathematical Analysis   

Proofs and Fundamentals : A First Course in Abstract Mathematics
by Ethan D. Bloch
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Hardcover (20 April, 2000)
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Reviews (1)

an excellent introduction to formal mathematics
This book is an excellent and thorough intorduction to the world of formal mathematics. Typically, a new mathematics student finds himself or herself picking up random mathematical concepts and techniques of proof along the way, without ever having the chance to sit down and go through the fundamentals of formal mathematics and proof. This book covers a little of everything, and has a thorough introduction to sets, functions, inverses, equivalence and order relations. There are also sections on introductory number theory, algebra, combinatorics, logic, and much more. It provides an excellent overview for the student who will be using these tools on a daily basis, for the layman who is interested in finding out what mathematics is really about, or for the seasoned mathematician who needs a good general reference book. There are also extensive and thorough sections on the construction and writing of mathematical proofs -- somthing on which many new and not-so-new mathematicians could use some improvement. ... Read more

Isbn: 0817641114
Sales Rank: 150222
Subjects:  1. Logic    2. Mathematical And Symbolic Logic    3. Mathematics    4. Proof theory    5. Science/Mathematics    6. Set Theory    7. Mathematics / Logic    8. abstract mathematics    9. proofs   

Mathematical Proofs : A Transition to Advanced Mathematics
by Gary Chartrand, Albert D. Polimeni, Ping Zhang
Average Customer Review:
Hardcover (28 May, 2002)
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Reviews (2)

Mathematical Proofs:A Transition to Advanced Mathematics
I purchased this book as a preparation to take advanced courses in mathematics.It is well laid out and explains the material clearly.I especially liked the chapter on "proofs found in calculus" which explains delta epsilon proofs in a manner that is easy and understandable.Big thumbs up!

Very Readable Textbook
This book is designed to prepare students for upper division math courses-like abstract algebra and advanced calculus-in which mathematical rigor and proofs are emphasized. The authors have made a serious effort to present the material with clarity and sufficient details to make it accessible to students who have completed two courses in calculus.Much of the material covered is fairly standard for such a textbook.Chapters 1-9 are devoted to basic topics from set theory and logic (including four proof techniques: direct proof, proof by contrapositive, proof by contradiction, and mathematical induction), equivalence relations, and functions, as well as a special chapter under the heading, "Prove or Disprove."Chapters 10-13 cover cardinalities of sets and proof techniques applied to results from number theory, calculus, and group theory.In addition, the authors have a web site which includes three additional chapters (Chapters 14-16) dealing with proofs from ring theory, linear algebra, and topology.Thus instructors using this book will have a wide choice of options in selecting the material they want to include after the basic concepts are covered.

The emphasis throughout the book is on proofs and proof techniques--how to recognize proofs, understand them and, above all, how to create and write them.The presentation is leisurely and thorough.Many examples are given, and discussions are always presented with all the details that students at this level would need to follow the argument.There are ample exercises at the end of each chapter (including those in the web site) that range in difficulty from routine to moderately challenging.The book also contains answers and hints to odd-numbered exercises.

There are two features of this textbook that I believe are helpful to students and that set this book apart from others at its level: the detailed way in which proofs are analyzed, and the inclusion of a chapter on how to write mathematics well.In most cases, before a proof is presented the authors offer a "proof strategy": a discussion pointing out what needs to be proved and how one might go about proving it.Also, many proofs are followed by "proof analyses" in which some of the interesting or unusual points of the proof are commented on.I believe that students would find these discussions very helpful.In particular, these discussions offer students concrete pointers from which they would learn how to cope with abstract mathematical proofs.

The chapter on writing mathematics (Chapter 0) is unique.While some mathematics textbooks encourage good writing and might devote a few paragraphs to the subject, the present volume offers a brief manual on mathematical writing.The authors begin by explaining why writing is important in mathematics and follow that by offering detailed instructions that would help students in improving their writing.From specific advice like, "Never start a sentence with a symbol" to explanations of "common words and phrases that are peculiar to mathematics," there is a wealth of material on writing from which students can learn.I believe that, by its very existence, this chapter on writing would have a positive influence on students writing.

This book can be used either as a textbook for a course such as the one described above or as a reference that students can consult on certain topics.

Fawzi M. Yaqub
Emeritus Professor of Mathematics
SUNY College at Fredonia ... Read more

Isbn: 0201710900
Sales Rank: 410796
Subjects:  1. Logic    2. Mathematical And Symbolic Logic    3. Mathematics    4. Philosophy Of Mathematics    5. Proof theory    6. Science/Mathematics    7. Set Theory    8. Mathematics / Advanced   

What Is Mathematics?: An Elementary Approach to Ideas and Methods (Oxford Paperbacks)
by Richard Courant, Herbert Robbins, Ian Stewart
Average Customer Review:
Paperback (01 May, 1996)
list price: $21.50 -- our price: $15.05
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Editorial Review

A 1996 revision of a timeless classic originally published in 1941. Highly recommended for any serious student, teacher or scholar of mathematics. ... Read more

Reviews (19)

the best bargain in introductory math books in existence
This book genuinely has more mathematical content, for around $15-$25, than most, maybe all, "bridge" texts for college math majors, costing 5 or 10 times as much.

This book was written by a master, for an intelligent person knowing only 1950's style high school mathematics (some trig, algebra, and geometry).

When I fiorst tried to read it as a youngster however I was not used to books that required actually thinking about each statement, before proceeding to the next. Hence I could not read it at the pace I thought normal.

So this is not a breezy read, but is an outstanding one.It has literally no competitor to my knowledge at the present time, in quantity of material, quality of material, and quality of exposition.

Even experts may learn something here about the most familiar topics.E.g. in presenting the proof of the well known fact that all integers greater than one have unique prime factorizations, the authors show how a clever use of induction avoids developing the characterization of a gcd, which usually precedes this theorem.I had never seen that before.

If you are looking for a miracle book that treats the reader like a baby, and still covers calculus, this is not it.But if you have the prerequisites of a good high school course of elementary math, and are willing to spend time on the arguments, there is no better book for beginners and intelligent laypersons.

Pretty good
For the most part I love this book. It is informative, and relatively simple to understand. This book is an "elementary approach to ideas and methods" for the whole field of mathmatics. In fact, this book is one of the reasons I changed my major to mathmatics.

However, there are two main problems with this book. First the quality of the print varies. Occasionally, whole sets of subscripts are blurred, which makes understanding the equation of the moment difficult, if not impossible.

Second, the order of steps for solving or understanding a problem are in an unexpected order, which is confusing. Often, I find that a difficult passage doesn't deal with difficult concepts, its just that the concepts are explained in an unusual way.

Aside from those problems, this is an extraordinary introduction to mathmatics.

Not an 'easy' read
This is a demanding book. One cannot read it listening to Bach. [a clash of complexities]. The construction of he book is `old style' [which is every seeming possible variation is mentioned] which has fallen into disfavor as confusing,

That written it is very complete and I really enjoyed many parts of this book. ... Read more

Isbn: 0195105192
Subjects:  1. General    2. History & Philosophy    3. Mathematics    4. Mathematics (General)    5. Science/Mathematics   

An Introduction to Mathematical Reasoning : Numbers, Sets and Functions
by Peter J. Eccles
Average Customer Review:
Paperback (11 December, 1997)
list price: $36.71 -- our price: $36.71
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Reviews (6)

Very well written book
I have a mathematics degree. Like most math majors, I struggled with proofs all through college. This book really has help me understand the art of writing proofs. The book is very well written and easy to read. This is just an awesome book!!!

Now I know how beautiful proofs can be
This book provides a nice introduction to mathematical reasoning and proofs. My intention on purchasing this book was to learn how to perform mathematical proofs. I believe it has achieved that purpose. The text is easy to follow and the author presents the work clearly.

Just buy this
I needed a book that covered fundamental background information behind mathematical proof techniques for an undergraduate univeristy level linear algebra class.

With this book, I was able to truly learn and understand the major concepts behind mathematical logic and proof. This text brings a whole new meaning to teaching the reader about being precise; and I mean the author does an extremely terrific job of doing just that. Wow!

Seriously, the focus here is on content so you won't find any sexy graphs or anything. The content is so good that I often felt that just by reading it I was propelled into a quasi- pseudo-lecture meeting.

After following this text, I can say that I now appreciate the act of being precise to the point that is required for mathematical proof. If you want to extend the knowledge of your 'white board' then just buy this thing. I am so glad I did.

BTW, I only needed the content from the first five chapters, I can't say much about the rest of the text. However, taking an inductive approach, I must assume that the other chapters are also very excellent. Yess, see it worked! ... Read more

Isbn: 0521597188
Sales Rank: 237922
Subjects:  1. Logic    2. Mathematics    3. Number Systems    4. Number Theory    5. Proof theory    6. Science/Mathematics    7. Set Theory    8. Mathematical foundations    9. Mathematics / General