- #26

Hsopitalist

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- #26

Hsopitalist

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- #27

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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 im 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!

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?

- #28

Hsopitalist

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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.

- #29

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From your list, Gelfand's are not textbooks but very short booklets aimed at highschool 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.

- #30

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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.

From your list, Gelfand's are not textbooks but very short booklets aimed at highschool 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.

- #31

Hsopitalist

Gold Member

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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

- #32

- 15

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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.

- #33

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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.

- #34

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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.

- #35

- 15

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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".

- #36

- 12

- 5

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

That's basically what I'm doing right now. Going to try and get the bulk of the foundation laid out over the summer, and then try to keep it fresh in my brain during the school year.

I'd love to keep you guys posted on my progress. It'll be a good way to motivate myself to stay committed!

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.

I've seen the Allendoerfer and Oakley books go for very cheap second hand. But as MidgetDwarf said, I'm going to try to put a pause on any more purchases. If I can't get to a point where I feel like I'm ready for calculus with my current resources, I may have other things to worry about, haha.

I imagine the book by Eccles and Hammack's free proof book will be more than enough to get a better grasp on proofs. I should've mentioned that in first semester we did a couple weeks on proof by contrapositive, contradiction and mathematical induction. The thing is, between the rush of being back in school for the first time in a number of years, and not having had done math in nearly a decade, a lot of it was learned at a very shallow level. I plan to dive much deeper this time around.

The books by Euclid and Euler were part of the additional materials I purchased after I initially made this post. I don't expect to make them part of the core part of this curriculum. But I do think I'll get a lot out of them after finishing some of the material I mentioned in the first post. I plan on waiting to go through Euclid's books until I get a copy of Hartshorne to accompany it with. This probably won't be until next summer.

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.

Couldn't agree more. I got a bit carried away with this new found excitement.

You got me very excited for the trig book by Gelfand. I am curious about one thing though. Since there aren't any problems to practice with in it, do you think Cohen's Precalculus with Unit Circle Trigonometry will be enough practice to hammer home those concepts? My ignorance might show a bit here as I'm unaware if the trigonometry that Gelfand covers and the things covered in Cohen's book are different from each other.

Lastly, that's the thing with teaching. You can know the material inside out, forwards backwards. But that doesn't make you a great teacher. I imagine there's a reason why out of the endless list of publications out there, I see the same few authors get recommended over and over.

- #37

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But sometimes, while taking a bath or walking. Things click. I can replay some of his lectures by memory years later. So I don’t think he’s that bad.

How do you like the books so far?

- #38

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But sometimes, while taking a bath or walking. Things click. I can replay some of his lectures by memory years later. So I don’t think he’s that bad.

How do you like the books so far?

Haha, I certainly didn't mean your teacher was bad. Sometimes being introduced to a new topic by a teacher who's extremely intelligent can be confusing. It's tough for them to put themselves in the students' shoes.

The books are great! So far I've only received the Gelfand books, Lang's Basic Mathematics, Euclid's books and Velleman's How to Prove It.

I'm working through Lang right now and it's exactly what I needed. With the rest of the shipments that are yet to come, I've got a very busy summer ahead of me!

- #39

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