Other Should I Become a Mathematician?

[dkotschessaa]

There is conflicting information out there. A lot of people tell you give you this sort of line that in "real life" (whatever that is) "grades don't really matter." But the fact is that there are a lot of opportunities and programs that will exclude you if you do not get high marks. At least that's what I'm seeing.

I believe grades don't really fairly reflect my understanding, because I am not a good test taker. I'm sure lots of people feel this way. I think my strengths will eventually come out in project oriented work and research, which I love doing. But in order to get into those programs you are often expected to have high marks. So it's a bit of a conflict.

I make every attempt to do a lot of problems and get As on everything. I'm usually a bit upset or annoyed when I don't, because I have pretty high expectations at this point in my life. But I get over it fairly quickly and put it into context. I ended up with a C+ in Calc III this semester and was not happy about it, because I felt I had a very pedantic professor who tests so short (about 7 questions) that it was really impossible to get As or Bs on them. That C might actually exclude me from some things I wanted to get into later. Hopefully my ability to connect with others and network and my strong work ethic and maturity will make up for some of that. Otherwise I have had no lower than a B in any math or physics class and no lower than an A in any non-math class.

I am being recognized by some professors at my school for my sheer enthusiasm and dedication to the subject. They aren't asking about my grades.

It's also been said though that "they" (people that care about your grades) do take the nature of your classes into consideration. A "B" in Analysis is probably as good or better than an "A" in Calc I.

Curious what Mathwonk thinks about this too, especially in regards to the conflict I have mentioned above.

-Dave K

 

[drummer - em]

hi good day pips can i ask your opinion about my thoughts..i am 30 yrs old and I've been interested in math lately, i have learned math accidentally..when one of my co teachers ask me to substitute to her math class for two weeks, so i decided to somehow read and study intermediate algebra.. and from now on I've enjoyed doing it..i enjoy reding and solving math books even I am alone...and i have decided to study my second degree in mechanical eng. can i still be a mathematician? does my age not a hindrance to become an engineer or i just need to stop coz i am too old for that dream? how can i improved in math? I've been dreaming math everyday..thanks hope to hear from you guys...Godbless you all

 

[dh363]

Whew! I just finished reading this entire thread! In my spare time over the past couple of weeks of course. Thank you all for being so helpful.

 

[grendle7]

Does anyone know how to subscribe to a thread, without having to post a reply?

 

[jbunniii]

Does anyone know how to subscribe to a thread, without having to post a reply?

Look under "thread tools" in the blue bar right above the first post on any page of the thread.

 

[homeomorphic]

Do you care a lot about getting an A/A+ or would you not mind getting a B+, simply because you haven't done 50+ problems (drilling/grinding can be fun but it can get tedious...) and thus couldn't finish within the required time? I had an exam today (stats) and lost three marks (out of fifty) because of a timing issue.
The instant my exam was taken from me, which was about a minute after "pens down", I figured out how to solve one of the problems. I also lost another 4-5 marks, mostly because of time issues. Now there's also potential mistakes in the things I thought I did correct!

Generally, I care much more about whether I am learning a lot than whether I get a good grade. As Mark Twain said, "Never let your schooling interfere with your education."

However, I will say that I do like to over-learn things. It helps with long-term retention. If you learn it really well, so that it is at your finger-tips, it will probably help with the test. So, I wouldn't worry overly about the grade itself, but it could be that you could benefit from learning things a little bit more thoroughly than you think is necessary.

 

[mathwonk]

I love and agree with the previous answer. However, when I needed some good grades to get a fellowship or some such, I tried to nail one. it isn't that hard. you learn everything presented, and then you practice old tests given by the same prof and then you study further from outside sources and you are likely to get an A. I did this this as a senior and got into grad school with a fellowship.

I.e. indeed learning is somewhat peripherally related to grades, but some things are given based on grades, so you need to know how to learn and also how to get grants.

I.e. don't complain that grants are given in a way that ignores knowledge, just do both, acquire knowledge, and survival skills.

 

[dkotschessaa]

So well said. I guess that even if there isn't always a direct correlation between grades and knowledge, grades are most of what they have to go on when evaluating for certain programs.

I'm sure you've heard this before, but my grades suffer from "dumb mistakes." I don't know how to stop making them, and I don't know if they are something that is eventually going to be ironed out or if I have to find another way to fix this. I really do take my time with everything, but they still seem to crop up.

-Dave K

 

[homeomorphic]

I'm sure you've heard this before, but my grades suffer from "dumb mistakes." I don't know how to stop making them, and I don't know if they are something that is eventually going to be ironed out or if I have to find another way to fix this. I really do take my time with everything, but they still seem to crop up.

I am a master at making dumb mistakes. That's part of why I did so much better when I got past high school math and lower division math. In the long run, it doesn't matter that much, as long as the mistakes are inadvertent ones. In "real world" situations (including research), you can check your work 20 times if you want to get it right.

 

[homeomorphic]

From childhood I was passionate about mathematics but I noticed I can not afford to become a mathematician.

Anyone can afford to be a mathematician to some extent. In America, all you have to do is do really well in high school and you can get a scholarship. Then, in grad school, you usually get paid. Even if you don't go to college, you can still teach yourself quite a bit on your own.

 

[mathwonk]

you might try becoming a mathematician who spends more time with her family. you could start a trend.

 

[dkotschessaa]

you might try becoming a mathematician who spends more time with her family. you could start a trend.

I recently started getting invited to gatherings with our math department, and it was funny to start finding out how many of the professors were married to each other. I had no idea, because most of the women kept their last names. So, I guess that's one way!

-Dave K

 

[mathwonk]

there are at least 5 couples in our department such that both spouses are either professors or instructors.

 

[Mépris]

Good to know. That was my intended course of action. (go outside the "syllabus" if I feel like it but then when there's exams, I focus on those)

A lot of what motivated my initial question was that I had some ~12 exams within the span of 3-4 weeks and they were all exams that are much in the vein of the usual standardised testing...

Does anyone here have any experience with the Jerry Shurman (at Reed College) notes on single variable calculus? I'm currently checking out Apostol and Spivak using the free previews available on Google Books and Amazon, before choosing which of the two to buy. Shurman says that he learned from them, Courant and Rudin.

Mathwonk, I read on another post that you used Sternberg and Loomis after Spivak back in the day. What do you think about this course compared to the modern alternatives - Apostol's second volume, I guess? Would one be correct in assuming that the current MATH 55 course at Harvard assumes (equivalent?) knowledge of both that book and Spivak?

 

[Mariogs379]

@mathwonk,

Bit of a specific question but I thought you might be a good source of advice. Here's my background/question:

Went to ivy undergrad, did some math and was planning on majoring in it but, long story short, family circumstances intervened and I had to spend significant time away from campus/not doing school-work. So I did philosophy but have taken the following classes:

Calc II (A)
Calc III (A)
Linear Alg. (B+)
ODE's (A)
Decision Theory (pass)
Intro to Logic (A-)

Anyway, I did some mathy finance stuff for a year or so but realized it wasn't for me. I'm now going to take classes at Columbia in their post-bac program but wanted to get your advice on how best to approach this.

They have two terms so I'm taking Real Analysis I in the first term and, depending on how that goes, Real Analysis II in the second term. I'm planning on taking classes in the fall semester as a non-degree student and was thinking of taking:

Abstract Algebra
Probability
(some type of non-euclidean geometry)

Anyway, here are my questions:

1) What do you think of my tentative course selection above?

2) How much do you think talent matters as far as being able to hack it if I ended up wanting to do grad school in math?

3) I'm also having a hard time figuring out whether math is a fit for me. By that, I just mean that I really like math, I'm reading Rudin / Herstein in my free time, but I've spoken with other kids from undergrad and it's clear that they're several cuts above both ability and interest-wise. Any thoughts on how to figure this out?

Thanks in advance for your help, much appreciated,
Mariogs

Thought I'd update. This 6 week real analysis class covers the first 6 chapters of Rudin. I'm finding the homework hard but we have a midterm on Monday; he showed us the one from last year and it looks *relatively* easy (definitely compared to the HW). Anyway, thinking I'm going to take RA II, and some classes in the fall, decide about applying to grad school the following year.

In short, material's harder than I appreciated but also much more interesting. I think I'll enjoy it even more once I get more comfortable with some of the concepts (I feel like I spend a lot of time trying to understand Rudin's language/terminology/general technical writing even when he's conveying a *relatively* basic idea. A good example is his def. of convergence; easy now, but was a bit confusing at first. Tho I think once I'm able to get the ideas more easily, it'll be even more rewarding.

Thoughts?

 

[eliya]

Thought I'd update. This 6 week real analysis class covers the first 6 chapters of Rudin. I'm finding the homework hard but we have a midterm on Monday; he showed us the one from last year and it looks *relatively* easy (definitely compared to the HW). Anyway, thinking I'm going to take RA II, and some classes in the fall, decide about applying to grad school the following year.

In short, material's harder than I appreciated but also much more interesting. I think I'll enjoy it even more once I get more comfortable with some of the concepts (I feel like I spend a lot of time trying to understand Rudin's language/terminology/general technical writing even when he's conveying a *relatively* basic idea. A good example is his def. of convergence; easy now, but was a bit confusing at first. Tho I think once I'm able to get the ideas more easily, it'll be even more rewarding.

Thoughts?

Wowza. Six chapters of Rudin in six weeks? How many times do you meet every week?

I'm not sure what you meant by ``thoughts?", I'll take it that you ask how to understand the material quickly. I don't think there's a tired and true method to expedite one's understanding other than practice in time. I'll also add that if you manage to understand the ideas in Rudin in 6 weeks, then you're doing fine. Also, this stuff takes a lot of time to understand. With that being said, try the following:

Write definitions, proofs, concepts, whatever you see fit really, in your own words. By explaining the ideas to yourself, you'll start figuring out how you understand things, and how to approach them. So next time you read a definitions or a proof, you'll be faster.

Get a few more books from your library. Sometimes Rudin is terse, and sometimes those proofs are hard. Other authors expand on the material more than Rudin. It'll be worth it to look some stuff up in those books. I recommend Charles Chapman Pugh's Real Mathematical Analysis. It has the same breadth and depth as Ruding, although sometimes the author does things with less generality.

Read about some of this stuff on Wikipedia. I tried to avoid Wikipedia for a long time, because I was afraid that I'll read an entry that was edited by some crank. All entries I've encountered were nicely written, explained the ideas in depth, and have a nice way of tying things together (how one theorem relates to another, why it's important, generalizations, etc.)

Good luck!

Especially if your first course in upper level math is with Analysis from Rudin. Rudin isn't a bad book, and in fact I like it quite a bit, however, it's a little hard for beginners

In fact, I think that practice and time will help you understand things more quickly

 

[mathwonk]

in my opinion loomis and sternberg is a show offy book (my book is harder than yours) and the two volumes of apostol or the two volumes of spivak, or of courant, are much better.

 

[grendle7]

^
Sounds awesome! Post here to tell us how things pan out. What is "summer B" though? A summer class for business students?

---

Does anyone have experience with the math departments at these colleges:
- Berea College
- Carleton College
- Reed College
- UChicago
- Colorado College
-Grinnell College
- University of South Florida

These are a few places I'm considering applying for next year. I don't know much about any of them except for what is found on their website and that a number of them are in cold, bleak places. And that they're quite selective...at least, for people who're non-US citizens requiring aid!

Just to let you know, it's MUCH harder to get into Berea, than Harvard. G'luck! Out of the "foreign students" pool of accepted students, only 30 aspiring applicants can be chosen, out of thousands. I'd still apply, if I were you. Just cross your fingers for good outcomes, from crazy probabilities. They usually prefer to accept "brilliant" foreign applicants who are living under crisis conditions, really deserve going to college, and/or won't ever have a chance at it; like that talented math-wiz living in Homs, Syria right now.

Either way, it's a great liberal arts school. In my opinion, you could get a great mathematics education there because it seems that their mathematics students graduate with a broad knowledge in mathematics, ranging from pure mathematics, applied mathematics, and statistics/probability; which is ideal, I think. Check out their http://www.berea.edu/cataloghandbook/dpc/mat-c.aspx! The only problem is, though, that they don't offer much variety in mathematics courses :b

And, have you considered, the best one of them all for math (in general), the University of Waterloo? It's in a town close to Toronto, Canada. I'd go there, if I didn't mind getting into debt; "Lulz."

By the way, unless you want to be chocking in debt after you graduate, then go to Colorado College! I'm infatuated with their block plan and great academic programs; and the MAGNIFICENT LOCATION; but it's totally not worth graduating with $130,000+ in debt.

Lol

 

[Mépris]

Just to let you know, it's MUCH harder to get into Berea, than Harvard. G'luck! Out of the "foreign students" pool of accepted students, only 30 aspiring applicants can be chosen, out of thousands.

Coincidence I came back to see this post. I read it before it was edited.

I think my grades may actually be just good enough to get me into Waterloo but it's really not worth the money...that I don't have. I don't know much about Colorado; it looked nice and has financial aid on offer, but it's very limited, as with most liberal arts colleges. I probably won't apply there. There's also the issue of limited coursework but few math/physics majors mean that one can try get some "independent study" thing going on. It doesn't mean grad-level courses, though.

Yeah, I read that about Berea. It's definitely going to be competitive but I believe it's free to apply, so I might as well give it a shot. There's also a list of those "free to apply to" colleges, somewhere on CollegeConfidential. It's easy to find - in case you can't find it, let me know and I'll try dig it up.

Another thing about liberal arts colleges is that bar a few (Amherst and Williams, being one of those), there just isn't much money to give to international students, which makes the competition even fiercer. It makes more sense to apply to larger colleges. Casting too wide a net is also not a very good idea. Too many essays, too much money on application fees, etc but some people can manage that just fine. ;)

This looks interesting:
http://en.wikibooks.org/wiki/Ring_Theory/Properties_of_rings

in my opinion loomis and sternberg is a show offy book (my book is harder than yours) and the two volumes of apostol or the two volumes of spivak, or of courant, are much better.

It's the post below, on another thread, that made me ask the question. I had also, per chance, stumbled upon the book, which is available for free on Sternberg's website.

In spite of its "show offy" nature, is the book any good? As for Spivak, are you referring to "Calculus on Manifolds" or is there another text which comes after "Calculus"?

In the old days, the progression was roughly: rigorous one variable (Spivak) calculus, Abstract algebra (Birkhoff and Maclane), rigorous advanced calculus (Loomis and Sternberg), introductory real and complex analysis via metric spaces as in Mackey's complex analysis book, general analysis as in Royden, (big) Rudin, or Halmos and Ahlfors, algebra as in Lang, and algebraic topology as in Spanier. Then you specialize.

 

[mathwonk]

It depends on your definition of "good". I have already stated that i think it is not as good as the other three I named.

Of course Loomis - Sternberg is very authoritative and correct and deep and well written. But the show offy aspect refers to very little attempt to make it accessible to anything like an average student, or to cover what is really needed by that student.

Differential calculus is done in a Banach space, possibly infinite dimensional, essentially the last case anyone will ever need. Most people will benefit far more from a careful treatment of calculus in 2 and 3 dimensions instead.

E.g. after giving all the definitions of differentiation in infinite dimensions, most applications are to finite dimensions. Even the brief discussion of calculus of variations is apparently influenced by Courant who devotes a chapter to it.

The treatment of the inverse function theorem again in Banach space is overkill, and gives little intuition that is actually needed in everyday practice. The implicit function theorem should be understood first for single valued functions of two variables.

Loomis is an abstract harmonic analyst. His own personal preference is to render everything as elegant as possible, not as useful or understandable.

But make up your own mind. These books are available in many libraries. Just because my course of lectures from Loomis left me feeling very disappointed, with little intuition, and almost deceived as to what is important in calculus, does not mean it may not help you.

If you read Loomis and Sternberg at least you will learn that a derivative is a linear map. That's a lot right there. Indeed that's about all i got from loomis, and it has been very helpful. But I recommend Fleming, Calculus of several variables more highly. Loomis used that book officially in his course, before writing his own.

If you want a very high powered book that also does things in banach space, but manages to be very useful, in my opinion, there is dieudonne's foundations of modern analysis. he also perversely adheres to a credo of making life harder for the reader by banishing all illustrations from his book. but it is good book with a lot of useful high level information not easy to find elsewhere. he explcitly states however that one should not approach his book until after mastering a more traditional course, (e.g. courant).

Another book Loomis used that I do not recommend either is the super show offy book by Steenrod Spencer and Nickerson. As one reviewer put it roughly, this book is more about the ride than the destination. However I do have all these books on my shelf, I just don't look at them all very much nor with the same pleasure.

Your last quote from me above is a historical account of life at Harvard in the 1960's, not a personal recommendation, indeed to some extent the opposite.

Spivak's second recommended book is indeed calculus on manifolds, an excellent place to learn the most basic several variable calculus topics, but very condensed. moreover he makes the proof of the general stokes theorem look very abstract and to me off putting. to understand this result, just work it out on a rectangle in the plane, as lang does in the back if his book, maybe analysis I.

now that i reflect, i am not familiar so much with sternberg's (second) half of the LS book. i only heard him lecture once and was quite impressed with his down to Earth and insightful approach. maybe that half of the book would suit me more.

but I'm not much into physics.

In my opinion you are spending more than enough time here asking for advice, i.e. "dancing around the fire", and need to get to work in the library reading some of these books.

 

[mathwonk]

well you provoked me to go look at LS and i did in fact like Sternberg's chapter 12 on integration.

This whole discussion is beginning to remind me of a friend telling me that his brother warned him off of reading a famous algebra book, so i myself also avoided it for years.

Finally I was required to read some of it and found it wonderfully clear. When I went back and asked my friend's brother he said he had never said it was bad, just "tedious". by which he seems to have meant overly detailed, just what I appreciated about it.

so please take what we have said with a grain of salt and try to get a good look at these books yourself.

Even Loomis' half of the book helped me in the section on "inifinitesimals" and his slick proof of the chain rule.

But the abstract implicit function theorem in terms of projections from a product of banach spaces, there left me wondering what Mumford even meant when he said the theorem simply says you can solve for some of the variables in terms of the others.

 

[mathwonk]

oh, also the intro to LS says plainly that apostol, spivak, and courant are suitable prerequisites for their book. if that includes both volumes of those books, i would agree.By the way, Jerry Shurman's calculus notes from Reed are to me, as an old professional, far too wordy, hence hard to get something out of in a reasonable amount of time. But a beginner might like them just for that reason, so you must be your own judge.

 

[RJinkies]

Hi sahmgeek...


Quote: 'I am seeking advice on a receiving a math degree (e.g. Master's + Secondary Ed cert) however I have very little formal math training beyond high school...

...Given that I would need to start from scratch, I wondered if taking the basics at a community college (Cal 123, Linear Algebra Abstract Algebra, Finite Math, and ODE) and, of course, doing very well...

...stay home with my 2 very young children and would most likely need to go back to school part-time. I might have some time during the day to work on math, but most of my free time would occur in the evening, 8pm and later. I am concerned that this isn't enough free time to really study this subject...

...I am not very concerned about my intellectual capabilities, but with my time constraints perhaps this is an unrealistic goal given the rigorous nature of math. I do, however, like the idea of studying math for it's own sake, even if the end result is purely for personal gain...

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

Well, I would think that there shouldn't be any problem for self-study, if you feel that you can put in the time to read the textbooks and do all of the problems.

With the rule of thumb for an Ivy League school, it's about 48 hours of class work a week with a full schedule. And you do that for 12-15 weeks to complete one semester or half of the textbook. One can figure out how to pace yourself pretty okay on your own. As long as you know what the good textbooks and supplementary textbooks [1-3 texts - old and new] should be, and the books fit best with your learning style.

It sounds like you could actually learn the subject better and on your own terms, setting your own schedule as long as you're motivated to get the most out of the textbook by reading all the pages and doing all the problems 95% of the time.

------

As for goals, that can change semester by semester as you master one more notch in the textbook ladder, and your interests may change, and perhaps your direction... If you want to take some courses later, by all means, but I'd probably do most of the work on my own, but it all depends how much time to spend with the family, and how quickly you want to zoom up the ladder of checking off the courses that you got a lettergrade in.


Quite a while back, people could teach high school math where i was with at least a minor in math and maybe a major in education or something else... [like a major in physics and a minor in math and some education courses]

-----

But it sounds like you want to do a BA/BSc in Math and well just know what the basics are, and then add the stuff that interests you to your liking. As you finish off one textbook and then go to the next tier, you get to choose your own path pretty much.

a. getting your Calculus I II III ... and IV [aka Vector Calculus]
b. Taking you Analysis courses and thinking of them as one stream of at least four semesters to like Real Analysis - say from RG Binmore/Apostol/Rudin/Hartle/Strogatz/Royden...
c. Linear Algebra - and up
d. Differential Equations - and up into PDE and Non Linear Dynamics/Chaos
e. Complex Analysis [Applied if you're for Physics, Pure for just math, or maybe both]

You can always figure out if you want to go into [most people might only do 5 courses worth [20%] of these...

f. Geometry - like Coxeter's book
g. Number Theory
h. Mathematical Logic and Set Theory
i. Abstract Algebra [helpful with Analysis to get into Topology]
j. Topology - Munkes and Guilleman as the main two books
k. Probability
l. Differential Geometry and Tensor Calculus - like Synge's book [what you'd want after Vector and for say Wheeler's Gravitation]
m. Mathematical Physics stuff [like if you took Symon and then Goldstein in physics] and then wanted to go into the mathematical side of LaGrange and Hamilton
n. Fluid Mechanics [if you're more physicist/engineering curious]
o. Continuum Mechanics [if you're more physicist/engineering curious]
-----

I tend to think of Grad School as basically what textbooks did you find cool in Third and Fourth Year, and sometimes the rest of those books are your grad school classes [like Royden] or the supplementary reading in those texts...

Me i would choose a Mathematical Physics like route where you can get the best of the Applied and the best of the Pure, i don't think people think of things as Pure Math anymore like Hardy...


I think of it as, spend the 400 hours on each textbook, master things on your own with completing the reading of one chapter as your self-mastery, and then doing all the problems in that book, as the proof of your self-mastery.

That way you don't get hung up on midterms and finals, you see the ladder of math or science as a bite filled chunk as a single chapter, accomplished usually in a week with maybe 20 hours of effort, getting to that goal of the last chapter and 400 hours clocked on your mental library card for your own textbook, using your own dining room table as your own little uni.

 

[RJinkies]

Hi PrinceRhaegar


Quote: my second semester of college as a mechanical engineering major, but I'm thinking about switching to math. The reasons are simple; recently I've found that I'm better at math than any other subject (especially physics...

...I just think math is cooler than any other subject I've seen so far. The reason I'm really hesitant to do so is because firstly, I have no idea what I'd do with my degree after I graduate, and secondly, and this may seem a bit shallow, I know that I'll likely be making more money as an engineer...

...In a perfect world I'd major in math and get a job as an engineer (or at least in an engineering company). This is because I love math and I feel like I'd get a TON of satisfaction out of doing useful stuff for the world while also doing what I love...


-------

Honestly, it sounds like the ideal path is to do both, and just take that extra one or two years for your B.Sc and do a double major

There are people out there that sound a lot like you and they do things like get a Mechanical Engineering degree and double it with a Physics Degree...

if you really wish to slow it down, and you got zero problems with the textbooks, you can almost accomplish it all, and think of it as engineering as a hobby and math as a hobby, and then think of the engineering stuff as your income...

-------

Some bizarre and brilliant souls in 5-6 years end up with a satisfying thing of doing four Bachelor degrees. [maybe 6-7 for ordinary mortals with the same goal]

a. Mechanical/Aerodynamic Engineering
b. Physics Degree
c. Math Degree
d. Electronics Engineering Degree

since there is considerable overlap and his future goals worked out that he used most all of it in his career... though he wasnt as deep as some that just took one and only one path...

But you can be 65%-80% fluent in two courses with a Double Degree.


so there is a LOT you can accomplish with an extra 1.5 years of your life, that these sorts of things are possible.

The Hardest thing is knowing how to self-study and how much effort to put into things, and not fearing failing or exams anymore... the second dilemma is what really makes you happy, and a career may or may not conflict, if you just put some extra time into things.

but sure if you go up a ladder in academia you do tend to end up stuck there, where people who get a physics degree who almost wanted to go to grad school in pure math, and they find they *had* to pick one or the other, but if it's a hobby, or circumstances are right, you can sometimes slide into both worlds... all depends how happy you are, and you like the results..

 

[mathwonk]

or perhaps, just start with one small significant goal, like learning calculus, and do it well. then go further.

 

[RJinkies]

mathwonk - 'perhaps, just start with one small significant goal, like learning calculus, and do it well. then go further...'


Agreed!


I just think it's good to know that the most important thing are probably the two streams

a. Calculus I II III IV - where you know Vector Calculus
b. some Foundations in Analysis that takes you by the hand up to Real Analysis.. [RG Binmore's three books/Hartle/Strogatz - which are all friendlier to start off with than Rudin]


which boils down to a promise that you'll get through 1-2 calculus/Vector books and two Analysis texts...


you could think of Linear and Abstract Algebra as stuff that gets 'analysis' heavy so (b) gets to be important...

and if you're doing (a) you get to see how it all works in the real world with differential equations.

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

but it does boil down to, how does one start off... and that's usually with a good algebra text. You suggested one of the classics of the 50s Welchons and Krickenberger [about 1953], and there was also the Dolciani books [about 1964] which was probably the only non-experimental text to come out of that Yale SMSG Special Math Studies Group...

Munem's algebra books in the 80s seemed like a easier path than dolciani also...

And well, i still think that the two old classics from the 1910s and 1930s still work out pretty damn good. Syl Thompson [Calculus made Easy], and JE Thompson [Calculus made Simple]. And we got Mathwonk and Martin Gardner to recommend the first [though Gardner's edition sounded totally unnecessary], and Feynman to recommend the second Thompson.

As for calculus, i think just realizing that mastering one chapter almost perfectly, is better than rushing through the book with 65% comprehension. And most any of the texts from the 1920s to the 1950s i think are great since like the older algebra books, are truly meant to be read front to back, without too many frills of abstraction/formalism/the new math]. And one can always rush to Courant/Spivak/Apostol after the easy books... for the 'deep stuff'.

And that's one of the deepest things i got from Nathan Parke III, about self-study that you go from vigour to rigor. Read the baby book on calculus, and then read the elegant book on calculus, or physics or anything else scientifical...

It's hard for lots of people to appreciate Hardy or Rudin, or Apostol without some gentle breaking in...

--------

and knowing one chapter deeply, by just spending enough hours on it, reading it, and rereading it, and doing all the problems, i think shakes people off from thinking about teachers, exams, and the course as a whole as one hurdle.

If you make just one chapter in physics or math your hurdle, and you take 10-20 hours jumping it, you don't need to worry about falling down, much later down the path...


I just that unis would still offer physics and math right from the basic nuts and bolts more often, rather than expecting people to learn it all perfectly and then some in high school.

 

[dimasalang]

i want to be a mathematician


but I am not very good in abstraction and analysis




is there a magic ingredient to be very good in math?

 

[chiro]

i want to be a mathematician

but I am not very good in abstraction and analysis

is there a magic ingredient to be very good in math?

Hey dimasalang.

The key ingredients are persistance, and thinking about continuously. If you do these and do what you can to learn and later teach what you have learned, you will surprise yourself.

 

[homeomorphic]

i want to be a mathematician


but I am not very good in abstraction and analysis




is there a magic ingredient to be very good in math?

I think the magic ingredient is the ability to conceptualize (although you might say that conceptualizing is just one possible style, and people have different styles). To ask why and find an intuitive answer whenever possible. For the best possible example of this in action, the book that allowed me to take this to the next level when I was an undergraduate was Visual Complex Analysis. I would say reading that book was one of the keys to my success (the other being a few years of trying to understand electrical engineering as deeply as possible). I read it before I took real analysis and then breezed through the class with probably over a 100% with some extra credit when everyone else in the class was struggling (although, I dropped all my engineering classes that semester to switch to math, so I also had a lot of time to spend on it).

Another tip is to go to office hours and talk to professors. I I didn't need too much help, but by talking to my professors a bit when I was stuck, I got to know my professors on a more personal level, which is helpful for getting recommendation letters for grad school. Anyway, generally speaking, one on one conversations are much, much better for communication purposes than lectures (and often written material) are, so it's always good to take advantage of that. Because I'm not very sociable, I think that still holds me back. I'm not that good at picking peoples brains and getting more of their intuition, which is often easier to find by talking to mathematicians in person, one on one.

 

[homeomorphic]

Also, I forgot to mention, if you need to get used to doing proofs, it's good to warm up by studying something like naive set theory, where the proofs are easier (and in set theory, you also get to learn some foundational concepts). For example, Halmos wrote a book on that.

Another route might be to try to study the foundations of geometry, which is also a bit easier than analysis, but is proof-based. I don't know a good book for that. I took a class like that which just used lecture notes.

 

[AnTiFreeze3]

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.

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?

EDIT: Mathwonk, if you do see this, maybe it would be a good idea to update your original post including MM as an active mathematician on this forum, seeing as that's true. When people first join this forum, and possibly read this thread, then they are reading your statements from 2006, not present day, and would be missing out on the information that we have the fine mathematic mind of MicroMass who is also capable of helping out around here.

 

[RJinkies]

Probably the only 'magic' is knowing just how much time to spend on reading and re-reading a chapter, and really grappling with each puzzle you face. You just realize that in most cases, spending a lot more time to know each example inside out, and tackling all the problems.

For me, i find just making a 'single chapter' your goal and putting in something like 20 hours into it, rather than 2-3 hours on things, works for me.

-----

I like to think of semester of math [usually half a textbook], not as one course, or consisting of 4 big exams for marks...

but as single chapters... or sub chapters...

so that half a math text is like 70 mini courses, with no exam, and no teacher.

-----
So one textbook to me is like 30 weeks of reading, with 400 hours of my time to
a. read it
b. keep re-reading it
c. studying all the examples, inside out [or getting out the schaums outline if needed etc]
d. doing ALL the problems
e. if i feel uncertain, do the problem again, or try it another way [or three], and don't surrender so easily

sometimes, you can do this with a textbook and 2-3 supplementary texts, but you really need to watch out about 'order'...

but that can be a bit brutal in a classroom situation or a demanding schedule... but if you eat sleep and breathe something 2-4 hours a day, six days a week, it's amazing what one can do in 3 or 4 months.

if you're less ambitious, finishing 'one chapter' is magic...
and if you got the stamina, go for the next one...

usually you can finish the book.

----
for me the real magic is figuring out what the best textbook for me is, and the 2-3 supplementary textbooks with it are.

I just care about the next rung on the ladder, i don't think about running up the ladder quickly, and i don't think about the next few floors either...

 

[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.

-------
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).]

-------

- 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]

-------

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.]

-------

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]

------

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.

------

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.]

-----

- 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.]

------

 

[RJinkies]

Part II

and then stuff on books on how to do proofs

-------

- 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.]

-------

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

[well respected]

------

- 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

-----
- 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]
-----

- 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]

-----

- 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.]

-----
-----

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
------

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

-------
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

-------

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.