Should I Become a Mathematician?

[AnTiFreeze3]

Not entirely sure if that was directed towards me, RJinkies, but if so, then that's more or less what I'm doing now, just maybe not to the extent of 2-4 hours a day. I'm still putting in time every day, but I'm not drowning myself in math.

I ultimately want to know if understanding proofs gets easier as you go on, or if there are any good books dedicated to the better understanding of proofs.

 

[micromass]

Is it normal to have to spend a relatively long time understanding proofs in a math book? I'm currently going through Basic Mathematics by Serge Lang (recommended to me by MicroMass), and I've noticed throughout the book that, besides the exercises, I'm spending the majority of my time rereading and going over proofs.

Yes, absolutely. I think you're doing it right, because that is how I study (not that my methods of studying are the best, it's very personal). In my case, I would study the proofs until I knew them inside out. That does not mean: memorizing the proof, but rather memorizing the method and trying to see if it works in other circumstances.

When I study, I always ponder about the theorems and proofs, for example, I ask myself:
- Did I use all the assumptions of the theorem? Were some assumptions unnecessary?
- Is the converse of the theorem true?
- Can I think of an actual example that illustrates the theorem?
- Is the method of proof used a lot? In what circumstances can I use it?
- How could I describe the theorem/proof in one sentence?
- What is the intuition behind the theorem?

These kind of questions are really helpful (to me).

Is understanding proofs just a skill that you develop over time, or would it be beneficial for me to pick up a math book that is solely made for better understanding proofs? If so, what would be a good book that would help me out with understanding proofs?

Yes, your proof skills will develop over time. The more proofs you actually do (and find yourself!), the better you will be at it.
I'm not really a big fan of proof books, as they isolate the proof from their natural context. It would probably be more benificial to read a good book on logic/set theory. Nevertheless, some good books are:

- "How to Think Like a Mathematician: A Companion to Undergraduate Mathematics" by Houston
- "Journey into Mathematics: An Introduction to Proofs" by Rotman
- "How to prove it: A structured approach" by Velleman

 

[Infinitum]

Not entirely sure if that was directed towards me, RJinkies, but if so, then that's more or less what I'm doing now, just maybe not to the extent of 2-4 hours a day. I'm still putting in time every day, but I'm not drowning myself in math.

I ultimately want to know if understanding proofs gets easier as you go on, or if there are any good books dedicated to the better understanding of proofs.

I read this book a couple years back, and I have thoroughly enjoyed it.

How to solve it, By Polya. http://www.amazon.com/dp/0140124993/?tag=pfamazon01-20

 

[RJinkies]

well, my guess there's two ways of doing it...

a. picking extra gentle books on analysis when starting out

b. getting 1 or 2 of the half dozen books on how to do proofs, which can start off as a slow and frustrating path for many, but if you get a book who's style speaks to you, that's another way.

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Here's some of my notes
[aka stuff i cut and pasted off the web]


- Introductory Mathematics: Algebra and Analysis (Springer Undergraduate Mathematics Series) (Paperback) - Geoffrey C. Smith - Second Corrected Edition - Springer 1998 - 216 pages

[The material and layout is different to most textbooks. It is probably a book for people who want to grasp the idea of mathematics rather than just pass an exam. As the author notes in the preface it is a 'gentle and relaxed introduction'. The mathematics is pure and the emphasis is on the idea rather than on how to solve particular problems in the life sciences or engineering. Topics covered include; Sets, functions and relations; Proofs; Complex numbers; Vectors and matrices; Group theory; Sequences and series; Real numbers; and Mathematical analysis. It is an excellent book for those interested in learning and understanding mathematics. The book also offers an interesting glimpse of the mathematical mind.]

[A splendid introduction to the concepts of higher mathematics]

[Geoff Smith's Introductory Mathematics: Algebra and Analysis provides a splendid introduction to the concepts of higher mathematics that students of pure mathematics need to know in upper division mathematics courses.]

[The text begins with material on set theory, logic, functions, relations, equivalence relations, and intervals that is assumed or briefly discussed in all advanced pure mathematics courses. Smith then devotes a chapter to demonstrating various methods of proof, including mathematical induction, infinite descent, and proofs by contradiction. He discusses counterexamples, implication, and logical equivalence. However, the chapter is not a tutorial on how to write proofs. For that, he suggests that you work through D. L. Johnson's text Elements of Logic via Numbers and Sets (Springer Undergraduate Mathematics Series).]

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- D. L. Johnson's - Elements of Logic via Numbers and Sets

so if you wanted to piece together an baby analysis library for self-study

you could do

1. - Introductory Mathematics: Algebra and Analysis - Springer - Geoffrey C. Smith
2. D. L. Johnson's - Elements of Logic via Numbers and Sets

supplemented with:
a. Bartle - Introduction to Real Analysis - 3ed - Wiley 2000 - Chapters 1-3
b. Burn - Numbers and Functions, Steps into Analysis - Cambridge 2000 - Chapter 1–6
[This is a book of problems and answers, a DIY course in analysis.]
c. Howie - Real Analysis - Springer 2001
supplemented by:
d. Mary Hart - A Guide to Analysis - MacMillan 1990 - Chapter 2 - too gentle
e. Burkill - A First Course In Mathematical Analysis - Cambridge 1962 - Chapters 1, 2 and 5 - too gentle
f. Binmore - Mathematical Analysis, A Straightforward Approach - Cambridge 1990 - Chapters 1–6 - too gentle
g. Bryant - Yet Another Introduction to Analysis - Cambridge 1990 - Chapter1,2 - too gentle
h. Smith - Introductory Mathematics: Algebra and Analysis - Springer 1998 - Chapter 3 - too gentle
i. Michael Spivak - Calculus - Benjamin 1967 - Parts 1,4,5 - more advanced
j. Bruckner, Bruckner and Thomson - Elementary Analysis - Prentice Hall 2001 - Chapter 1–4 - more advanced]

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If you took a class in calculus and didnt know anything about proofs, another way could be:

- Advanced Calculus: A Friendly Approach - Witold A.J. Kosmala - Prentice-Hall - 700 pages - 1998

[I have copies of Rudin, Apostol, Bear, Fulks, and Protter, but this book beats them all as an introductory text. If you are looking for a self-study text, or if you want a reference companion to help you understand Rudin or Apostol, try this book first. You won't be disappointed.]
[The author of this book has used " a friendly approach " to present the stuff so that readers will actively be engaged in learning with less strain. This has not in a sense simplified the difficult elements of Calculus but bringing along the readers to think and reason while studying the subject.]

[Designed to be readable and intimidation-free, this advanced calculus book presents material that flows logically allowing readers to grasp concepts and proofs. Providing in-depth discussion of topics, the book also features common errors to encourage caution and easy recall of errors. It also presents many proofs in great detail and those which should not provide difficulty are either short or simply outlined.]

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something mathwonk said a few years ago is in my note with another book...

- Allendoerfer, C.B. and Oakley, C.O. Principles of Mathematics - McGraw-Hill 1963
[MAA recommendation] - Calculus and Precalculus: School Mathematics
[mathwonk recommended this for help with logic and reading proofs and writing proofs]

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Mathematical Analysis: A Modern Approach to Advanced Calculus - Second Edition - Tom M. Apostol - Addison-Wesley 1957/1974

my freaky notes has this remark about Apostol's book:

[This book is more detailed, and the dependency of the material is less strict - it's easier to open this book to a specific topic and understand it without having to cross-reference earlier theorems.]

What you'll need to acquaint yourself with is:

a) learning math on your own. You need to be able to sit down with a textbook, read it, understand every line, and be able to apply it. This is very hard for most folks in college. As a college student, your job is to teach yourself. The professor only facilitates. Most people not only don't know this, they also have the very hardest time teaching themselves math.

b) you need a gentle introduction to proofs. The bright folks can and do figure out simple proofs on their own. Most high school and elementary college math completely omits proofs (because students balk). As a result, very basic things about proofs are not completely understood by the bright math student starting out. You need to bone up on this stuff - at first, it will seem really simple, maybe even an insult to your intelligence. It is not. Spending just a few weeks understanding very elementary proving techniques, learning all of the abstract terminology and rules about sets, logic, etc., will be truly invaluable to you.

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a note i got on Bartle

- Introduction to Real Analysis, Third Edition*- Robert G. Bartle and Donald R. Sherbert - Wiley 1999

[Way better than Pugh. Don't let real analysis be your first proofing class - do your first proofs in elementary number theory or geometry, then when you have a repertoire of proofing tools and some skill in proofing, then take real analysis. You cannot learn proofing and real analysis at the same time. First learn to proof, then take real analysis. If not you will be miserable]

Nice Preparation before Real Analysis might be:
[a. Polya - How to Solve It - [problem solving strategies]
[b. Velleman - How to Prove It - [technique to work out proofs]
[c. Bryant - Yet Another Introduction to Analysis [a good grasp of fundamentals in analysis]
[Plough through Bartle first, then consult Rudin. It's a bit easier that way.]

------------

and...

- The Way of Analysis (Jones and Bartlett Books in Mathematics)*- Robert S. Strichartz

[This textbook on real analysis is intended for a one- or two-semester course at the undergraduate or beginning graduate level. It gives a thorough account of real analysis in one or several variables, from the construction of the real number system to an introduction to the Lebesgue integral. Written in a lively and informal style, the text provides proofs of all the main results, as well as motivations, examples, applications, exercises, and formal chapter summaries.]

[This is the kind of textbook you can bring with you on a car trip and easily study along the way. It takes an informal writing style and from the beginning is focused on making sure you, as the reader, understand not just the theorems and proofs, but the concepts of real analysis as well. Every new idea is given not only with a What or a How, but with a Why as well, preparing the reader to ask themselves the same questions as they progress further.]

[This is not to say the book is without rigor though. The theorems and the proofs are still there, just enriched by the other material contained within the book, and anyone mastering this book will be well prepared for future analysis courses, both mathematically and in their way of thinking about the subject.]

[Good for novices in mathematics]

Strichartz's book contains many clear explanations, and most importantly, contains informal discussions which reveal the motivations for the definitions and proofs. I believe the 'informalness' of the book with the insights make this book a very appropriate text for those taking their first rigorous mathematics class. And this text is definitely much better than many of the texts that target that audience.]

[The format of the book is more disorganized than the standard texts like Rudin, but makes it more likely that it will be read and thoroughly digested, instead of sitting on the shelf.]

[This is certainly the most intuitive Analysis book on the market. It is well written and the author presents the proofs in a way that should be accessable to most readers. He usually tries to use similar proof techniques over and over again giving the student the practice he needs and seldom uses the rabbit in a hat style some other authors seem to prefer. Although these arguments make this book well suited for self-study, lack of solutions to the exercises is annoying. In any case this book offers a nice change of pace to the standard terse presentation of most Analysis books.]

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- Elementary Analysis: The Theory of Calculus (Hardcover) - Kenneth A. Ross - Springer 2003 - 273 pages - [originally 1980]

[Designed for students having no previous experience with rigorous proofs, this text on analysis can be used immediately following standard calculus courses. It is highly recommended for anyone planning to study advanced analysis, e.g., complex variables, differential equations, Fourier analysis, numerical analysis, several variable calculus]

[The style of this book is a bit similar to Spivak's Calculus in that the author is a bit wordy. I find Ross' presentation more direct and less pretentious than Spivak - and far less intimidating.]

[This is definitely the best introductory analysis book I know of for self-study. A student who masters the material in this book will be well prepared to tackle Rudin and other classic works in real analysis.]

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[RJinkies]

Part II

and then stuff on books on how to do proofs

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- How to Think LIke a Mathematician: A Companion to Undergraduate Mathematics - Kevin Houston
Cambridge 2009 - 278 pages

[easy to follow, pragmatic]
[Chartrand goes much deeper though]
[in the same spirit as Chartand, Velleman, Solow]
[Get Chartrand and Exner and Houston which seem like the best for proofs and abstract math troubles]

[Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician.]

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- Mathematical Proofs: A Transition to Advanced Mathematics*- Gary Chartrand - Addison-Wesley

[well respected]

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- How to Read and Do Proofs: An Introduction to Mathematical Thought Processes - Daniel Solow - Wiley 1982 [1990 Second Edition]

[This book is the "magic decoder ring" for terse proofs. This book should be passed out to every undergraduate taking the first mathematical analysis course. Numerous examples and exercises are included. The typesetting and notation are very readable. The great strength of this book is that the proofs used for exercises are restricted to the level of algebra and set theory. This makes it easy to concentrate on the technique of proof rather than the specific results. Also check out Polya's book "How to Prove It" and Velleman's book of the same name.]

[MAA - 2 star recommendation] - Analysis: Foundations of Analysis

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- How to Prove It: A Structured Approach - Daniel J. Velleman

[I wish I had such a book before taking advanced calculus - Believe it or not, I graduated with a BS in math without being able to write proofs all that well. I wished that I had this book a year or so before taking advanced calculus/introductory real analysis). Actually, this book can be handled by a person just finishing high school. When you have to take advanced calculus, topology or abstract algebra you will not be struggling to learn how to write proofs, you will be spending more time on learning concepts and little effort on the actual methods and techniques of proofs. Set Theory is the foundation on which mathematical proofs are based. This book emphasizes set theory.]

[Advance to Rudin's Principles of Mathematical Analysis and keep Velleman aside. Now one can work on complete proof of materials in Rudin with rigor and study how he constructs logical structures step by step]

[I am a high school math teacher and when I left college I was quite upset with myself that I had this fancy math degree and couldn't prove anything. I picked up this book and today I'm working on my PhD in mathematics. This book inspired me to that. Mastery of this book, will certainly lead to a mastery of proof-writing in mathematics. I totally 100% recommend you buy this book if you are interested in mathematical proofs.]

[I recommend only buying this book if you have a lot of time to invest. If you are looking for light reading or a quick review this is the wrong book. It took me about 2-4 hours to fully digest each chapter.]

[Before reading this book, I had no idea how to prove anything, I would stare blindly at a problem without knowing where to start.]

[This is an excellent book for the early undergraduate student. It is actually two books in one. The first half is a careful review of Logic and the essentials of Set Theory with an emphasis on precise language. Thereafter a structured development of proof techniques is clearly presented using these tools. The second half of the book is a detailed presentation of introductory material about functions, relations, and a few aspects of more advanced set theory. These chapters serve as a wonderful introduction and show applications of the proof techniques developed earlier. I have referred back to this book often in my own study of analysis and number theory. I recommend it highly. It will be very useful to any undergraduate proceeding through a mathematics curriculum. I recommend studying it early in the first semester, and re-reading it as time goes on.]

[Starts off good, and then goes off on a tangent.]

[I bought this book in the hopes that it would help me improve my proof writing skills. Being only a high school graduate (a month ago), I had practically no knowledge of set theory. The initial proof structures were great, and I enjoyed following the proofs from the premises and, through logical steps, to the desired conclusion. However, then the Set Theory came in. I can understand why a certain amount of set theory was necessary in order to be able to talk about certain types of proofs, but he goes so far into set theory in the book, that by a certain point, instead of following the logical flow of the proofs, I was trying to remember abstruse terminology he had mentioned briefly and trying, successfully for the most part, to understand what the actual proof meant, and why it would make sense that it was correct. Its possible that the reason I feel this way is because when I do proofs, I usually need to understand it intuitively first and then go from there, and it could be the case that this isn't possible with more abstract proofs. Overall, it was a good read, but unfortunately, he went a little too far into the set theory than was necessary. Reading it twice would fix that problem though. Another criticism is that there are no solutions to the exercises.]

[Similar to the book - The Nuts and Bolts of Proofs - Antonella Cupillari]
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- The Nuts and Bolts of Proofs, Third Edition - Antonella Cupillari - Academic Press - 192 pages

[I own the second edition of this book and find it incredibly well done. I am a math major, and this book was recommended to me by a caring professor to help aid my transition between computational mathematics and the more abstract area of Mathematical Proofs.]

[If you are having trouble with proofs, there is no better book]

[It is a complete and easy to follow introduction to proofs. It quickly goes over the basic properties of numbers and symbols, then goes into direct proofs. It then explains the logic of using the contrapositive instead of using direct proof by showing truth trees of the statements. After that she goes over special types of theorems.]

[Everything is well documented and there are tons of examples. In her examples Antonella first explains the peculiarities of the proof she is about to attempt, and then she does the proof. That discussion is enlightening and her proofs are easy to follow.]

[I found going through this book was invaluable to my mathematics career. The level of difficulty of this book is very EASY, so it is perfect for going though on ones own. You will likely have much more difficult proofs in class, but by going through this book will become familiar with the techniques and ideas of Proofs, which is where most students have difficulty!]

[The book is small and does not feel like a textbook. It has about 80 pages of text and the rest of the book is solutions to exercises. The new edition is larger, but still relatively small, and so hopefully it has this same feel, and if nothing else is light enough to carry around!]

[When I was going through this book, I would carry it around in my purse with me - that is how small and portable and useful this treasure is.]

[similar to - How to Prove It: A Structured Approach - Daniel J. Velleman]

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- Reading, Writing, and Proving: A Closer Look at Mathematics (Undergraduate Texts in Mathematics) - Ulrich Daepp

- Mathematical Reasoning: Writing and Proof - Second Edition - Ted Sundstrom

- Introduction to Mathematical Structures and Proofs - Corrected Edition - Larry J. Gerstein - Springer 2001 - 360 pages

[This textbook is intended for a one term course whose goal is to ease the transition from lower division calculus courses, to upper level courses in algebra, analysis, number theory and so on. Without such a "bridge course", most instructors in advanced courses feel the need to start their courses with a review of the rudiments of logic, set theory, equivalence relations, and other basic mathematics before getting to the subject at hand. Students need experience in working with abstract ideas at a nontrivial level if they are to achieve what we call "mathematical maturity", in other words, to develop an ability to understand and create mathematical proofs. Part of this transition involves learning to use the language of mathematics. This text spends a good deal of time exploring the judicious use of notation and terminology, and in guiding students to write up their solutions in clear and efficient language. Because this is an introductory text, the author makes every effort to give students a broad view of the subject, including a wide range of examples and imagery to motivate the material and to enhance the underlying intuitions. The exercise sets range from routine exercises, to more thoughtful and challenging ones.]

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There you go...

i picked my notes clean for all sorts of books that can be helpful for figuring out proofs, and a possible collections of books if you want to start off easy in analysis and don't know squat about proofs as well.

Sorry about the length, but i hope someone finds a few of the things as useful as i did.

 

[AnTiFreeze3]

Wow, thanks for all of the help and recommendations, all of you. I went to go get some food and come back with all of this.

RJinkies, I found it useful, and I'm sure others will too when they browse over this thread, so don't worry about it being long or anything like that.

 

[RJinkies]

Antifreeze3, I'm glad you liked a chunk of my notes, took about 2 hours to get all the facts out, but i decided not to give up!

Micromass - thanks for the book recommendations, i think it's more than surreal that we were both mentioning the same textbooks at the same time!I feel pretty good that someone other than me, is tossing up a thumbs up on some of those books...

- How to Think Like a Mathematician: A Companion to Undergraduate Mathematics" by Houston
- Journey into Mathematics: An Introduction to Proofs by Rotman
- How to prove it: A structured approach by Velleman

I think i 'quoted' a little about the mixed feelings about Velleman... and Cupillari might be the easier and more enjoyable book.

Houston-Chartrand-Solow-Cupillari were books that excited me when i was making the list

the interesting thing is that books that get a brutal reputation, like Rudin, are actually way easier if you get an easier book like Ross or Strichartz and then make rudin your supplementary and follow up textbook!

same goes for physics when people take Jackson for Electrodynamics, if you read 2 or 3 of the intermediate books after Purcell or Griffiths, like Lorrain and other classics, then those 'scary' books arent so nasty...

it's just that people don't realize that one of the greatest things to collect are intermediate texts. Like how going from Dolciani's Algebra to Courant or Spivak, its nice to toss in a Sylvanius Thompson Calculus Made Simple, and a JE Thompson Calculus for the Practical Man as the 'intermediate' pathway.and good books for analysis that are neglected a little are:
Bartle - Introduction to Real Analysis
Burkill - A First Course In Mathematical Analysis - Cambridge 1962
Binmore - Mathematical Analysis, A Straightforward Approach - Cambridge 1990

I bought Binmore's book [second edition - copper] and it was the first analysis book that 'spoke' to me, it held my hand making up for the crappy curriculum and textbooks earlier on. I soon found the first edition of the same Binmore book [purple] , and then got his 2 sequels which tosses you important ideas and prepares you for real analysis, with Royden or some books on Topology later on.

I think that's one thing that make me think nicely about Spivak's calculus book, it was one of the RARE first year calculus books that would just dump a ton of recommended reading at the back and give comments about other textbooks and things for future reading.

I would always dislike textbooks that the writer seemed to feel he was the sole authority and wouldn't dare recommend 'further reading', let alone supplementary reading.

And if the book was really awesome, mention in the forward what textbooks are ideal before tackling said textbook...


What sold me was his recommended reading lists at the end. It was something that i wanted when i thought Hardy's Pure Mathematics was too difficult for me, and Rudin too terse and dry.

Then again, i learned 'parts' of calculus with Swokowski and Thomas/Finney, with unread copies of Syl Thompson, JE Thompson, Sherman K Stein [early 70s], and Hardy's Pure Mathematics in 60's paperback!?, Courant in my bedroom... The only person who knew anything about Analysis was the teacher!

I made up for my shaky background, by searching out the neatest textbooks on my own, before, and embarassingly late to figure out how to self-study and relax.

I didnt realize that going slow, and taking your time and just spending a ton of hours one chapter at a time, making it a puzzle, feels almost foolproof considering what i did in my youth lol I'd rather read 30 pages of hardy and figure it out really well, all on my own, than zoom through a crappy similar text, and study for 75% mastery and get a C-, just for the sake of a teacher holding my hand for 12 weeks...
The only crime is making good books, go out of print, or changing the cover...

 

[RJinkies]

oops...

I left one book off the listin my notes about Houston i said

[Get Chartrand and Exner and Houston which seem like the best for proofs and abstract math troubles]so here's Exner
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An Accompaniment to Higher Mathematics - Corrected Edition - (Undergraduate Texts in Mathematics) - George R. Exner - Springer 1996 - 215 pages

[looks like one of the best paths for people who have never done a proof before]

[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 Transition Course bridging the gap between calculus and higher mathematics.]

[Good book on proofs]

[I really appreciate An Accompaniment 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 fourth 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.]

[Great Introduction]
[This is simply a great text for introducing undergraduate students to the basics of upper-level mathematics. Stresses the importance of examples and definitions in proof discovery. While probably inappropriate for graduate students, it makes a great primary text for any first course in proofs at the undergraduate level and is written primarily to students in this situation. Overall, the book seems to be extremely appreciated by students transitioning from calculus to upper-level mathematics.]
[First Corrected Edition] 1996 - 215 pages

 

[RJinkies]

oh yes...

K.G. Binmore, Mathematical Analysis: A Straightforward Approach, New York, Cambridge University Press, 1977, 1981
[MAA recommendation] - Analysis: Elementary Real Analysis

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Foundations of Analysis: Book 1, Logic, Sets and Numbers - K.G. Binmore
[concise intro to logic/set theory for analysis by famous economist]

Foundations of Analysis: Book 2, Topological Ideas - K.G. Binmore
[Concepts of point set topo for Banach space analysis. If eps/delta in calculus not clear/not fun, read before you take analysis, you might have better time - I did]
--------and i think i'll toss some of those notes for what i grouped together as something as a nice prep for analysis...
a. Bartle - Introduction to Real Analysis - 3ed - Wiley 2000
b. Burn - Numbers and Functions, Steps into Analysis - Cambridge 2000
c. Howie - Real Analysis - Springer 2001
d. Mary Hart - A Guide to Analysis - MacMillan 1990
e. Burkill - A First Course In Mathematical Analysis - Cambridge 1962
f. Binmore - Mathematical Analysis, A Straightforward Approach - Cambridge 1990 -
g. Bryant - Yet Another Introduction to Analysis - Cambridge 1990
h. Smith - Introductory Mathematics: Algebra and Analysis - Springer 1998
i. Michael Spivak - Calculus - Benjamin 1967
j. Bruckner, Bruckner and Thomson - Elementary Analysis - Prentice Hall 2001

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a. Bartle

[this is the green - then do the blue bartle]
[people do the Green Bartle/Intro to Real - then - the Blue Bartle/Elements of Real]

Nice Preparation before Real Analysis might be:
[a. Polya - How to Solve It - [problem solving strategies]
[b. Velleman - How to Prove It - [technique to work out proofs]
[c. Bryant - Yet Another Introduction to Analysis [a good grasp of fundamentals in analysis]
[Plough through Bartle first, then consult Rudin. It's a bit easier that way. - Jon A. Middleton - Get Maturity in Pure Mathematics for Grad School]

[some prefer apostol much more]

[It's Not That Good - This problem is the discipline's fixation on abstraction and technique which alienates some less capable and prepared students. In many of the examples and proofs, the authors leave out important information, expecting that the already stressed and overloaded graduate student will figure out on their own. Many of the examples are not instructive at all, but very frustrating because they are too complicated. There is in many places of the text too much information left out, and in other places points/claims made with no explanation]

[The proofs themselves are terse, so without an instructor who understands the gaps, you may not connect the steps solo.]

[What a breath of fresh air after dealing with Pugh's book! The language is clear. The proofs are concise and easy to follow. The illustrations are good without being overwhelming. I cannot say enough good things about this book. Poor math teachers are obsessed with the most general case and introduce it first. A good teacher starts with a specific case, relates it to what the student already knows, and then begins to generalize it slowly, layer by layer until the most general case is achieved. This is how the mind works, this is how mathematics really developed over time, and this is how math should always be taught! Bartle and Sherbert do a outstanding job of this]

[Way better than Pugh. Don't let real analysis be your first proofing class - do your first proofs in elementary number theory or geometry, then when you have a repertoire of proofing tools and some skill in proofing, then take real analysis. You cannot learn proofing and real analysis at the same time. First learn to proof, then take real analysis. If not you will be miserable]

[It is not an Introduction to Real Analysis as the author assumes the reader has familiarity with most of the topics. There are very few Examples and the worst thing about this book is coming across the statement like 'We leave it to the reader to show that ...' or another one like 'We leave it to the reader to write out the detail of the proof'. How can the author call the book an 'Introduction' when they take you half way and leave you there? I regret buying this book. There are few explanations, few examples and many exercises. This book is horrible for a beginer in Analysis. I wish I never bought this book.]

[One of the best books in the subject - I have read this whole book for a Phd qualifying exam, mastering all the proofs and solving almost all the exercises, except for the sections on numerical methods. I can say that this book is a masterpiece. The proofs are clear and easy to follow, and the book flows smoothly. I can say that it is a classic in its field as Royden's Real Analysis]

[This book is very helpful to those student who want a advanced calculas process and need a basement to the study of real analysis. This book has many example which are very helpful to the student and we can have a chance to think about the process to the solution. Best textbook of what i have read this year.]

[This textbook is terrible for self-study.]

[Not for the faint-hearted]

[The book is well written, easy to understand and full of pertinent exercises. It is a 'must-have' book.]

[some prefer Rosenlicht more]

[helpful for Qualifying Exams for Graduate School]

[this book has very good notation (i.e. writes theta dependence on epsilon when it comes to limits). the pace is also very appropriate for those who haven't seen rigorous calculus in R yet.]

b. Burn

[Interesting and refreshing approach]

[I worked through this book several years ago and I remember enjoying its style of pointing out an interesting property of a particular function, and then showing, step by step, that a whole class of functions have that property; that is, the theorems are built up from examples, instead of the other way round. I also think each step was quite manageable - there were no big gaps where I was left scratching my head not knowing what to do. It is not meant as a reference book, as you're more likely to find sketches or hints to parts of proofs, rather than complete proofs. I don't know if it's ever been used as a textbook, but if it were, students couldn't just sit back and absorb knowledge - they would have to figure things out.]

[Best Undergraduate Single Variable Real Analysis Text by Far - Sandy Lemberg]

[This beautiful book is by far the best undergraduate single variable real analysis text I have seen. It covers all the basic topics in impeccable detail. Each chapter opens by listing a few references, labelled "Preliminary", "Concurrent", and "Further" Reading. The main part of each chapter consists of "questions" which guide the student through a complete theoretical development of the material and which the student is invited to work through.]

[The last part of the chapter contains a complete working out of all the "questions". At the end of the book is an extensive bibliography, containing all books mentioned at the beginning of the chapters and many others.]

[All in all, the text contains an exhaustive and perspicuous treatment of material which often is presented in a less transparent way in other texts such as Rudin. I also prefer it by far to other excellent recent books such as those by Ross or Abbott. The format engages the reader in a unique way that other books don't. This book was developed for use in the math program at the University of Warwick and as far as I know, it is still in use there.]

[Unfortunately, it is less well known in the US. I cannot recommend this book highly enough. Once you see a copy for yourself, I think you will understand why.]

c. Howie

[Written in an easy-to-read style, Real Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates.]

[This is an introductory text of real analysis and it is kind of British Style (in term of the way they proved the theorems). Also, some advanced topics like "Metric" and "Generalized Riemann Integral" are not covered. If you really want to learn real analysis yourself, try Robert Bartle's "Introduction to Real Analysis", Manfred Stoll's "Introduction to Real Analysis", Apostol's "Mathematical Analysis" and Rudin's "Principle of Mathematical Analysis". Stephen Abbott's "Understanding Analysis" is also an excellent real analysis text.]

d. Mary Hart

[In the first year of my maths degree I was lost... until I found this book. It's unbelievable! It makes sense, it has nice little historic interest bits and most importantly it'll answer all the exam questions. You won't need another analysis book. I actually love it... yes, I do realize it's just a textbook but trust me, you'll love it too.]

e. Burkill

[After 45 years this is still the best first year analysis book on the market, with more stimulating problems that Rudin. Also written in a transition to university maths style.]

f. Binmore

[already dealt with this one]

g. Bryant

[Please take the time to go through this before diving into analysis. It will go quickly, provide a road map, and save you time in the long run.]

h. Smith

[i went into this one before - but here's more detail]

[The material and layout is different to most textbooks. It is probably a book for people who want to grasp the idea of mathematics rather than just pass an exam. As the author notes in the preface it is a 'gentle and relaxed introduction'. The mathematics is pure and the emphasis is on the idea rather than on how to solve particular problems in the life sciences or engineering. Topics covered include; Sets, functions and relations; Proofs; Complex numbers; Vectors and matrices; Group theory; Sequences and series; Real numbers; and Mathematical analysis. It is an excellent book for those interested in learning and understanding mathematics. The book also offers an interesting glimpse of the mathematical mind.]

[A splendid introduction to the concepts of higher mathematics]

[Geoff Smith's Introductory Mathematics: Algebra and Analysis provides a splendid introduction to the concepts of higher mathematics that students of pure mathematics need to know in upper division mathematics courses. Smith's explanations are clear and laced with humor. He gives the reader a sense of how mathematicians think about the subject, while making the reader aware of pitfalls such as notation that varies from book to book or country to country and subtleties that are hidden within the wording of definitions and theorems. Since the book is written for first-year British university students who are reading pure mathematics, Smith's approach is informal. He focuses on conveying the key concepts, while gradually building greater rigor into the exposition. The exercises range from straightforward to decidedly non-routine problems. Answers to all questions are provided in an appendix or on a website devoted to the book whose address is listed in the book's preface. That website also contains a list of known errata, extra, generally more difficult, exercises on the material in the book, and discussions of topics related to those in the book. The book is suitable for self-study. Students preparing to take or review advanced mathematics courses will be well-served by working through the text.]

[The text begins with material on set theory, logic, functions, relations, equivalence relations, and intervals that is assumed or briefly discussed in all advanced pure mathematics courses. Smith then devotes a chapter to demonstrating various methods of proof, including mathematical induction, infinite descent, and proofs by contradiction. He discusses counterexamples, implication, and logical equivalence. However, the chapter is not a tutorial on how to write proofs. For that, he suggests that you work through D. L. Johnson's text Elements of Logic via Numbers and Sets (Springer Undergraduate Mathematics Series).]

[Once this foundation is established, Smith discusses complex numbers. After describing the types of problems that can be solved using natural numbers, integers, rational numbers, and real numbers, he justifies the introduction of complex numbers by showing that they are necessary to solve quadratic equations. After deriving the Quadratic Formula, Smith describes the algebra of complex numbers, their rectangular and polar forms, and their relationship to trigonometric, exponential, and hyperbolic functions. Throughout the remainder of the book, he draws on the complex numbers as a source of examples.]

[The next portion of the book is devoted to algebra. Smith discusses key concepts from linear algebra, including vectors, the Cauchy-Schwarz and Triangle inequalities, matrices, determinants, inverses, vector spaces, linear independence, span, and basis, that are widely used in mathematics. In addition to looking at their algebraic properties, Smith examines their geometric interpretation. He continues this practice with permutation groups, which he uses to introduce group theory, the branch of mathematics in which he does his research. Group theory is a deep topic, on which Smith and his wife, Olga Tabachnikova, have written a text for advanced undergraduates, Topics in Group Theory (Springer Undergraduate Mathematics Series). In this text, he confines the discussion to subgroups, cosets, Lagrange's Theorem, cyclic groups, homomorphisms, and isomorphisms.]

[Smith introduces analysis with a chapter on sequences and series. After providing another proof of the Triangle Inequality, Smith focuses on limits, thereby giving the reader a first exposure to quantifiers. He also discusses some properties of the real numbers, introducing the concept of boundedness, the Completeness Axiom, and Cauchy sequences. The aforementioned exposure to quantifiers makes the subsequent definitions and proofs of theorems about continuity and limits of functions easier to grasp. He concludes the book with a discussion of how the real numbers can be constructed using Dedekind cuts and Cauchy sequences.]

[There is a book by Ian Stewart and David Tall, The Foundations of Mathematics, that covers similar ground. It is devoted to building up the properties of number systems, which is a useful foundation for courses in analysis. However, it will not prepare you as well for courses in algebra as Smith's text, which I recommend enthusiastically.]

i. Michael Spivak

[This is a book everyone should read. If you don't know calculus and have the time, read it and do all the exercises.]

[a quirky book]

[Some reviewers have been puzzled as to the style of this book, deep mathematics for the unsophisticated reader. This is explained by its origin in the 1960's when many bright high school students were not offered calculus until college. Hence some top colleges experimented with very high level introductions to calculus aimed at gifted and committed students who had never seen calculus. Possibly Spivak took such a course, but certainly his book was used as the text for one at Harvard, and was still used more recently at a few schools still offering this course, such as University of Chicago.]

[Unfortunately today, due to the somewhat misguided AP movement, which is oriented to standardized test performance rather than understanding, almost all mathematically talented high school students take calculus before college, receiving significantly inferior preparation to what they would receive in college. The result is that many top colleges where the Spivak type course originated, no longer see the need to offer it.]

[This means that gifted freshmen at schools such as Harvard and Stanford are now asked to begin with an advanced honors calculus course for which Spivak is the ideal prerecquisite, although those same schools do not offer that prerecquisite. Thus if you are a high school student hoping to become a mathematician and planning to attend many elite colleges, almost the only way to be adequately prepared for an honors level mathematics program is to read this book first. It may be that a book like Stewart or even Calculus Made Easy, is useful as a first introduction to calculus, but it will not get you to the level you need for a course out of Apostol vol. 2, or Loomis and Sternberg.]

[note: i think that could be mathwonk's comment actually...]

[Anyone who has ever read Rudin knows he was a poor bookwriter. Spivak’s Calculus is hands-down the worst book I have ever read in my entire life. I still have my copy because I can’t figure out a good enough way to destroy it. For those who know a little math, I would add this: His treatment of manifolds, a wonderful, graphically intuitive topic, is 3 full pages of definition. I had never seen them before this. I never had such an easy subject made so difficult by such bad writing. Hardy is almost as bad as Spivak! Some bastard gave me that book when I was 14 or 15, and it was supposed to be my self-taught introduction to number theory. I didn’t get the subject at all until a much better teacher with a much better book made it clear. After that, it became my specialty.]

[Stewart is great and all, but Spivak is better because he doesn't waste time on pointless crap. He tells you a little bit, and explains some with an example or two. After that, it's pretty much your job to do the rest.]

[rigorous first year analysis]

[asks good questions]

j. Bruckner, Bruckner and Thomson

[Among the best math books I've ever read - I am not an analyst, but this book is fun to read. This book does something that few others textbooks accomplish - it tells an interesting and compelling story. I didn't really understand measure until reading this book, which does a great job of laying out the various competing ideas of the time and how they evolved into the current notion. Further - and perhaps most important for a math book - is that the proofs are very clear and complete. It's true that many important concepts are left as exercises, but those that are covered in the text are covered well. In contrast, I have suffered too many math texts that attempt to cover every important result but with only short uninformative proofs]

[used by SFU in British Columbia for Math 320 - Advanced Calculus of One Variable]

[Simon Fraser used Goldberg [not heard about as much now but a pretty typical book and a bit gentler than rudin] and Bruckner - as their main two texts, where UBC the main one in BC used Rudin] - both interesting choices slightly out of the mainstream

------
------

a good wack of those and Apostol and Rudin can really be tackled without choking on a peach pit...and one more

Understanding Analysis - Stephen Abbott

[PhD University of Virginia 1993]

[nice complement with Pugh]

[an absolute gem - wonderful]

[some people skip Bartle and go right into this one]

[people do the Green Bartle/Intro to Real - then - the Blue Bartle/Elements of Real]
[liked by the Strichartz people]

[If one were to do analysis from [easy to hard - dumbass to Princeton - Calculus made Easy to Courant] this is the way to appreciate analysis on four levels:

a. Abbott - close as you get to comic books/a great text that illuminated numerous side issues
b. Strichartz - really down to easy explanations for numerous abtruse topics/solid text/wisdom and informative
c. Apostol - not as brilliant as Rudin/more wordy than concise/definately easy to understand/wonderful text/a slightly different set of skills than Rudin's work actually
d. Rudin - brilliant/concise/requires almost an impossible level of mathematical maturity
This would be the way to do analysis without tears. - Suggestion by Richard Deveno of Alameda, Calif.]

and that's about it for training wheels for analysis and proofs...

 

[dustbin]

I figure I will ask here, rather than cluttering the main page with another of these topics...

I'm currently studying calculus (Stewart) at my college. I have found it pretty unchallenging (there are exceptions of course to some concepts and problems- but I can pick up on these as well without issue) so I decided to start into Apostol. I've been reading all of the corresponding topics for my Stewart-based class in Apostol, which has proven to shed a clear light on all of the subjects. I really like to read Apostol, it makes a lot of sense (is explained very well), and the read feels just challenging enough.

However, I can do very few of the problems in Apostol. Any of the problems involving calculation, I can do (that I've encountered). I can then do maybe a select few (or couple) of the more theoretical problems. Sometimes I know the "why" but I have little to no idea of how to put it on paper in a way that would give me a good score were it graded. I have the solutions to some of the answers worked out (from MIT, Caltech, etc.) which have helped, but they are still a little overwhelming for me.

This is a little frustrating because, when I read Apostol, I feel like I really understand it and feel confident that I will be able to do the problems. The first few problems make me feel good and then I get smacked in the face. I still continue to read it and attempt problems since my understanding of the material has clearly shown in my Stewart-based classes (I have a 100%)... but I really want to work the more theoretical/rigorous problems and material.

Is the missing ingredient logic and proof? I was reading Principles of Math (mathwonk's suggestion) and got through the first few chapters and then quit because I do not much care for the style. I'm checking out a handful of other proof and logic books from my library (Eccles, Houston, D'Angelo, and Chartrand). Is there anything else I should be studying to really be able to conquer the Apostol problems??

 

[micromass]

I figure I will ask here, rather than cluttering the main page with another of these topics...

I'm currently studying calculus (Stewart) at my college. I have found it pretty unchallenging (there are exceptions of course to some concepts and problems- but I can pick up on these as well without issue) so I decided to start into Apostol. I've been reading all of the corresponding topics for my Stewart-based class in Apostol, which has proven to shed a clear light on all of the subjects. I really like to read Apostol, it makes a lot of sense (is explained very well), and the read feels just challenging enough.

However, I can do very few of the problems in Apostol. Any of the problems involving calculation, I can do (that I've encountered). I can then do maybe a select few (or couple) of the more theoretical problems. Sometimes I know the "why" but I have little to no idea of how to put it on paper in a way that would give me a good score were it graded. I have the solutions to some of the answers worked out (from MIT, Caltech, etc.) which have helped, but they are still a little overwhelming for me.

This is a little frustrating because, when I read Apostol, I feel like I really understand it and feel confident that I will be able to do the problems. The first few problems make me feel good and then I get smacked in the face. I still continue to read it and attempt problems since my understanding of the material has clearly shown in my Stewart-based classes (I have a 100%)... but I really want to work the more theoretical/rigorous problems and material.

Is the missing ingredient logic and proof? I was reading Principles of Math (mathwonk's suggestion) and got through the first few chapters and then quit because I do not much care for the style. I'm checking out a handful of other proof and logic books from my library (Eccles, Houston, D'Angelo, and Chartrand). Is there anything else I should be studying to really be able to conquer the Apostol problems??

Apostol (and Spivak) are both known for their challenging exercises. It is perfectly normal that you are not able to do most exercises. I actually think you're already doing a good job if you can "see" intuitively why something is true.

Would it help to post your solutions (or thoughts) in the homework forum. We will certainly help you to rigorize your arguments. I think the best way of learning proofs is by doing them, making mistakes and being corrected.

 

[dustbin]

Apostol (and Spivak) are both known for their challenging exercises. It is perfectly normal that you are not able to do most exercises. I actually think you're already doing a good job if you can "see" intuitively why something is true.

Would it help to post your solutions (or thoughts) in the homework forum. We will certainly help you to rigorize your arguments. I think the best way of learning proofs is by doing them, making mistakes and being corrected.

Thanks for the suggestion, MM. I will certainly start posting up to get help with the problems in the help forum. I am fairly comfortable with induction and epsilon delta proofs, but beyond that I certainly need a lot of work. I'm starting through Chartrand's book (which I like so far) and really deconstructing/going through all the proofs given in Apostol and Stewart.

 

[RJinkies]

dustbin - I'm checking out a handful of other proof and logic books from my library (Eccles, Houston, D'Angelo, and Chartrand).

those other two books are good too, but D'Angelo and Eccles are a bit more advanced...but if you got the other books, they are good to supplement once you're a few chapters into the other ones...



[on a side note, most people think Apostol's book on analysis is a great second text on the subject if you start with one easier...]

but if anyone's tackled both texts, do you run through his calculus book and then tackle his analysis book in the next semester, or have some done both books at the same time... I'd think that both books would be a second tackling of calculus and a second tackling of analysis in the ideal world... you need a bit of intuition starting off...

[i'd like to hear what people tried in calc or analysis before tackling those tomes]

[I heard of people doing fine with Syl P Thompson's calculus book and then going into Apostol's calculus pretty okay... which says a lot for thompson being great preparation...]

[I know some people that really want to prepare well for Apostol or Rudin and they tried this text
- Advanced Calculus: A Friendly Approach - Witold A.J. Kosmala - Prentice-Hall - 700 pages - 1998 - going for the intimidation-free approach... anyhoo, just my three cents]

----
Mathematical Thinking: Problem-Solving and Proofs - Second Edition - John P. D'Angelo and Douglas B. West - Prentice-Hall 1999 - 412 pages

[For anyone interested in learning how to understand and write mathematical proofs, or a reference for college professors and high school teachers of mathematics.]

[Offering a survey of both discrete and continuous mathematics, Mathematical Thinking begins with the fundamentals of mathematical language and proof techniques such as induction. These are applied to easily-understood questions in elementary number theory and counting. Further techniques of proofs are then developed via fundamental topics in discrete and continuous mathematics. The text can be used for courses emphasizing discrete mathematics, continuous mathematics, or a balance between the two. It contains many engaging examples and stimulating exercises.]

[Extremely Useful - Great Read]

[I ran into the first edition of this book ten years ago when taking courses at George Mason University, and really loved it. I still love it.]

[It covers proofs from all basic 'pieces' of mathematics and gives the reader a good feel for the 'proofology', both in technique and fundamental nomenclature and results, that a student is expected to know when taking the first analysis and abstract algebra courses. It's not perfect though.]

[The author gives solutions or hints for one-third to one half the problems depending on the chapter, which is more than enough for self-study. I would disregard the whiny one star review that is posted for this book; it is typical of someone who wants to be spoonfed mathematics.]

[Difficult but well worth it]

[I'm using this in an undergraduate introduction to proofs class with a focus on analysis. As a freshman, it seems a bit overwhelming at times - I wouldn't recommend it to most freshmen or even sophomores. I do feel like this does a more than adequate job preparing me for more advanced math, and goes far above and beyond similar 'proofs and problem solving' style books.]

[The best reference for Proofs]

[This is an advanced book, with a lot of information on every page. I use it as a reference book, since it has hundreds of wonderful proofs and problems, along with thorough and concise definitions for just about every major branch of mathematics.]

[It's highly recommended for anyone who is *serious* about mathematical proofs. Although the book is packed with material, it's a small book, so it's one of the first I choose to take with me when I travel.]

[pretty hardcover]

[Used at University of Pennsylvania Math 202]

[they use it with - Howard Eves and Carroll Newsom - An Introduction to the Foundations and fundamental Concepts of Mathematics - Revised edition - Holt, Rinehart and Winston 1965
-----

and

-----
An Introduction to Mathematical Reasoning (Paperback) - Peter J. Eccles

[User-Friendly! Almost makes learning analysis fun!]

[If you are struggling with a first analysis course or any course that uses proofs, this is the book for you! It introduces basic analysis topics like logic, sets, and the real numbers. And it is written in almost plain english! Moreover, the author focuses on teaching proof writing.]

[Fabulous So Far]

[I'm at the end of my first discrete mathematics course and have struggled to find clear explanations of how to write a proof, meaning how to choose what method and how to choose what the next statement should be to lead to the desired conclusion. I'm only on chapter five and it is a breath of fresh air to read this. Rather than just showing the completed proof Eccles shows the "scratch" work that goes into making the proof, discusses the reasoning and alternative paths, and then has the final proof that is easily understood.]

[For a student who is just learning mathematical proofs, this book is just horrible. The examples are awful and the author shortcuts many proofs. For example only part of a proof is proven. Not only that, when giving the answer to a problem, instead of writing out the reason to why, it's just a one worded sentence. I'm in a class with about 20 students and we all agree this is probably one of the worst mathematical reasoning book out there. We got more help from using online resources then the book. For someone out there who knows the material then this book is a good review but for people learning the material do not get this book.]

[Chris Gray approved]

[Logic/set theory based introduction to problem solving and proofs, with chapters on various techniques: induction, finite and infinite sets, counting, and number theory. My current fav.]
----

hope you enjoy the notes, on the other two...

 

[dkotschessaa]

Just a few things I saw in the notes there... "How to Solve it" (in it's various editions) is the classic G. Polya book on problem solving. While it contains examples, it is more philosophical and is based on getting you into a particular mindset of problem solving. It's excellent. I took my time reading it - almost a year off and on while I let the concepts sink in. There is a section on proofs, but it won't teach you anything like set theory.

"How to Prove it" is Daniel Vellemen's book, which I'm using now. It's excellent. Lots of examples, and a very logical structure. I'm going through it before I take my first abstract math class to avoid the "culture shock" of such a class. The first couple of chapters introduce logic and set theory, and then different proof techniques are explained, and then some more advanced concepts in set theory. It's nothing like Polya's book, but it's a great companion to it. The title is possibly an homage to Polya(though there is no mention of this), but sometimes people seem to get them mixed up.

Oh, and if you get this book, get the latest edition, because the first one had no answers or hints to any of the problems. I found that very frustrating. Fortunately I was able to swap it out for the newer edition at my library.

 

[dustbin]

I agree with you about D'Angelo's text, RJinkies. Hopefully I will have the time to come back to it at a later point, though. It looks like a very interesting text. I read a bit of Eccles and did not have an issue... but perhaps this is because I have read material on logic, proof, etc. before from Apostol, Allendoerfer, and some brief touchings on it in my college algebra class. I've found so far that Chartrand is great for me. I have started working through it since I was able to purchase it for <$10 with shipping. Hopefully I will be cross-enrolling in an intro to proofs/higher maths course at the local university this fall.

Thanks for the tip on Kosmala. It looks like an interesting read... I just requested it from the library.

It is interesting how difficult of a jump it is to make from the standard mathematics education to the more rigorous material. I have been the top of my class in all math courses up to this point and, while I feel I'm quickly picking up on this new material, it is still a difficult transition. Any advice is always appreciated.

 

[homeomorphic]

It is interesting how difficult of a jump it is to make from the standard mathematics education to the more rigorous material. I have been the top of my class in all math courses up to this point and, while I feel I'm quickly picking up on this new material, it is still a difficult transition. Any advice is always appreciated.

I didn't even do that well up to that point, but it was actually an easy transition for me. It was much more natural to try to figure out how everything worked than to do it by rote, which drove me insane. So, when I changed majors to math and started doing upper division stuff, I felt like I was being freed from my chains. Only lasted a couple years, though, and then it got hard again.

 

[miglo]

do grades in lower division math classes count as much as upper division when your trying to get into grad school? or is it all based on GPA? i know having research experience helps a lot but I am at a community college at the moment waiting to transfer very soon, and i don't think community colleges have any research opportunities, unless i haven't looked in the right direction. i ask because i definitely plan on shooting for at least a masters in the subject

for lower division i guess that would be anything below calculus, the whole 3 semester calculus sequence plus intro to linear algebra and differential equations (the college i attend bundles both linear algebra and diff equations in one class)
I am guessing upper division begins with intro to analysis, or a class aimed at helping students learn what proof based mathematics is.

 

[homeomorphic]

do grades in lower division math classes count as much as upper division when your trying to get into grad school?

They count a little, but not as much. I wonder if getting a C in linear algebra and diff eq is a factor in why I only got into one grad school, despite strong recommendation letters and very good upper division grades. Probably not, I think. I'm guessing it's probably just that other applicants had taken more math classes or had research experience and that sort of thing.

or is it all based on GPA?

GPA doesn't matter very much. Most programs just require a 3.0 minimum, but that's it. Good overall GPA is sort of a sign of a consistent, hard worker, which they like. But mathematical ability is more important.

for lower division i guess that would be anything below calculus, the whole 3 semester calculus sequence plus intro to linear algebra and differential equations (the college i attend bundles both linear algebra and diff equations in one class)
im guessing upper division begins with intro to analysis, or a class aimed at helping students learn what proof based mathematics is.

Yeah, pretty much.

 

[dustbin]

@miglo:

Check out REU's (Research Experience for Undergraduates) to find opportunities for research. There are several sites that have lists and other resources.

@homeomorphic:

I do not find the material itself difficult... I am just having to work very hard at gaining comfort in proofs (both reading and writing). Prior to a few months ago I had never even seen or worked one out (save for my trig teacher proving the quadratic formula). I am particularly terrible at brevity... I just finished a proof using Rolle's Theorem which was two paragraphs long. I compared it with another answer to the problem which was only 3 sentences. At least my answer was correct I just need more practice and experience.

 

[RJinkies]

- for lower division i guess that would be anything below calculus, the whole 3 semester calculus sequence plus intro to linear algebra and differential equations (the college i attend bundles linear algebra and diff equations )
- I am guessing upper division begins with intro to analysis, or a class aimed at helping students learn what proof based mathematics is.

It depends where, usually the lower is

calculus i ii iii iv
linear
diff eqs
intro to analysisbut some linear and diff equations can be considered upper if they deal with some analysis and extra stuff.

sometimes you can see the regular linear being second year [sometimes the 200 levels], yet the honours classes can be the upper division [at the 300 levels]

same with diff eqs, regular classes could be 200 level, and honours at the 300 level.

-----

with analysis, some like to bunch it with the calculus classes, others as a separate course of half a year or a whole year, and then your next class will be a 300 level/upper division onegeometry classes can be upper or lower too depending how intense, and some schools that do really lite courses on abstract algebra can be lower division.

often they'll use linear algebra as a prerequisite, though I am not sure it should really matter that much.

------

For my money, all one needs to really focus on is
a. calculus i ii iii iv
b. basic analysis and more analysis and then probably more analysis

all the rest is filler...



and neat if you're doing good,

and a drag if it's painful... [where you're skipping too fast and not going deep enough]


heck with a super duper calculus text, and two supplementary texts maybe you got (a) and (b) both


as for proof and rigour, you can face that at any stage, first year extra hard calculus texts, second year linear with tons of abstract spaces and forcing you to generalize/do proofs, or you can get hit hard with analysis classes or abstract algebra with it...

A lot can really depend on your choice of textbooks...

and a really limited and inflexible curriculum i think is why you get people who face these 'hard' things somewhere up the ladder, and sometimes the higher the costs of education [and textbooks] the curriculum gets worse by being more bare bones...

---

if i had my way, a uni would be a library where you get a duotang for all the math texts, year 1 2 3 4, and the reading lists [and options]

and a duotang for the physics texts, year 1 2 3 4, and the reading lists...

exams? what exams, doing one chapter and all the problems, is your damn exam lol

spend 40 hours on a chapter, and no not pass go, till the time clocks says 40 hours...

might take 8 years to get your degree, but it'd be like 'speed learn' in the Prisoner, 100% Entry, 100% Pass.

 

[homeomorphic]

I do not find the material itself difficult... I am just having to work very hard at gaining comfort in proofs (both reading and writing). Prior to a few months ago I had never even seen or worked one out (save for my trig teacher proving the quadratic formula). I am particularly terrible at brevity... I just finished a proof using Rolle's Theorem which was two paragraphs long. I compared it with another answer to the problem which was only 3 sentences. At least my answer was correct I just need more practice and experience.

The interesting thing about brevity is that one of my profs said he was really impressed with my brevity, yet I never gave any thought to it. I think it's probably because my thought process is very conceptual. In trying to understand something deeply, you usually want the simplest explanation possible. When you make that rigorous, often, the proof ends up being short. That doesn't always happen, but it's my theory as to why my proofs tended to be shorter than most people's. I tend not to plow through stuff using technical brute force. It could also be that after a while, I knew which steps I could skip when I wrote the proof down, since they were clear enough.

 

[miglo]

@miglo:

Check out REU's (Research Experience for Undergraduates) to find opportunities for research. There are several sites that have lists and other resources.

so even though I am at a community college i can still apply for REU's? i always thought that only applied to undergraduates at universities.

 

[dustbin]

Miglo, some are only for undergraduates at universities. I have found some that are open to all undergrad students. When you look, just look through the requirements and such info about the application. Some do not specify, which I assume means they are open to all undergrads...

 

[mathwonk]

here is a problem from an 1895 high school algebra book, Treatise on Algebra, by Charles Smith:

{a^2 (1/b - 1/c) + b^2 (1/c - 1/a) + c^2 (1/a - 1/b)} / {a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}. simplify. any takers? (show work.)

 

[micromass]

here is a problem from an 1895 high school algebra book, Treatise on Algebra, by Charles Smith:

{a^2 (1/b - 1/c) + b^2 (1/c - 1/a) + c^2 (1/a - 1/b)} / {a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}. simplify. any takers? (show work.)

That was fun! Everybody should definitely try this one.

Here's my solution:

Let

$$\frac{a^2 (1/b - 1/c) + b^2 (1/c - 1/a) + c^2 (1/a - 1/b)}{a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}$$

be our expression. We multiply numerator and denominator by abc to get

$$\frac{a^3 (c - b) + b^3 (a - c) + c^3 (b - a)}{a^2(c - b) + b^2(a - c) + c^2(b - a)}$$

Rearranging gives us

$$\frac{a^3 (c - b) - a(c^3-b^3) +bc(c^2-b^2)}{a^2(c - b) - a(c^2-b^2) +bc(c-b)}$$

We can eliminate c-b from numerator and denominator to get

$$\frac{a^3 - a(c^2+bc+b^2) +bc(c+b)}{a^2 - a(c+b)+ bc}$$

Rearranging again gives us

$$\frac{a(a^2 - c^2)-bc(a-c)-b^2(a-c)}{a(a-c)- b(a-c)}$$

Eliminating a-c and we get

$$\frac{a(a+c)-bc-b^2}{a- b}$$

Rearranging again and we get

$$\frac{(a^2-b^2)+c(a-b)}{a- b}$$

Eliminating a-b yields

$$a+b+c$$

 

[AnTiFreeze3]

I got the same thing as well, Micro.

I would show my work, but I did it on some graph paper that I had nearby.

That was fun to do though. I solved mine a little differently than you, so I might take of picture of my work and show it that way. I haven't taken the time to get Matlab or LaTeX or anything like that yet, so I don't want to just type in all of my math and have it be a huge, ugly mess.

 

[RJinkies]

looked at a little bit of the problem, the pattern is interesting:ax+by+cz
------------
a+b+c


not making it messy, now that's a challenge...

 

[AnTiFreeze3]

...

Rearranging again and we get

$$\frac{(a^2-b^2)+c(a-b)}{a- b}$$

Eliminating a-b yields

$$a+b+c$$

I'm not sure that I understand this step (even though it looks very simple). I did it a different way, so the way I got to that answer was different. What I'm not seeing is how you have a-b in the denominator, yet three separate occasions of a-b in the numerator (a² - b²; and a-b), yet when you essentially cancel them out, you are somehow left with a + b + c.

In my mind, when you cancel out the a-b on the bottom with any of the three pairs of a-b on top, you are either left with:

(a-b) +c(a-b), or

(a² - b²) + c

What do I seem to be missing, or not understanding?

 

[micromass]

I'm not sure that I understand this step (even though it looks very simple). I did it a different way, so the way I got to that answer was different. What I'm not seeing is how you have a-b in the denominator, yet three separate occasions of a-b in the numerator (a² - b²; and a-b), yet when you essentially cancel them out, you are somehow left with a + b + c.

In my mind, when you cancel out the a-b on the bottom with any of the three pairs of a-b on top, you are either left with:

(a-b) +c(a-b), or

(a² - b²) + c

What do I seem to be missing, or not understanding?

You know that $a^2-b^2=(a-b)(a+b)$

So

$$\frac{(a^2-b^2)+c(a-b)}{(a-b)}=\frac{(a-b)((a+b)+c)}{a-b}=a+b+c$$

 

[AnTiFreeze3]

I'm not going to try to right this with brevity like Micro, but instead I want to explain my thought process, because I feel that I may have done something incorrect.

My solution:

{a²(1/b - 1/c) + b²(1/c - 1/a) + c²(1/a - 1/b)}
____________________________________________________________________

{ a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)

I noticed from the start that the binomials would cancel out, so long as I was able to manipulate the problem and get them next to each other, so I didn't see a reason to get rid of the fractions, since I knew they would cancel out anyways with their respective opposites. I then simplified a, b, and c to get rid of any multiplication in the denominator, and I used the commutative property to rearrange the denominator:

{ a(1/b - 1/c) + b(1/c -1/a) + c(1/a - 1/b)}
___________________________________

{(1/a - 1/a + 1/b - 1/b + 1/c - 1/c)}

The denominator then cancels out to equal 1, so I am left with:

{ a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}

This next step is where I have broken math. I recognized what the answer should be, but I think that I may have cheated in order to get to that final result. As a result, I did this:

{(a + b + c)(1/a - 1/a + 1/b -1/b + 1/c - 1/c)

Then, similarly as before, the fractions cancel each other out, so I was ultimately left with this:

a + b + c

I didn't peak at Micro's answer, and actually came to the correct answer myself. Regardless of that, I still feel as if that last step isn't allowed. Is it even possible to solve it correctly using the process that I used?

EDIT:

I messed it up in the first step, which is why I ended up in a situation where I couldn't correctly solve it.

 

[jbunniii]

I'm not going to try to right this with brevity like Micro, but instead I want to explain my thought process, because I feel that I may have done something incorrect.

My solution:

{a²(1/b - 1/c) + b²(1/c - 1/a) + c²(1/a - 1/b)}
____________________________________________________________________

{ a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)

I noticed from the start that the binomials would cancel out, so long as I was able to manipulate the problem and get them next to each other, so I didn't see a reason to get rid of the fractions, since I knew they would cancel out anyways with their respective opposites. I then simplified a, b, and c to get rid of any multiplication in the denominator, and I used the commutative property to rearrange the denominator:

{ a(1/b - 1/c) + b(1/c -1/a) + c(1/a - 1/b)}
___________________________________

{(1/a - 1/a + 1/b - 1/b + 1/c - 1/c)}

Can you explain what you did to get to this step? It is clearly not correct, because:

The denominator then cancels out to equal 1

Actually, it cancels to 0, not 1.

 

[AnTiFreeze3]

Can you explain what you did to get to this step? It is clearly not correct, because:
Actually, it cancels to 0, not 1.

I already mentioned that I messed up the first step, and that that is what threw off my whole solution. Thanks though.

EDIT: Although, if you are curious as to what was going through my mind, I embarrassingly forgot that I needed to simplify it before I could just eliminate a, b, and c. The rest of my problems stemmed from that.

I think it was coincidental that my answer ended up being a + b + c, even after making two big mistakes. Or maybe it wasn't coincidental, and I have just inadvertently invented a new form of Algebra where you break rules until you get the answer.

 

[mathwonk]

Very impressive micromass and Antifreeze! nice solutions!

micromass and Antifreeze are very strong, but we can also make progress using some basic principles to help us.

Here is a hint for other possible solutions: Generalized factor theorem: if f is an irreducible polynomial, and if f = 0 implies g = 0, then f divides g. (This is a basic result in “algebraic geometry”, and generalizes the basic result that x-r is a factor if r is a root.).)

For instance, suppose a-b = 0, then what about a^3(c-b) + b^3(a-c) + c^3(b-a), does it vanish too? Then what?

Now how did we guess to try a-b=0? Recall the "rational root theorem"? It says you look for roots of form X-r by trying factors r of the "constant term.

As miromass observed, we can rewrite the top of the fraction after simplifying,

as a^3(c-b) - a(c^3-b^3) + bc(c^2-b^2). Think of this as a polynomial in a.

thus the constant term has prime factors ±b,±c, ±(c-b),±(c+b).

(also other products of these factors, possibly.)

So we should try setting a equal to those factors. e.g. a=b iff a-b = 0.

 

[Dowland]

I don't want to start a new topic for this question, so i post it here:

How important is (euclidean) geometry in the higher (that is at the university) mathematics education? I'm currently in high school and feel that I've barely touched the subject, only simple computations with area, proportions, and some volume problems, together with a few "angle games".

I'm thinking of maybe getting the following book: https://www.amazon.com/dp/0201508672/?tag=pfamazon01-20

But maybe it's all too much, and not so important? I've enjoyed the little euclidean geometry I've done, but if I don't have very much use of it in the basic calculus and linear algebra courses, I'll probably skip it (for now).

Thoughts on that?

(Sorry for possible language errors, english is not my native, hope it's all readable )

 

[mathwonk]

its important, that's a good book: here' a cheaper one:
http://www.abebooks.com/servlet/Sea...rtby=17&sts=t&tn=elementary+geometry&x=74&y=9

and here's the all time great original version from euclid:

https://www.amazon.com/s/ref=nb_sb_...-keywords=euclid+green+lion&tag=pfamazon01-20and an excellent companion volume:

http://www.abebooks.com/servlet/SearchResults?kn=hartshorne+euclid&sts=t&x=52&y=12