Relearning Math: Comparing Resources for the Best Learning Experience

[norxport]

Hey guys, first post. This might get sort of long, so brace yourselves. tl;dr at the end.

I’m a college student who just finished up the semester and has 4 months (and any free time during the following school year) to go all out and relearn all of the fundamentals of math up to but excluding calculus. I just placed a hefty order on Amazon, but I still feel confused about the content I chose. After tons of research, I’ve essentially narrowed down my list of resources in the following categories to the following books (they should be arriving by early-mid May).

Geometry:

Lang & Murrow - Geometry
Kiselev - Geometry Vol I & II
Moise - Elementary Geometry from an Advanced Standpoint
Gelfand - The Method of Coordinates
Gelfand - Trigonometry

Algebra:

Gelfand - Algebra
Gelfand - Functions and Graphs
Lang - Basic Mathematics (not sure if this is considered algebra or just thoroughly covers concepts that are needed for algebra)

Logic & Proofs:

Eccles - An Introduction to Mathematical Reasoning
Polya - How to Solve It
Velleman - How to Prove It

I chose these books because they were basically the most recommended for beginners in each topic. People said they were rigorous, meaty, and were great for adults who aren’t learning math for the first time. It really looks like they are, but I’m still a bit concerned.

I’d like to be able to work through the books I’ve ordered over the next year and spend my next summer trying to work through Spivak’s Calculus. The thing I’m kind of unsure of is how these books compare to other recommendations I’ve seen? For example, tons of people recommend Jacob’s geometry books, and comparing the page count alone, it looks like it’s a lot more content than the Kiselev books, yet it’s recommended at a lower frequency. Another example is comparing Gelfand’s Algebra, and Lang’s Basic Mathematics, to a more standard textbook like one from Blitzer or Jacob’s Elementary Algebra. Each of the latter books are many times the size of the former books combined, and even a comparison of the table of contents leaves a lot to be desired. Now I get that it’s an unfair comparison as there’s a ton of filler in the Blitzer books, but things like recursive formulas, logarithms, matrices, aren’t covered at all in Gelfand’s book, and Lang’s book although looks like it spends a few pages on each of those topics, there isn’t anything about recursion. All of those topics are covered at great length in Blitzer’s Algebra and Trigonometry 6th Edition. What I’m trying to get at is, should I have these other books as well to fill in these gaps with? I keep reading people saying that Gelfand and Lang are all you need to get ready for Spivak, but It seems like there’s a ton missing.

Another thing that’s puzzled me is how the books in my list compare to the Art of Problem Solving (AoPS) series. I’ve read they’re geared towards competition math, but they get great reviews. They’re regarded as a difficult set of beginner/intermediate level math books as well. How would someone who completes that curriculum compare to someone who goes through the list I’ve provided? Would one be better than the other at going further down the pure math path? Would the other have troubles adapting to competitive math later on? Is one type of student (competition vs pure math) generally regarded as a better mathematician than the other? I’ve noticed a lot of the AoPS books also have tons and tons of problems to solve. Would it be useful to just use them for the problems? Or will the things I learn not mesh well with them since they’re more focused on contests? I read through a short discussion regarding this on another site, not sure I'm allowed to post links so I won't, and it left me even more confused. There was barely any agreement throughout the whole discussion comparing the two approaches (AoPS vs Gelfand/Lang/Kiselev in this example). Two users went back and forth arguing over which is better than the other.

I guess I’m basically worried that I might not be learning all that I need to, or that things are explained in a different/better way elsewhere. I just want to make sure I’m using the best resources available to me. After all, if AoPS is the most popular curriculum for gifted students, they must be doing something right, right?

tl;dr I want to spend my summer relearning math. I ordered some of the most recommended books for algebra/geometry but it looks like they don’t cover as much as other books. Is it just because I'm not knowledgeable enough yet that I'm not comparing books properly just by looking at their table of contents? Will the books in my list really help me learn as much as Jacobs, AoPS, and Blitzer considering they cover way more stuff?

Thanks for any guidance you may be able to provide!

 

[MidgetDwarf]

The Lang book is good. But the geometry book from Moise is not appropriate for you at this point. It’s an advanced book on geometry. I have a bachelors in mathematics and I remember going through it my last semester In undergrad.

Instead, for geometry. I would recommend Moise/Downs Geometry (this is an intro). Very well written book and proofs are explained. Like most of Moise books.

The issue with this book is that it lacks constructions. Although, there is a section devoted to impossible constructions, the history, etc.

However, the books Planimetry(Kisselev) covers these shortcomings.

I would not get the Lang geometry if you get these two.

Then once you get used to Moise writing style, you can read his excellent Calculus book.

For algebra/trig. I just went through Axler:Pre-Calculus and Lang: Basic Mathematics together. Been years.

For the proofs book. I hate the formatting of how to prove it. Author talks to much. Not enough symbols for me.

I would get Hammok. Book of Proofs, and Levin: Discrete Mathematics, for intro proofs . Both books can be downloaded for free and legally by the authors.

 

[sysprog]

Please note, @norxport, that @MidgetDwarf knows how to break his text into paragraphs -- not bothering to do that makes it less likely that people will read the more lengthy of your stuff.

 

[norxport]

The Lang book is good. But the geometry book from Moise is not appropriate for you at this point. It’s an advanced book on geometry. I have a bachelors in mathematics and I remember going through it my last semester In undergrad.

Instead, for geometry. I would recommend Moise/Downs Geometry (this is an intro). Very well written book and proofs are explained. Like most of Moise books.

The issue with this book is that it lacks constructions. Although, there is a section devoted to impossible constructions, the history, etc.

However, the books Planimetry(Kisselev) covers these shortcomings.

I would not get the Lang geometry if you get these two.

Then once you get used to Moise writing style, you can read his excellent Calculus book.

For algebra/trig. I just went through Axler:Pre-Calculus and Lang: Basic Mathematics together. Been years.

For the proofs book. I hate the formatting of how to prove it. Author talks to much. Not enough symbols for me.

I would get Hammok. Book of Proofs, and Levin: Discrete Mathematics, for intro proofs . Both books can be downloaded for free and legally by the authors.

Thanks for the detailed response.

I wish I had known that before I bought the Moise book. I read around and a few people had recommended it for relearning high school geometry so I figured it would've been decent.

Do you suppose I could go through it without much trouble after finishing Kiselev? Or are there other more advanced concepts being used that I won't get introduced to without studying further with other resources?

I have a copy of Kenneth Rosen's Discrete Math book, and it goes over proofs in the first chapter. Hopefully I can supplement one with the other.

Please note, @norxport, that @MidgetDwarf knows how to break his text into paragraphs -- not bothering to do that makes it less likely that people will read the more lengthy of your stuff.

I tried to break it up into relevant ideas as much as I could but I think I had so many questions that the paragraphs sort of got out of control. I'll try to be more conscious of my formatting next time.

 

[MidgetDwarf]

Yes, its a very common error between his two geometry books. However, the one people recommended is the Moise/Downs. I own the first edition. Hmm. I would not recommend right after Kisselev, but it is doable. Theres some Analysis and Algebra (Modern) hidden within the book. It is also devoid of diagrams. Can be a bit dry.

You would learn such things as a field, characterization and what it means in mathematics. Some intro to Point Cure (I am misspelling this) model of Geometry. Its a very neat book, but does require mathematical maturity. Problems are easy compared to the information given within the text.

Personally, I would read the book, try it out, but not beat my pencil and paper too much. That is what Analysis is for lol.

If you can, I would return the book and order the Moise/Down geometry book instead. Its a very great combination. Kisselev could be a bit dense sometimes, but Moise really talks you through it.

I do not really like the books by Rosen. Although they are not bad books. I would just download the free proof books I mentioned. IRC, Levin has a neat explanation of logic quantifiers, and he does a problem solving approach. He starts every section by giving you a problem you are not supposed to be able to do. However, by working through the section, you can tackle it. So it kinda leads you to answer why you are learning this so to speak. Really breaks down "counting proofs." Something that I always have trouble with, because I do not really do these type of combinatoric proofs.

If you really want to put your proof writing abilities into action, then a great place to start is Number Theory. But you have enough there to study.

 

[MidgetDwarf]

Also, purchase a nice compass. You can be a bit cheap with the straightedge (fold an edge of paper in half), but not so much the compass.

 

[norxport]

Yes, its a very common error between his two geometry books. However, the one people recommended is the Moise/Downs. I own the first edition. Hmm. I would not recommend right after Kisselev, but it is doable. Theres some Analysis and Algebra (Modern) hidden within the book. It is also devoid of diagrams. Can be a bit dry.

You would learn such things as a field, characterization and what it means in mathematics. Some intro to Point Cure (I am misspelling this) model of Geometry. Its a very neat book, but does require mathematical maturity. Problems are easy compared to the information given within the text.

Personally, I would read the book, try it out, but not beat my pencil and paper too much. That is what Analysis is for lol.

If you can, I would return the book and order the Moise/Down geometry book instead. Its a very great combination. Kisselev could be a bit dense sometimes, but Moise really talks you through it.

I do not really like the books by Rosen. Although they are not bad books. I would just download the free proof books I mentioned. IRC, Levin has a neat explanation of logic quantifiers, and he does a problem solving approach. He starts every section by giving you a problem you are not supposed to be able to do. However, by working through the section, you can tackle it. So it kinda leads you to answer why you are learning this so to speak. Really breaks down "counting proofs." Something that I always have trouble with, because I do not really do these type of combinatoric proofs.

If you really want to put your proof writing abilities into action, then a great place to start is Number Theory. But you have enough there to study.

Ah, I see. I think at this point it's probably best to just wait for it to get delivered and see how intimidated I get after I flip through it. I just checked and Moise/Downs Geometry looks to be roughly $90 CAD more than the one I purchased.

I just downloaded Hammok and Levin's books, thanks for those recommendations.

Also, purchase a nice compass. You can be a bit cheap with the straightedge (fold an edge of paper in half), but not so much the compass.

Just looking at compasses now. What do you consider the price point for a nice compass to be? I saw two Staedtler ones online. The one with an extension arm included is $12 and another without is $35. I imagine the price is based on the quality of the build. Would the $35 one really be necessary, even without the extendable arm?

 

[MidgetDwarf]

https://www.thriftbooks.com/w/geometry_floyd-l-downs-jr/257493/item/5956128/#isbn=0201050285&idiq=5956128

here is a cheap edition. I am not sure which edition it is. but you won't loose to much. if you add $3.20 with an extra book, you get free shipping. I think its the second. I have only seen the first (the one I own).

For the compass, I have vintage that were given to me. The closet I seen to decent quality is the one that Rotring produces with the metal wheel. I believe $20ish.

 

[MidgetDwarf]

are you in America?

 

[norxport]

I'm in Canada. But that link you shared looks like they ship here as well! For that price, it seems like a no-brainer.

Looking up the Rotring compass now.

 

[MidgetDwarf]

I'm in Canada. But that link you shared looks like they ship here as well! For that price, it seems like a no-brainer.

Looking up the Rotring compass now.

Great. Hopefully nothing changed to much from the first to that one. Typically, older editions of books are better.

Take for instance Thomas Calculus With Analytic Geometry 3rd to the editions after (5th I think). It Is not the same book. The 3rd edition of Thomas is a gem, although an applied book. But the ones past the (5th?) are junk.

Another website to look for older books is abebooks.com

also, for future studies in mind. Springer has made available some of their science/math books. Abbot: "Understanding Analysis", and Axler: Linear Algebra Done Right are free for download right now. I would download them, and read them in a year or so when you are ready.Maybe you like programming? Those are also free, but I do not know what's good on that list. I downloaded everything lol/

 

[norxport]

Great. Hopefully nothing changed to much from the first to that one. Typically, older editions of books are better.

Take for instance Thomas Calculus With Analytic Geometry 3rd to the editions after (5th I think). It Is not the same book. The 3rd edition of Thomas is a gem, although an applied book. But the ones past the (5th?) are junk.

Another website to look for older books is abebooks.com

also, for future studies in mind. Springer has made available some of their science/math books. Abbot: "Understanding Analysis", and Axler: Linear Algebra Done Right are free for download right now. I would download them, and read them in a year or so when you are ready.Maybe you like programming? Those are also free, but I do not know what's good on that list. I downloaded everything lol/

I've seen that mentioned quite a bit. The newer editions of quite a few math books being recommended were regarded as watered down, or not as rigorous.

Abebooks has come up in my searches a few times. Although they carry a solid selection, the shipping costs and wait times often have me resorting back to Amazon.

I think I saw the free Springer books posted on these forums a couple days ago. I'm going to take a look at them now, but I imagine a lot of them will be too advanced for me. That being said I didn't see the programming ones, definitely going to search for those. One of the reasons I'm trying to relearn math is in hopes of it translating to better programming skills, at least in terms of my mindset while approaching a problem.

 

[jtbell]

I’m a college student who just finished up the semester

Which semester? (sorry, I'm feeling a bit grumpy tonight before going to bed...)

 

[sysprog]

Which semester? (sorry, I'm feeling a bit grumpy tonight before going to bed...)

You're up until 2:00 AM ? well then grumpy is no wonder -- my HS girlfriend used to get 'grumpy' sometimes, so I think I know what it means -- I tend to just take naps so I'm up whenever . . .

 

[norxport]

Which semester? (sorry, I'm feeling a bit grumpy tonight before going to bed...)

Sorry, I should've been more clear. Second semester of first year of an information security program.

 

[jamesson]

I defer to others here regarding geometry and proofs, but for precalculus and algebra I highly recommend Larson. Free worked solutions online for odd-numbered problems.

 

[sysprog]

Sorry, I should've been more clear. Second semester of first year of an information security program.

Please look at the Orange Book even though the industry requirements in general aren't the same as those of DoD -- https://en.wikipedia.org/wiki/Trusted_Computer_System_Evaluation_Criteria

 

[sysprog]

Ah, I see. I think at this point it's probably best to just wait for it to get delivered and see how intimidated I get after I flip through it. I just checked and Moise/Downs Geometry looks to be roughly $90 CAD more than the one I purchased.

I just downloaded Hammok and Levin's books, thanks for those recommendations.
Just looking at compasses now. What do you consider the price point for a nice compass to be? I saw two Staedtler ones online. The one with an extension arm included is $12 and another without is $35. I imagine the price is based on the quality of the build. Would the $35 one really be necessary, even without the extendable arm?

Staedtler-Mars is a clear yes in my view . . .

 

[Mondayman]

I didn't like Basic Mathematics. It's not that it's poorly written, it just feels rushed. The book by Axler is better organized for precalculus. And there's Mathematical Proofs by Chartrand/Zhang, it's better than the book by Velleman in my opinion.

I have Rosen's books on number theory and discrete mathematics. They aren't bad and can be had for a decent price. If you got started on these topics, you would get good practice with proofs before moving on to Spivak.

I have also been told it doesn't hurt to have a cookbook calculus text like Stewart to go alongside Spivak if it's your first go around with calculus, because doing the computational stuff is good for motivating the material and building intuition.

 

[norxport]

I defer to others here regarding geometry and proofs, but for precalculus and algebra I highly recommend Larson. Free worked solutions online for odd-numbered problems.

Thanks for the recommendation. I certainly might need it for supplemental material, or at least for extra problems to practice with!

Please look at the Orange Book even though the industry requirements in general aren't the same as those of DoD -- https://en.wikipedia.org/wiki/Trusted_Computer_System_Evaluation_Criteria

I had actually read up about the Rainbow Series a little while back, but forgot all about them. Thanks for bringing them back to my attention. This one in particular looks like it's a lot more applicable to the fields I'm interested in pursuing.

My only concern is the possibility of the information being dated. On that Wikipedia page, it states that the Orange Book was canceled in 2002, and then reissued in 2014. Either way, I imagine there's a ton of knowledge to be gained.

Staedtler-Mars is a clear yes in my view . . .

I'm going to place an order tonight. Really torn between the Staedtler Mars 552 and the Rotring set that comes with the extension. Going to be reading some more reviews and going with one of the two.

I didn't like Basic Mathematics. It's not that it's poorly written, it just feels rushed. The book by Axler is better organized for precalculus. And there's Mathematical Proofs by Chartrand/Zhang, it's better than the book by Velleman in my opinion.

I have Rosen's books on number theory and discrete mathematics. They aren't bad and can be had for a decent price. If you got started on these topics, you would get good practice with proofs before moving on to Spivak.

I have also been told it doesn't hurt to have a cookbook calculus text like Stewart to go alongside Spivak if it's your first go around with calculus, because doing the computational stuff is good for motivating the material and building intuition.

I thought about going with a more standard precalculus text, but only only worry is that it won't go over the fundamentals in as much detail. Even though I finished 2 semesters of discrete math with strong grades, I still have to acknowledge that I have a huge amount of holes in my algebra skills. I haven't been in a high school math class in roughly 10 years. Never took math in grade 12 because I figured I'd never need it (one of my biggest regrets).

So what I tried to do was pick material that would not only reacquaint me with things I have forgotten, but reteach it to me with great depth. Luckily, after making this post I did order Cohen's Precalculus with Unit Circle Trigonometry and Jacob's Elementary Algebra in order to have more problems to work through.

I was debating between those two and going with something from the Art of Problem Solving (AoPS) line up, but since no one bothered to chime in about those, I figured they might not be as good as others make them out to be.

As for Rosen's book, I had already purchased it last September as it was the course textbook. I really enjoyed it personally, but having nothing else to compare it to, I'm not sure if that's saying much.

I have about a year to go until I can start to worry about Spivak, but when I do, I'm sure I'll accompany it with a more computational text to get some practice. As of right now, I feel nervous about my basic algebra skills lol. April 2021 is my goal to be able to get started with my calculus studies.

 

[sysprog]

You seem to me to be well-oriented on very good tracks. Rotring is good. I think that you'll never regret Staedtler-Mars.

 

[sysprog]

If you want a great, albeit for most people difficult, book, please check here: https://www-cs-faculty.stanford.edu/~knuth/taocp.html

 

[norxport]

If you want a great, albeit for most people difficult, book, please check here: https://www-cs-faculty.stanford.edu/~knuth/taocp.html

That series is on my holy grail list. Hopefully once I graduate, I'll be able to understand the first chapter of the first volume

 

[sysprog]

That series is on my holy grail list. Hopefully once I graduate, I'll be able to understand the first chapter of the first volume

It's not easy . . .

 

[MidgetDwarf]

I personally don’t care for the art of problem solving books. But I’m into pure mathematics.You can just skip Spivak, and go straight to Analysis (Abbot) but I am not sure how that would fit into your career goals.

For an applied math book. Moise Calculus or Thomas Calculus 3rd edition.

Although Moise is a middle ground between Spivak and Courant. Kind of like if Spivak and Courant were to have a baby you would get Moise.

 

[Hsopitalist]

I did this last year and I'd recommend EdX courses online, algebra and then precalculus, the ones you can get college credit for. Worked well for me. In addition to the other stuff mentioned, of course. Logic and proofs aren't in those, of course

 

[norxport]

I personally don’t care for the art of problem solving books. But I’m into pure mathematics.You can just skip Spivak, and go straight to Analysis (Abbot) but I am not sure how that would fit into your career goals.

For an applied math book. Moise Calculus or Thomas Calculus 3rd edition.

Although Moise is a middle ground between Spivak and Courant. Kind of like if Spivak and Courant were to have a baby you would get Moise.

I'm not sure how much I downplayed my lack of mathematical maturity in my earlier posts, but I get anxiety just thinking about analysis. I've never even had a single lesson in calculus yet. If miraculously, calculus doesn't seem completely impossible after finishing my current reading list, I may have an early introduction to analysis.

As for my career goals, I don't expect any direct benefits from any of this self-study. I'm simply hoping it translates to a better thought process while coding, and more importantly, I'm doing this because this past year I've realized how much I actually love math and regret not having taken all of these classes when I was younger.

That description of Moise's calculus sounds phenomenal. Once I get to that level, I'll be sure to check it out!

I did this last year and I'd recommend EdX courses online, algebra and then precalculus, the ones you can get college credit for. Worked well for me. In addition to the other stuff mentioned, of course. Logic and proofs aren't in those, of course

Funny you should mention EdX. I'm currently doing Harvard's CS50 course on there and it's going very well. I did a quick browse through their math courses and they do look solid. Having said that, for MOOC style learning in regards to math, I often see people recommend MIT Opencourseware. Any input on the comparison of the two platforms?

 

[Hsopitalist]

Funny you should mention EdX. I'm currently doing Harvard's CS50 course on there and it's going very well. I did a quick browse through their math courses and they do look solid. Having said that, for MOOC style learning in regards to math, I often see people recommend MIT Opencourseware. Any input on the comparison of the two platforms?

The EdX courses I took, thru ASU, were great because they used the Pearson format and basically force you to work problems. But it's a non-threatening system, at least to me anyway. For minimal cost you can register on the site and actually get college credit at the end of each class if you do the tests and if that's important to you.

I've been at this for about a year and I'm telling you math looks completely different from this side than it did when I started, I'm really enjoying it. Right now I'm working through Chris McMullen's calculus based workbook for physics, volume 1. It's got a very gentle reintroduction to important algebra, calculus and trigonometry in the first hundred pages.

Oh, and my brother is a programmer. He tells me there's a clear difference in his line of work between those who worked their way through math and those who didn't, fwiw.

 

[econreader]

Everything by Lang in general is very good but can/should be supplemented for additional reading. If your goal is to read Spivak, why not read a standard calculus textbook first? I liked Lang's "A First Course" very much. It's a bit lengthy though, so you'll probably want to skip some parts.

From your list, Gelfand's are not textbooks but very short booklets aimed at high school pupils. They are fine but you shouldn't expect to get as much from them as from a standard textbook. Polya's is a weird book, because most of it is a dictionary. If you want to practice proofs, then studying geometry, discrete mathematics, number theory or abstract algebra helps. I liked the book by Eccles in particular for learning abstract mathematics.

I think you may still want to keep the book by Moise, because it also spends the first chapters on introduction to abstract mathematics. But I agree that it is a second book on geometry rather than a first.

 

[norxport]

The EdX courses I took, thru ASU, were great because they used the Pearson format and basically force you to work problems. But it's a non-threatening system, at least to me anyway. For minimal cost you can register on the site and actually get college credit at the end of each class if you do the tests and if that's important to you.

I've been at this for about a year and I'm telling you math looks completely different from this side than it did when I started, I'm really enjoying it. Right now I'm working through Chris McMullen's calculus based workbook for physics, volume 1. It's got a very gentle reintroduction to important algebra, calculus and trigonometry in the first hundred pages.

Oh, and my brother is a programmer. He tells me there's a clear difference in his line of work between those who worked their way through math and those who didn't, fwiw.

I just checked out the courses from ASU. It looks like they have precalculus and college algebra/problem solving ones open for enrolment right now.

I think I just got so many resources now that I might wait until I get through the more elementary stuff and use the precalculus course they offer to supplement Santos' precalculus book that I downloaded.

As for what your brother said, I already feel it. This year that just passed, I had 2 programming courses. Although I did very well, it took me a lot more effort to get through some of the more challenging assignments than my classmates who just graduated high school and took calculus and advanced functions in grade 12. That's part of what motivated me to dig deeper into math and make sure I fill in the gaps by 3rd year.

Everything by Lang in general is very good but can/should be supplemented for additional reading. If your goal is to read Spivak, why not read a standard calculus textbook first? I liked Lang's "A First Course" very much. It's a bit lengthy though, so you'll probably want to skip some parts.

From your list, Gelfand's are not textbooks but very short booklets aimed at high school pupils. They are fine but you shouldn't expect to get as much from them as from a standard textbook. Polya's is a weird book, because most of it is a dictionary. If you want to practice proofs, then studying geometry, discrete mathematics, number theory or abstract algebra helps. I liked the book by Eccles in particular for learning abstract mathematics.

I think you may still want to keep the book by Moise, because it also spends the first chapters on introduction to abstract mathematics. But I agree that it is a second book on geometry rather than a first.

I actually placed an order for a few other supplementary texts that might give me more problems to work through. I chose not to go with the AoPS content as no one really brought them up, but compared to my first list, I've also added the following:

Euler - Element's of Algebra
Euclid - 13 books Vol I-III, translated by Thomas Heath (the Dover editions)
Jacobs - Elementary Algebra
Jacobs - Geometry
Cohen - Precalculus with Unit Circle Trigonometry

My approach has kind of changed as well due to how much the list of material increased since making this post. I plan on using Lang and Gelfand as a main source of what I should be learning, and dig for those topics in the supplemental books for extra problems. Of course if the initial explanations by Lang and Gelfand get confusing, it'll be nice to have multiple perspectives as well.

I noticed the page count of Gelfand's books as well. I figured they were so short because they weren't filled with a bunch of pictures, and popups and whatever else algebra books printed in the 2010s contain. Again, if they aren't detailed enough, I plan to supplement them with Jacob's Elementary Algebra. I will say that I'm glad you claim they are aimed at high school students though. As of right now I consider my algebra skills to be at a grade 9 level at the most. To paint a clearer picture, I had to have a friend teach me FOIL while we were learning mathematical induction last semester.

I regret not looking deeper into Polya's book before purchasing it. Until you mentioned it, I didn't realize it was just a bunch of definitions. It looks like there's a page or two of problems at the end but they look rather lacklustre. Luckily it was one of the cheapest books in the list, so I hope I'll get at least some use out of it. Between the book by Eccles, the free discrete math book I downloaded by Oscar Levin (thanks MidgetDwarf), and the copy of Rosen's discrete math book I currently own, Polya will undoubtably be on the back burner for quite some time.

The book by Eccles that you mention helped you learn abstract mathematics, do you mean the one that I ordered?

Lastly, I definitely will be keeping the book by Moise. First book or not, I found a pdf online and did a quick skim. It looks like it'll be an incredible resource down the road.

 

[Hsopitalist]

One last thing. I gave myself a year to do algebra, precalculus and trig. Starting with those two EdX courses made a HUGE difference there. Now I'm on to calculus I'm doing at least ten problems per day and the steady exposure really helps.

And I use a trig text by Lial. I've found it quite useful in explaining the really basic stuff.

Good luck and get to work! Keep us posted on your progress

 

[econreader]

I noticed the page count of Gelfand's books as well. I figured they were so short because they weren't filled with a bunch of pictures, and popups and whatever else algebra books printed in the 2010s contain.

Well, trigonometry is the only one which is a bit like a textbook. It is still kind of short (but I guess the length is normal for a school textbook). Of course, Gelfand being one of the top mathematicians, one can still learn a lot from anything he writes. Same with Polya. I just wouldn't expect these books to replace read & drill from more conventional sources. I don't know much about precalculus books since I learned it while in secondary school, but you may want to check one of the books by Allendoerfer and Oakley which is often recommended here. I read their book as a refresher some time ago and I can confirm that it is awesome. Also, it's very cheap for a second hand copy.

Yes, I read the same book by Eccles as what you ordered. I was skeptical at first but it is actually quite good unlike some "introduction to proofs" books which are often superficial. I supplemented it with Norman Biggs' Discrete mathematics for a couple of topics, but since you have one by Rosen, you should be able to use it instead.

The books by Euclid are mind-blowing but will require a very serious commitment - so I personally put them aside till I have free time to study them. Same with Euler - it's fun to read a genius (and actually quite easy to follow this one), but he goes into topics which are not so immediately pressing to learn, so I'd keep it aside and turn to it for fun reading and use the time to focus on the shortest reading path to calculus.

 

[MidgetDwarf]

I just checked out the courses from ASU. It looks like they have precalculus and college algebra/problem solving ones open for enrolment right now.

I think I just got so many resources now that I might wait until I get through the more elementary stuff and use the precalculus course they offer to supplement Santos' precalculus book that I downloaded.

As for what your brother said, I already feel it. This year that just passed, I had 2 programming courses. Although I did very well, it took me a lot more effort to get through some of the more challenging assignments than my classmates who just graduated high school and took calculus and advanced functions in grade 12. That's part of what motivated me to dig deeper into math and make sure I fill in the gaps by 3rd year.
I actually placed an order for a few other supplementary texts that might give me more problems to work through. I chose not to go with the AoPS content as no one really brought them up, but compared to my first list, I've also added the following:

Euler - Element's of Algebra
Euclid - 13 books Vol I-III, translated by Thomas Heath (the Dover editions)
Jacobs - Elementary Algebra
Jacobs - Geometry
Cohen - Precalculus with Unit Circle Trigonometry

My approach has kind of changed as well due to how much the list of material increased since making this post. I plan on using Lang and Gelfand as a main source of what I should be learning, and dig for those topics in the supplemental books for extra problems. Of course if the initial explanations by Lang and Gelfand get confusing, it'll be nice to have multiple perspectives as well.

I noticed the page count of Gelfand's books as well. I figured they were so short because they weren't filled with a bunch of pictures, and popups and whatever else algebra books printed in the 2010s contain. Again, if they aren't detailed enough, I plan to supplement them with Jacob's Elementary Algebra. I will say that I'm glad you claim they are aimed at high school students though. As of right now I consider my algebra skills to be at a grade 9 level at the most. To paint a clearer picture, I had to have a friend teach me FOIL while we were learning mathematical induction last semester.

I regret not looking deeper into Polya's book before purchasing it. Until you mentioned it, I didn't realize it was just a bunch of definitions. It looks like there's a page or two of problems at the end but they look rather lacklustre. Luckily it was one of the cheapest books in the list, so I hope I'll get at least some use out of it. Between the book by Eccles, the free discrete math book I downloaded by Oscar Levin (thanks MidgetDwarf), and the copy of Rosen's discrete math book I currently own, Polya will undoubtably be on the back burner for quite some time.

The book by Eccles that you mention helped you learn abstract mathematics, do you mean the one that I ordered?

Lastly, I definitely will be keeping the book by Moise. First book or not, I found a pdf online and did a quick skim. It looks like it'll be an incredible resource down the road.

Glad you liked Moise. I find his exposition very clear, informative, and sometimes funny. However, his Elementary Geometry From An Advance Standpoint, can be sometimes dry. Its a good book tho, but many people confuse the audience which the book is aimed for, ie., the elementary part. This kind of reminds me of Komogorov: Introduction To Real Analysis. This book is not really meant for a first course in analysis, however, it has some weird chapters that are aimed for intro courses., ie., the intro to sets.

 

[MidgetDwarf]

I would hold off on buying any more sources. You have enough to last at least 2 years. The Gelfand book on trig is very neat, although it lacks problems/topics. But everything contained within it is lucid. There is a neat proof of the Pythagorean theorem, that is not so trivial. It uses some advanced Theorems about geometry. However, he dumbs it down that even a kid can understand. Although, the proof given in Gelfand is more intuitive than mathematical rigor. You have a pre-calculus book. Combine that with Gelfand, and you are good at intro trig. One can dive deeper, but I think its good enough.

This same proof was required in my Modern Geometry class, of course, the more rigorous proof. But the teacher explained it so confusing (we used his lecture notes) during a previous lecture. Needless to say, I was the only person in my class to understand and write a correct proof, since I seen the intuitive explanation in Gelfand 5 years prior.

 

[econreader]

This kind of reminds me of Komogorov: Introduction To Real Analysis. This book is not really meant for a first course in analysis, however, it has some weird chapters that are aimed for intro courses., ie., the intro to sets.

I haven't read it myself but I know that the Russian original is based on material taught in the third part of an analysis course at the top national university. No idea why the English edition was titled "Introduction".