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Did I choose the right books for relearning math (Geometry, Algebra, Logic and Proofs)?

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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 wanna 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!
 

Answers and Replies

  • #2
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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.
 
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  • #3
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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.
 
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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.
 
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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.
 
  • #6
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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.
 
  • #7
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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?
 
  • #8
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https://www.thriftbooks.com/w/geome...93/item/5956128/#isbn=0201050285&idiq=5956128

here is a cheap edition. im 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.
 
  • #9
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are you in America?
 
  • #10
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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.
 
  • #11
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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/
 
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  • #12
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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.
 
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  • #13
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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...)
 
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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 . . .
 
  • #15
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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.
 
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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.
 
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  • #18
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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 . . .
 
  • #19
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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.
 
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  • #20
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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 cancelled 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.
 
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  • #21
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You seem to me to be well-oriented on very good tracks. Rotring is good. I think that you'll never regret Staedtler-Mars.
 
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  • #24
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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 . . . 😌
 
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  • #25
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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.
 

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