How should I claim mathematics credit for self study?

[Calaver]

Hello everyone, I am currently a sophomore in high school (year 10 by the US system), with two years before I graduate to university. Last year, I began to learn calculus and physics on my own because they interested me. There is more to this story, but basically I seem to be at a crossroads now with a question that has been pestering me for a while. How should I go about claiming credit for these self studies, or should I even worry about it at this point? It should be noted that I hope to major in physics and/or possibly mathematics, and attend UT Austin due to the fact that it seems to be a fairly good in-state university for me to attend (and thus saving me money in the long run).

You can stop reading at this point if you have an answer, or if you think I shouldn't worry about it, as the options I list below begin to make this a very long thread. But I've come up with a few options that you might want to take a look at:

1. I could take the AP exam for Calculus AB this year, and claim credit with my local community college for Calculus (I just linked to the College Board's site because the CC's site is a maze). I could then take the CC's Calculus II, III, IV, Diff EQ, and possibly Linear Algebra (if it is offered) classes throughout my junior and senior year. http://www.utexas.edu/student/admissions/acc/2013-2014/natsci-3.pdf , effectively eliminating any lower division math classes (with the exception of possibly Linear Algebra) that I would need to take for a physics (or possibly even mathematics) degree.

2. I could take the AP exam for Calculus BC next year, and follow approximately the same trail with the exception of taking only Calculus III and IV at the CC because Calculus II credit would be covered by the BC exam. I would be able to take Diff EQ too (and once again Linear Algebra if it is offered), but this would require doubling up with one of the CC's math courses my senior year.

3. I could follow the more "normal" path, and take the Calculus BC exam my senior year, and http://ctl.utexas.edu/studenttesting/exams?field_subject_area_tid=8&field_exam_type_tid=3&combine=calc at UT (the equivalent of the CC's Calculus I and II courses), but nothing else.

In short, Option 1 would save me money but might take a great deal of my time that could be spent in these last two years of high school studying things that interest me, whereas Option 2 is the equivalent of Option 1 in terms of credit and money (with the exception of possibly the Diff EQ and Linear Algebra courses), but would give me more time to learn things that are of pure interest to me. Option 3 would be more costly, but would leave me with a ton of time.

It should be noted that taking these courses would be very interesting too, but I would have a lot less freedom to "dabble" in other areas.

So, if you've read my lengthy post this far, thank you for your time! If you don't mind, I'd like to hear your opinions on which option I should take, and also of any other options that might be possible. Or, if you don't think I should worry about it at all, then please tell me. Once again, thanks for your time!

By the way, I omitted what may have been some important details due to the length of my post (what I have studied thus far, how I am testing my knowledge, etc.), so if there is any other information that would help your answer, just ask and I'll try to cover the details of it.

 

[symbolipoint]

Ask your high school and your local community colleges about actually ATTENDING a course or two at the community colleges for earning credit. You would then, after earning the credit using the c.c., be able to show documentation (transcripts) of having the actual credit. If you have already studied the material as you stated, then repeating that as an enrolled student and getting a good grade should be a very strong possibility. Just do not try to fool yourself about retaining what you previously studied - be sure you work hard, keep up, and then you would be able to re-learn what you already did on your own. Also be sure to keep up in your regular classes at the high school.

 

[Calaver]

Ask your high school and your local community colleges about actually ATTENDING a course or two at the community colleges for earning credit. You would then, after earning the credit using the c.c., be able to show documentation (transcripts) of having the actual credit. If you have already studied the material as you stated, then repeating that as an enrolled student and getting a good grade should be a very strong possibility. Just do not try to fool yourself about retaining what you previously studied - be sure you work hard, keep up, and then you would be able to re-learn what you already did on your own. Also be sure to keep up in your regular classes at the high school.

Do you mean I should take the CC's Calc I and II courses (or their prerequisites) instead of the AP exams? I was thinking that the AP exams would be a good way to get me onto the mathematics track at the CC, but maybe as you suggested this is not the best option. The reasoning behind taking the AP exams is not only for the options I stated, but also in case I end up going to a university other than UT Austin. I have heard (perhaps incorrectly) that AP exams would be a more universal credit than the possible struggle of transferring CC credit to a different university that has no formal link to said CC. Although, I haven't really considered a path similar to your suggestion before, and it seems like it might be a better option, so thanks for the advice!

 

[Choppy]

Taking a test rather than going through the full course (in which ever direction) can be a dangerous move in my opinion. It would be one thing if you were not planning on really building on these courses for the rest of your education. But if you are interested in majoring in physics or math, an introductory calculus sequence is a critical cornerstone of the education. The problem is that it's possible to have huge holes in your foundation that won't be covered by an exam, and those can come back to haunt you later on.

The other issue is that taking a calculus sequence + linear algebra and an introductory differential equations course is going to be extremely demanding on a high school student, even if you do get through it. When are you going to have time for a girlfriend or boyfriend? How much time are you going to have for a part time job?

 

[MidgetDwarf]

Atleast in California, community college courses are free for students who are enrolled in an accredit high school. I'm sure many states offer this program or programs similar. I would attend classes at a CC and finish my GE, before it is time to graduate from high school.

 

[CalcNerd]

CLEP is another option. I believe it is a $100 exam or so. You can clep out of many types of classes. Calc I and II are available (actually same exam, your score determines if you get any credit, Calc1 or Calc1&2 credit). Your score is kept on file for several years. Most accredited colleges accept Clep exams over anything else ie they may accept AP exams, but most DO accept CLEP.

 

[MidgetDwarf]

I believe that the UC system in California does not accept Clep credit, i am not 100% sure, more like 90%.

I would advise to retake Calculus 1 and Calculus 2 at the university when you get there, if you are not going to take them at a local community college.

From personal experience, I go to a community college in California, and I have seen many fellow classmates that are deficient in algebra, trig, and geometry.

I work at the tutoring center and I noticed many students who took AP calculus and received credit for AB and BC, have problems with higher maths, such as Discrete Mathematics, Linear Algebra (especially this one), and Differential Equations, and Cal 3. It seems to me, from personal experience while tutoring, that the AP test was plug and chug in nature and no conceptual understanding is needed to pass. Therefore, these students never learned how to think about math. As a result, I have seen many students exit the STEM field as a result of doing poorly.

 

[Calaver]

Thanks for the replies everyone!

CLEP is another option.

I actually hadn't thought about that option, but the exams look like they could be a possibility. As an example, it seems that they are offered year round, so I could take one in the fall and then enroll in the subsequent CC course in the spring.

Most accredited colleges accept Clep exams over anything else ie they may accept AP exams, but most DO accept CLEP.

It looks like this may not be true, at least according to the majority of websites that popped up, and in fact it seems as if it is the opposite, i.e. AP exams are more widely accepted than CLEP. However, it looks as if my state university accepts the CLEP credit, and I would get the same credit from CLEP as the AP Calculus exam (assuming I got equivalent scores). But, once again, if I decide not to attend that university, then I might be out of luck with CLEP but able to take credit for AP.

Taking a test rather than going through the full course (in which ever direction) can be a dangerous move in my opinion.

Yes, I understand the dangers associated with this. I've found that I can get through many calculus problems (assuming it's an area that I've studied, i.e., not in depth multivariate theory problems), whether they are in physics or math, from various books/other sources that I have looked at. However, I'm not sure if these problems have necessarily tested my understanding. Do you have any sources, be it books or websites, that you believe would aptly test my understanding? I was also planning on putting some of my worked problems in the homework forums on here, so that they could be critiqued. Would you recommend this as another method to ensure my conceptual understanding?

It seems to me, from personal experience while tutoring, that the AP test was plug and chug in nature and no conceptual understanding is needed to pass. Therefore, these students never learned how to think about math.

On the same note that I asked Choppy, do you recommend any sources that I could look through to ensure that my conceptual understanding was sound?

To both Choppy and MidgetDwarf (or anyone else who wants to comment for that matter), I am currently working through Rosenlicht's Introduction to Analysis (currently spending time on the exercises in Chapter III on metric spaces since they are the foundations for analysis and many other areas of mathematics) and checking my proofs/solutions via several course websites that have worked solutions on them. Also, after I finish this book, I may move on to Rudin, or at least buy it as a reference (there seems to be a debate on whether it should be read thoroughly or merely referenced for the major points). Do you think that this Introduction to Analysis book is a suitable reference to ensure my theoretical and rigorous understanding of analysis (and therefore calculus)?

 

[Intraverno]

Just out of curiosity, what kind of extracurriculars and other classes are you taking? It's great that you're self studying your subjects of interest, but it's very important that you keep your options open and take classes in things you're not necessarily interested in as of now to see if you like them. Extracurriculars are also of course very very important for college applications.

The reason I ask is I had an almost identical plan going into high school (currently a freshman), but with sports year round and all accelerated honors Biology/Algebra 2/Science Research and Technology classes and AP Environmental Science to boot I don't have any time to self study. I can barely balance friends/relationships, grades (goal is to keep at least a 4.0 gpa every quarter and get an A as my final grade in all classes), and sports as is.

 

[Calaver]

I don't really want to turn this into a credentials thread, so I'll try to be brief.

Just out of curiosity, what kind of extracurriculars and other classes are you taking?

I think that I have a sufficient amount of other things going on; researching stuff that interests me isn't all that I do. I'm not skimping on any of my classes (all honors/A's), and participate in several extracurriculars (more actively in some than in others). The issue is, I attend a small private school that doesn't offer too many advanced courses, and all of the public schools in my area are no better than said private school (and are worse in several aspects).

It sounds like you're very lucky to be able to take Algebra 2, Science Research, and an AP class your freshman year - my school doesn't have such an advanced track.

 

[Intraverno]

I am lucky, yes, I go to a public school that has a program for advanced students with STEM interests. I couldn't be more thankful to live in an area that has such an opportunity, and I'm sorry that your school doesn't offer a good selection of advanced classes, there was a similar problem in my middle school and I know how frustrating it can get (though to a lesser extent than it must be for you ofc)

 

[Calaver]

I am lucky, yes, I go to a public school that has a program for advanced students with STEM interests. I couldn't be more thankful to live in an area that has such an opportunity, and I'm sorry that your school doesn't offer a good selection of advanced classes, there was a similar problem in my middle school and I know how frustrating it can get (though to a lesser extent than it must be for you ofc)

Well, I'd still much rather be in my current situation, in the 21st century with the Internet in a developed country than in mostly any other time period in history. The opportunities that I have received in terms of being able to understand nature (even just rudimentary knowledge) are enormous, and I find myself incredibly lucky to even be able to learn about such concepts, even if I can't or shouldn't attempt to claim formal credit for them at this point.

Additionally, my school isn't bad by any means, in fact it and the schools around it are extremely good by many standards. However, when it is compared to one of the best as yours seems to be, it can seem to fall short. Thus, it was remiss of me to paint my school and the area in a bad light, and I should have more accurately compared the two situations instead of zooming in on the gap between them.

 

[MidgetDwarf]

Not sure with analysis, I have not self studied it. I am going to go through Rosenlytch and Sherbet together for analysis, then tackle Rudin or Apostol. I think I may do either Courant or Spivak calculus, before starting analysis. Currently I am going through Tolstov: Fourier Analysis, Physics, and learning logic from the first few chapters : Principles of Mathematics by Oakly. Tolstov is a fun little book, very well written, and you have to work the book out as you go. I learned what it means for a series to be uniformly convergent and monotonic from this book, I knew the definition, but did not understand it.

Before going through analysis, I suggest you learn a bit of Linear Algebra. Linear Algebra can be abstract, and it is a good introduction to higher mathematics.
I would recommended 3 books on linear algebra. I would use Anton: Elementary Linear Algebra,Paul Shields Elementary Linear Algebra, Friedberg/Insel/ Spence: Linear Algebra. All 3 can be purchased for under 15 dollars shipped.

I forgot to mention that Friedberg is intended for a second course. It does explain everything needed in the text. However it may be above your level if you are not familiar with the basics of linear algebra. I would still buy it.

Anton walks the fine line between, a pure and applied math book. Definition/theorem. My only problem with the book is that when I was a student, the author, atleast to my eyes, was not clear when he used a column vector or row vector to preform some operation in the earlier chapters. Took me some playing around to figure it out. Better then most introductory books at this level.

Shields is an easy book, but well written. Mathwonk recommends this book heavily. It was written by a professor at Stanford. The book sticks to R^3 and does not go higher. It is a good supplement to Anton. It will give you intuition and give you examples in R^2/R^3 that you can generalize to R^n. I understood what a linear transformation was from this book.

Do you the basics of logic? If not either buy a discrete math book, I like Epps as noob, and Rosen now. Epps is more reader friendly. Or buy Principles of Mathematics by Oakly. Many previous generations of American Mathematicians learned logic from Oakly, then took a calculus course at the level of Spivak.

 

[bacte2013]

Not sure with analysis, I have not self studied it. I am going to go through Rosenlytch and Sherbet together for analysis, then tackle Rudin or Apostol. I think I may do either Courant or Spivak calculus, before starting analysis. Currently I am going through Tolstov: Fourier Analysis, Physics, and learning logic from the first few chapters : Principles of Mathematics by Oakly. Tolstov is a fun little book, very well written, and you have to work the book out as you go. I learned what it means for a series to be uniformly convergent and monotonic from this book, I knew the definition, but did not understand it.

Before going through analysis, I suggest you learn a bit of Linear Algebra. Linear Algebra can be abstract, and it is a good introduction to higher mathematics.
I would recommended 3 books on linear algebra. I would use Anton: Elementary Linear Algebra,Paul Shields Elementary Linear Algebra, Friedberg/Insel/ Spence: Linear Algebra. All 3 can be purchased for under 15 dollars shipped.

I forgot to mention that Friedberg is intended for a second course. It does explain everything needed in the text. However it may be above your level if you are not familiar with the basics of linear algebra. I would still buy it.

Anton walks the fine line between, a pure and applied math book. Definition/theorem. My only problem with the book is that when I was a student, the author, atleast to my eyes, was not clear when he used a column vector or row vector to preform some operation in the earlier chapters. Took me some playing around to figure it out. Better then most introductory books at this level.

Shields is an easy book, but well written. Mathwonk recommends this book heavily. It was written by a professor at Stanford. The book sticks to R^3 and does not go higher. It is a good supplement to Anton. It will give you intuition and give you examples in R^2/R^3 that you can generalize to R^n. I understood what a linear transformation was from this book.

Do you the basics of logic? If not either buy a discrete math book, I like Epps as noob, and Rosen now. Epps is more reader friendly. Or buy Principles of Mathematics by Oakly. Many previous generations of American Mathematicians learned logic from Oakly, then took a calculus course at the level of Spivak.

I personally read Oakley/Allendoerfer, Rosenlicht, Rudin, Apostol, Friedberg, and many books in the discrete mathematics for my current research in the theoretical computer science. I would strongly recommend Georgi Shilov's Elementary Real and Complex Analysis (or K. Hoffman's Analysis in Euclidean Space), which I think is far better than other analysis books I just mentioned. His book presents the basics of analysis in rigorous yet clear manner. He also starts with the general metric space just like Rudin and Apostol. I would not recommend Rosenlicht at all.

I did not like Friedberg since his exposition is that of Strichartz's Way of Analysis, although Friedberg can certainly be used as a first introduction to the linear algebra as he maintains both the computational and theoretical aspects of the linear algebra. I think Hoffman/Kunze (although I only read first few chapters) and Shifrin would be the better bet than Friedberg.

As for the discrete mathematics, I strongly love Knuth et al.'s Concrete Mathematics and Grimaldi's Discrete and Combinatorial Mathematics, which are more rigorous yet insightful than Epps and Rosen. I especially found Rosen very wordy and does not get to the points in a sufficient manner.

 

[MidgetDwarf]

I personally read Oakley/Allendoerfer, Rosenlicht, Rudin, Apostol, Friedberg, and many books in the discrete mathematics for my current research in the theoretical computer science. I would strongly recommend Georgi Shilov's Elementary Real and Complex Analysis (or K. Hoffman's Analysis in Euclidean Space), which I think is far better than other analysis books I just mentioned. His book presents the basics of analysis in rigorous yet clear manner. He also starts with the general metric space just like Rudin and Apostol. I would not recommend Rosenlicht at all.

I did not like Friedberg since his exposition is that of Strichartz's Way of Analysis, although Friedberg can certainly be used as a first introduction to the linear algebra as he maintains both the computational and theoretical aspects of the linear algebra. I think Hoffman/Kunze (although I only read first few chapters) and Shifrin would be the better bet than Friedberg.

As for the discrete mathematics, I strongly love Knuth et al.'s Concrete Mathematics and Grimaldi's Discrete and Combinatorial Mathematics, which are more rigorous yet insightful than Epps and Rosen. I especially found Rosen very wordy and does not get to the points in a sufficient manner.

Yes, there are better books then Epps. However, I found Epps to be a good transition from plug and chug mathematics to understanding formal. I liked Rosen for the coverage of material, but enjoyed Epps exposition extremely more.

Thanks for the sound advice regarding Analysis text. I think I will go through both volumes of Courant or Spivak ( have both, and I can get both parts of Apostol Calculus for free from an instructor) , and just go straight through Rudin or Apostol after. The university I will be attending uses Sherbet as an introductory analysis textbook. I think there is an honors version that uses Apostol.

I read the reviews of Strichart'z. Never seen or heard of the book till now. It seemed to be extremely informal and not mathematically rigorous from the reviews. I did not found Friedberg verbose at all. Proof/Theorem approach. Exercise were at a good level, atleast for me. Proofs were great and informative, no missing steps. I could even proof some of the theorems before seing the authors proof. I felt I was able to this, because the ideas leading up to the text were clear.

Thinking of going through Axler next or Shilov Linear Algebra.

I did read Hoffmans book, and it was OK. Found Friedberg to explain the concepts more intuitively. The price difference was also a contributing factor. I got a first edition of Friedberg for 99 cent from the school library sale.

 

[Calaver]

Not sure with analysis, I have not self studied it. I am going to go through Rosenlytch and Sherbet together for analysis, then tackle Rudin or Apostol. I think I may do either Courant or Spivak calculus, before starting analysis. Currently I am going through Tolstov: Fourier Analysis, Physics, and learning logic from the first few chapters : Principles of Mathematics by Oakly. Tolstov is a fun little book, very well written, and you have to work the book out as you go. I learned what it means for a series to be uniformly convergent and monotonic from this book, I knew the definition, but did not understand it.

Before going through analysis, I suggest you learn a bit of Linear Algebra. Linear Algebra can be abstract, and it is a good introduction to higher mathematics.
I would recommended 3 books on linear algebra. I would use Anton: Elementary Linear Algebra,Paul Shields Elementary Linear Algebra, Friedberg/Insel/ Spence: Linear Algebra. All 3 can be purchased for under 15 dollars shipped.

I forgot to mention that Friedberg is intended for a second course. It does explain everything needed in the text. However it may be above your level if you are not familiar with the basics of linear algebra. I would still buy it.

Anton walks the fine line between, a pure and applied math book. Definition/theorem. My only problem with the book is that when I was a student, the author, atleast to my eyes, was not clear when he used a column vector or row vector to preform some operation in the earlier chapters. Took me some playing around to figure it out. Better then most introductory books at this level.

Shields is an easy book, but well written. Mathwonk recommends this book heavily. It was written by a professor at Stanford. The book sticks to R^3 and does not go higher. It is a good supplement to Anton. It will give you intuition and give you examples in R^2/R^3 that you can generalize to R^n. I understood what a linear transformation was from this book.

Do you the basics of logic? If not either buy a discrete math book, I like Epps as noob, and Rosen now. Epps is more reader friendly. Or buy Principles of Mathematics by Oakly. Many previous generations of American Mathematicians learned logic from Oakly, then took a calculus course at the level of Spivak.

Thanks for all the recommendations! I actually have the book you're going through, Tolstov, sitting on my desk to read, perhaps after I finish my current reading of Rosenlicht, or perhaps later. I have read the first few pages just to get a "feel" for what the subject is all about.

As for analysis, I found the book I'm going through to be pretty good for an introduction to more abstract math. It will by no means teach me everything about the subject or even get close (it is called Introduction to Analysis for a reason), but has taught me to think more abstractly about topics (and like I said previously, I'm only on Chapter 3, which is about 1/4 of the way through the book). I would highly recommend it, as it builds up from set theory, to the real number system, to metric spaces, and then on to continuous functions and what we call analysis.

I plan to do Linear Algebra next, or soon after, so thanks for all your recommendations on the subject! Do you recommend that I go through all three books simultaneously, or read most of one first before tackling the others?

 

[Calaver]

I personally read Oakley/Allendoerfer, Rosenlicht, Rudin, Apostol, Friedberg, and many books in the discrete mathematics for my current research in the theoretical computer science. I would strongly recommend Georgi Shilov's Elementary Real and Complex Analysis (or K. Hoffman's Analysis in Euclidean Space), which I think is far better than other analysis books I just mentioned. His book presents the basics of analysis in rigorous yet clear manner. He also starts with the general metric space just like Rudin and Apostol. I would not recommend Rosenlicht at all.

I did not like Friedberg since his exposition is that of Strichartz's Way of Analysis, although Friedberg can certainly be used as a first introduction to the linear algebra as he maintains both the computational and theoretical aspects of the linear algebra. I think Hoffman/Kunze (although I only read first few chapters) and Shifrin would be the better bet than Friedberg.

As for the discrete mathematics, I strongly love Knuth et al.'s Concrete Mathematics and Grimaldi's Discrete and Combinatorial Mathematics, which are more rigorous yet insightful than Epps and Rosen. I especially found Rosen very wordy and does not get to the points in a sufficient manner.

Thank you too for all the recommendations.

Why don't you recommend Rosenlicht?

I have heard of Knuth, but not Grimaldi for discrete mathematics. Is there one that you would recommend reading before the other?

 

[bacte2013]

Thank you too for all the recommendations.

Why don't you recommend Rosenlicht?

I have heard of Knuth, but not Grimaldi for discrete mathematics. Is there one that you would recommend reading before the other?

Rosenlicht is very brief and not clear in general (particularly the series and sequences). I personally think his book is a good book as a brief review for someone who already studied the analysis. I strongly recommend either Hoffman or Shilov (much, much better than Rudin and Apostol combined). Plus, I found that Russian textbooks are generally very clear, motivating, and challenging at the same time.

Grimaldi is more or less at the same level as Knuth, being rigorous and challenging; it is definitely advanced than Rosen and Epps, although Grimaldi is bit older than both books. If I were you, I would just start reading either Knuth or Grimaldi as I think it is not a good idea to worrying about the prerequisites.

If you are really interested in the discrete mathematics and want to learn more about it, then I would recommend to actually read books that treat the separate topics of discrete mathematics (i.e. combinatorics, number theory, algorithm analysis, etc.). Here are some of my favorite books on those topics:

Combinatorics: Principles and Techniques in Combinatorics (C.C. Chen)
Graph Theory: A First Course in Graph Theory (Gary Chartrand), Graph Theory (Bondy & Murty)
Introductory Algorithms: Algorithm Design (J. Kleinberg),
Computability, Complexity: Introduction to the Theory of Computation (M. Sipser), Nature of Computation (Moore)
Number Theory: An Introduction to the Theory of Numbers (Niven et al.)

 

[bacte2013]

Yes, there are better books then Epps. However, I found Epps to be a good transition from plug and chug mathematics to understanding formal. I liked Rosen for the coverage of material, but enjoyed Epps exposition extremely more.

Thanks for the sound advice regarding Analysis text. I think I will go through both volumes of Courant or Spivak ( have both, and I can get both parts of Apostol Calculus for free from an instructor) , and just go straight through Rudin or Apostol after. The university I will be attending uses Sherbet as an introductory analysis textbook. I think there is an honors version that uses Apostol.

I read the reviews of Strichart'z. Never seen or heard of the book till now. It seemed to be extremely informal and not mathematically rigorous from the reviews. I did not found Friedberg verbose at all. Proof/Theorem approach. Exercise were at a good level, atleast for me. Proofs were great and informative, no missing steps. I could even proof some of the theorems before seing the authors proof. I felt I was able to this, because the ideas leading up to the text were clear.

Thinking of going through Axler next or Shilov Linear Algebra.

I did read Hoffmans book, and it was OK. Found Friedberg to explain the concepts more intuitively. The price difference was also a contributing factor. I got a first edition of Friedberg for 99 cent from the school library sale.

I did not read Courant and Spivak, but some of graduate mentors of my research group strongly recommend Courant for learning both the theories of calculus and the mathematical physics. I still stand by Shilov as better alternative to Rudin and Apostol altogether. I like Apostol but his problems are mediocre..
Oh! I would recommend E. Landau's Foundation of Analysis for a treatment of the construction of number system, which is briefly stated in many analysis books.

I agree Friedberg is a well-rounded book, but his arrangement of topics and problem sets are weird to me. I like Axler, which I am reading it now, and I am also planning to advance to Shilov once I finish it. If you already read Friedberg, then Axler should not be difficult to get through in a relatively quick manner.

 

[MidgetDwarf]

Thanks for all the recommendations! I actually have the book you're going through, Tolstov, sitting on my desk to read, perhaps after I finish my current reading of Rosenlicht, or perhaps later. I have read the first few pages just to get a "feel" for what the subject is all about.

As for analysis, I found the book I'm going through to be pretty good for an introduction to more abstract math. It will by no means teach me everything about the subject or even get close (it is called Introduction to Analysis for a reason), but has taught me to think more abstractly about topics (and like I said previously, I'm only on Chapter 3, which is about 1/4 of the way through the book). I would highly recommend it, as it builds up from set theory, to the real number system, to metric spaces, and then on to continuous functions and what we call analysis.

I plan to do Linear Algebra next, or soon after, so thanks for all your recommendations on the subject! Do you recommend that I go through all three books simultaneously, or read most of one first before tackling the others?

If you are ok reading analysis textbooks, i.e., working out most of the problems and proving all the statements on your own (NO HELP), then you can skip Paul Shields and Anton all together. Friedberg should be the book you start at. If you have trouble completing Friedberg when you do go through it, then try Shields/Anton.

 

[Calaver]

Rosenlicht is very brief and not clear in general (particularly the series and sequences). I personally think his book is a good book as a brief review for someone who already studied the analysis. I strongly recommend either Hoffman or Shilov (much, much better than Rudin and Apostol combined). Plus, I found that Russian textbooks are generally very clear, motivating, and challenging at the same time.

Grimaldi is more or less at the same level as Knuth, being rigorous and challenging; it is definitely advanced than Rosen and Epps, although Grimaldi is bit older than both books. If I were you, I would just start reading either Knuth or Grimaldi as I think it is not a good idea to worrying about the prerequisites.

If you are really interested in the discrete mathematics and want to learn more about it, then I would recommend to actually read books that treat the separate topics of discrete mathematics (i.e. combinatorics, number theory, algorithm analysis, etc.). Here are some of my favorite books on those topics:

Combinatorics: Principles and Techniques in Combinatorics (C.C. Chen)
Graph Theory: A First Course in Graph Theory (Gary Chartrand), Graph Theory (Bondy & Murty)
Introductory Algorithms: Algorithm Design (J. Kleinberg),
Computability, Complexity: Introduction to the Theory of Computation (M. Sipser), Nature of Computation (Moore)
Number Theory: An Introduction to the Theory of Numbers (Niven et al.)

Hmm...I'm reading Rosenlicht right now and I do agree that he is brief, but personally I find no unclear concepts in his book. I suppose that it is because I didn't go into his book expecting that he would go down all possible paths with analysis, because of the name Introduction to Analysis. I've found that the exercises (mostly proofs to do) make the reader think sufficiently about the concepts that he may have just brushed over in the text to the point where the missing information in the text is almost negligible due to the expansive nature of the exercises. Once again, by no means do I expect this to be my longest look into analysis, so that is possibly why I am being more lenient.

I looked at Shilov's (analysis) book, and it seems like that it may be the one I go to for my next look at the subject.

As for discrete mathematics, thanks for the list! It's on my list of things to study, but unfortunately I haven't gotten time to even dip my feet into the waters yet (unless you count applied programming stuff - writing a program to find solutions to a system of equations, to calculate basic interest, etc., nothing too theoretical). I'll probably start out by getting one of the more general books you mentioned (Knuth or Grimaldi), and move on from there to other areas that are of interest.

 

[Calaver]

If you are ok reading analysis textbooks, i.e., working out most of the problems and proving all the statements on your own (NO HELP), then you can skip Paul Shields and Anton all together. Friedberg should be the book you start at. If you have trouble completing Friedberg when you do go through it, then try Shields/Anton.

Well as I said earlier I'm only reading an introductory analysis text, and can prove about half of the statements in the text on my own and do most of the problems (mostly more proofs or "show this"), checking (comparing, in some sense) my answers (proofs) with solutions that have been posted online on various course websites that are currently using the text. With that said, I'll probably start with Shields/Anton just to be sure I get a good introduction to the material.

 

[MidgetDwarf]

https://www.amazon.com/dp/0470458216/?tag=pfamazon01-20

https://www.amazon.com/dp/0879011211/?tag=pfamazon01-20

http://www.abebooks.com/servlet/Boo...rchurl=sts=t&tn=linear%20algebra&an=friedberg

For the friedberg book. I have the first edition. I think the cover of the 2nd ed. is ghastly. But is just a cover. I could not find the first edition for you.