Best Pre-Calculus Textbook for Depth, Problems, and Understanding?

>>8903319
honestly just go into calculus already and fix what you're lacking as you go
ocw.mit.edu 18.01sc

What the fuck is pre calculus

>>8903338
pretty much just trig and more algebra practice

>>8903319
Algebra by Gelfand and Shen
Functions and Graphs by Gelfand, Glagoleva, and Shnol
The Method of Coordinates by Gelfand, Glagoleva, and Kirillov
Trigonometry by Gelfand and Saul

>probability, combinatorics

Wait until after you've done calculus.

>vectors, matrices

Introduction to Linear Algebra by Marcus and Minc (out of print but easy to pirate)

>>8903344
If you haven't got a good gras of that by the ebd of HS you should kys.

Idk OP, just open the calc textbook and see if you lack something.

>>8903338
2nd degree and higher algebra, trig, exponentials and logarithms

>>8903336

It really doesn't help. Again, you only learn "what you need" but you don't learn to manipulate functions with ease. I got 5s in AP Calculus AB and BC and a 6 in IB HL Mathematics without really "getting" pre-calculus any more than I had to. Couldn't make the AMC cutoffs, occasionally had stupid intuitions about manipulating functions, etc. Probably because I've been "plugging and chugging" with only a modicum of adaptive problem-solving for most of my mathematics career.

I want to change that, starting with difficult pre-calculus problems and moving onwards to Apostol or something.

>>8903350

>Algebra by Gelfand and Shen
>Functions and Graphs by Gelfand, Glagoleva, and Shnol
>The Method of Coordinates by Gelfand, Glagoleva, and Kirillov
>Trigonometry by Gelfand and Saul

This is a good list. If I had more time than a summer, I would do this instead. But I like the textbook that I found by Stitz-Zeager, and maybe that would suffice.

>Wait until after you've done calculus.

I don't think that is a good idea. I understand that a solid intro to probability class will require heavy use of calculus, but a lot of classes that I will be taking will require familiarity with basic combinatorics to solve applied problems.

>Introduction to Linear Algebra by Marcus and Minc (out of print but easy to pirate)

Is this as the same level as the first four books that you've suggested? Because if it is, then I'll look for a used copy.

>>8903364
oh, are you interested in more serious mathematics?

if you already had a shitty plug and chug calculus class, and you want to get correct intuitions, it might just be time you grabbed Apostol and started working through it. it's not too hard.

if you're really interested in math I might even suggest starting analysis and linear algebra already. since you asked about linear algeba, Hoffman & Kunze is a great, serious book. you don't sound like someone who would appreciate shitty "linear algebra" for engineers stuff

>>8903364
also olympiads are very specific. to be good at IMO you need to train for IMO and learn a lot of shit. that's not a good way to learn general math at this point

>>8903373

>if you already had a shitty plug and chug calculus class, and you want to get correct intuitions, it might just be time you grabbed Apostol and started working through it. it's not too hard.

>if you're really interested in math I might even suggest starting analysis and linear algebra already. since you asked about linear algeba, Hoffman & Kunze is a great, serious book.

Thanks, I'll keep this in mind when I get to them.

>if you're really interested in math I might even suggest starting analysis and linear algebra already. since you asked about linear algeba, Hoffman & Kunze is a great, serious book. you don't sound like someone who would appreciate shitty "linear algebra" for engineers stuff

There's a difference between "here's the rules, here's the examples, now solve some similar problems" vs. "we're going to make you understand the basics, give you a few examples, and then you're on your own to crunch numbers and find shortcuts that will make you extremely familiar with quantitative methods". I'm interested in "Art of Problem-Solving"-style math, too.

>>8903376

I don't want to be an IMO competitor. I'm not looking to ace every problem. But it would be nice to be competent enough to feel comfortable with mathematical models and problem-solving to solve a commendable amount of them.

>>8903378
don't really understand what you mean, what are you after exactly?

>>8903371
which book is that?
is it in line with OP's message?

>>8903401
fuck

>>8903371
Gelfand's books are short and you could work through each of them in about a week.

>Is this as the same level as the first four books that you've suggested

No, Gelfand's books were written for gifted middle/high school kids as a supplement to their schooling. They're great for their problems that force you to understand the material and cover stuff you don't usually see in school. Marcus is a freshman linear algebra book for STEM majors.

>>8903364
>I got 5s in AP Calculus AB and BC
>I want to change that, starting with difficult pre-calculus problems and moving onwards to Apostol or something.

You're falling for the classic trap of being afraid of moving forward in mathematics. Usually the stuff you're shaky about will be clarified in later advanced courses. Instead of looking for a difficult precalculus book, look for a book on proofs like "A Transition to Advanced Mathematics" then study modern/abstract algebra and analysis.

>>8903319
>as I re-enter university

How long has it been since you last math course?

>>8903319
make sure you can factor anything

make sure you can PROVE trigonometric identities

make sure you know how to do basic vector problems

>>8903319
I'd suggest Mathematical Thinking: Problem solving and proofs

Professor Leonard on youtube has some great stuff.

>>8903404

>You're falling for the classic trap of being afraid of moving forward in mathematics. Usually the stuff you're shaky about will be clarified in later advanced courses. Instead of looking for a difficult precalculus book, look for a book on proofs like "A Transition to Advanced Mathematics" then study modern/abstract algebra and analysis.

I don't consider mathematics to be a linear progression of one topic to the next. After a certain point, there's horizontal and lateral development. For example, university-level calculus (I mean calc 1/2/3) is way easier than any of the pre-calculus problems you'd have to solve in problem-solving competitions until you start dealing with analysis. Unless you had a brutally hard calc 1/2/3 progression, you won't even know that you're shaky about those kinds of problems until you start hitting applied mathematical problems.

The way I see it, mathematics is about progression in two different kinds of thinking: 1) the ability to make theoretical models that build upon one another (proofing-based reasoning, i.e., found in analysis); and 2) the ability to apply theoretical understandings to practical problems (quantitative-based, i.e., applying pre-calculus, calculus, linear algebra, difEQ, etc., to solve in physics or chemistry, especially those that require creative solutions). I want to progress in both.

I appreciate your advice so far, and I will follow it, but don't mistake it for hesitance to move forward. Hell, I might work on both types at the same time, switching between the two on even and odd days.

>>8903412

About two years.

>>8903681

This is a good restatement of what pre-calculus should be about. Plus understanding the basis and the manipulations of various functions (logarithms, trigonometric functions, exponents, etc.), basic competence with matrices, vectors, and complex numbers.

>>8903404

>No, Gelfand's books were written for gifted middle/high school kids as a supplement to their schooling. They're great for their problems that force you to understand the material and cover stuff you don't usually see in school. Marcus is a freshman linear algebra book for STEM majors.

I meant more in terms of "breadth and depth" that I outlined in the beginning of the post, not in terms of the normal progression of mathematics.

>some combination of depth, quality/quantity of problems, and overall readability/understandability

I should probably ask tutors for their favorite textbooks because they'll know what the brainlets and what the brainmores tend to read.

>>8903681

Also, outside of pre-calculus, what course would cover complex number calculations that involve DeMoivre's Theorem, Euler's Identity, applications to vectors/matrices, basic combinatorics and probabilities, etc.?

>>8903319
Look at Cohen's precalculus book. It has challenging problems and is comprehensive, although I don't think it has probability. The book you suggested looks good too.

>>8904206

Cohen is another book that I considered. I don't remember why I thought Stitz-Zeager was better than Cohen, but if I can't get Stitz-Zeager, I will probably look towards Cohen. I think Cohen has more problems but less breadth... something like that. Thanks for reminding me.

>>8904181
>what course would cover complex number calculations that involve DeMoivre's Theorem, Euler's Identity, applications to vectors/matrices

There aren't any dedicated courses on complex numbers and geometry at that level. The topics are typically scattered throughout precaclulus, calculus, modern geometry, and complex variables. But there are books the organize the material together at your level:

Complex Numbers and Geometry by Liang-shin Hahn
Complex Numbers from A to... Z by Titu Andreescu and Dorin Andrica (More of math competition focus)
Introduction to the Geometry of Complex Numbers by Roland Deaux

>>8904438

Thanks for the advice. I think I'll just stick with the pre-calculus book that I found.

are you gonna waste all your time with babby tier "math" or are you gonna move on to calculus already? leithold's calculus 7 has all the precalculus you need in the appendix. DONT GET STUCK ON THE BASIC SHIT.

>>8903371
>Stitz-Zeager

Excellent choice. One of the more sensible uses of set theory I've seen in any textbook. A nice supplementary text which also uses a sensible approach to set theory is Richard Hammack's Book of Proof, which is free.

http://www.people.vcu.edu/~rhammack/BookOfProof/

>no one posted Basic Mathematics, Lang

This is how you know /sci/ is shit

>>8906320

Was about to post, good lad

>>8906320
This is what I recommend all the time. Although it is funny when people sometimes go "its too hard, this isn't for someone just learning". It's honestly one of the best books I've seen around for lower level math.

>>8906405
>>8906405

>I see some names thrown around here, like Precalculus Mathematics in a Nutshell by George Simmons and Basic Mathematics by Serge Lang, but I've found that while those books are better than your average high school fare, they're lacking according to the standards I set previously.

have you ever read OP's message?

So /sci/ I need to git good

I'm planning to do Apostol calculus

>read book of proof
>read a transition to advanced mathematics
>read precalculus shit if I lack something
>start doing apostol

what do you think?

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Which of these 2 books is better? To do before working through a calculus book. Algebra and Trigonometry -Beecher, Precalculus - Stewart

>>8906903
Do you really need to post it in 2 threads + make one?

>>8906909
yes

>>8903319
https://econsphdtutor.wordpress.com/free-stuff/

>>8906903
>Read a section on the same topic in both
>Use whichever you liked more

>>8906221

Just emailed Stitz asking about a hard cover copy. Apparently they split it into two textbooks for printing convenience purposes. And they sell it cheaply at exactly the cost margins. Good stuff, almost makes me shed a tear.

>>8906320
>>8906385
>>8906405

Dumb faggots I already mentioned Basic Mathematics by Serge Lang in the original post. It's a quality textbook but it doesn't cover the basics of combinatorics AND vectors AND complex numbers AND probability, which is a dealbreaker for me. Brainlets should learn to read first before doing math.

>>8906488

That's a good plan, I might just steal it. Good books too. Thanks!

>>8907019
>I might just steal it
oh

if you find a good pdf of Apostol without fonts getting bold, thin and stuff, pls share

>>8903319
>pre calc
>depth

kek

>>8907019
>It's a quality textbook but it doesn't cover the basics of combinatorics AND vectors AND complex numbers AND probability, which is a dealbreaker for me

Just get a book for each topic. Basic combinatorics is done in any probability book worth its salt so just get Hamming's "Art of Probability". For vectors, get Schuster's "Elementary Vector Geometry" or Robinson's "Vector Geometry". And >>8904438 or Durell and Robson's "Advanced Trigonometry" are the best you'll find on complex numbers without it turning into full blown complex analysis.

>>8907146

You might be in luck. I remember I've got an Apostol pdf somewhere and it might be good. Maybe in the next hour when I'm around my computer?

I dunno what's a good way to send it or upload it

>>8907425
>I dunno what's a good way to send it or upload it
just upload it on mega

>>8906488
Just
>read a transition to advanced mathematics
>start doing Courant & John

>>8907440

>courant

the shitty Springer released a fuckton editions of the books. Wich one should I download/buy?

>>8907455
It's all the same content.

>>8907146

Shit I just realized that my copy has weird font too. Lol it makes it look like a book for brainlets

maybe this apostol thing wasn't a good idea

>>8906488
>>8907440

Okay bros so let's veer off topic a bit. Why the hell are Apostol, Spivak, and Courant recommended as "calculus" books? Aren't they like rudimentary analysis books that decided to cover calculus? I don't get the point or the obsession.

>>8908316

If you want something rigorous but don't want to miss the applications, they're your best bet.

>>8903319
I'm looking for kinds the same advise and also programming courses and/or youtube playlists

>>8908328

I just want a calculus book to explain how things work and why. For example, engineers be like "multiply both sides by dx and solve the difEQ!" while mathemaricians be like "brainlet. that's not a variable", and I have no idea what the notation really means.

What is the benefit of doing some analysis-based book instead of a particularly well-written calculus book that proved crucial theorems, provides some exercises in less important proofing, and teaches you the ins and outs of quantitative calculations with calculus?

>>8908844

You'll get nothing from the autistic elitists here. You are right. Stick with Stewart.

>>8908316
no, they aren't analysis, they're calculus
>>8908844
you're very confused about what a calculus book normally is. they don't prove shit, for example.

>>8908844

It's good practice for when you get to an actual analysis but you don't have to read a rigorous calculus book. Some people like a challenge.

>>8909781
>>8909811

I don't mind reading a rigorous calculus book. I just want to know what I'm really doing while I'm doing it so I know how to cover all of my bases and judge my level of progress.

>>8909781
>>8909811

How does this calculus book measure up?

http://faculty.etsu.edu/knisleyj/calculus/final.pdf

See for why: >>8908844

Just do step papers and look at khan academy/Paul's notes for shit you haven't covered yet. Then move onto Slovaks calculus, artins algebra, Ireland and Rosen's number theory and rudins analysis.

>>8903373
As an engineer who tooK LA, how does Gilbert Strang's third edition hold up?

So, am I being too autistic if I'm going through Euler's Algebra and the Gelfand books, instead of just a precalc book like OP?

I've already taken stuff up to PDEs, but I wanted to cement my baby math knowledge

>>8910348

I get the feeling that what you're really looking for is in abstract algebra and number theory books.

>>8903378
>im interested in AOPS style math too

so why not just do the AOPS precalculus book then? It's objectively god-tier and it goes indepth, teaching you the stuff and making you prove all the major theories and tons of random-ass corollaries yourself, making sure you understand. It does probably have too much of a focus on olympiads for your taste though since some of the problems come from USAMO etc. But ya if you're looking to build just raw mathematical problem-solving/reasoning skill, AOPS books were literally made for this reason.

If you're still in high school I'd honestly do this. Problem solving is the most fun part of math honestly, couldn't give less of a fuck about analysis or topology etc.

>>8910532
and yes it does give you a stronger and important base, if higher math is what your end goal is

https://artofproblemsolving.com/articles/calculus-trap
That being said IDK how old you are, if you are already in college and just doing higher math you'll just have to roll with what you have

>>8903319
Get one that has a solutions manual to go with it and do a bunch of problems. You have to do problems to learn math.

Gelfand books look pretty good too, but have only skimmed through them. Lang's book looks interesting as well, but again I have only skimmed through it.

>>8903350
>probability, combinatorics
Literally no reason to wait desu.

>>8910549
>t. CS brainlet

>>8903319
https://www.amazon.com/Probability-Enthusiastic-Beginner-David-Morin/dp/1523318678/ref=pd_sim_14_19?_encoding=UTF8&pd_rd_i=1523318678&pd_rd_r=N5E2TC23FRPXVPDS7VYY&pd_rd_w=K70HK&pd_rd_wg=r5vS9&psc=1&refRID=N5E2TC23FRPXVPDS7VYY


for probablity self study

>>8910660
>he thinks combinatorics is advanced math

>>8910538
>AOPS
>Oh. You didn't know that the solutions are extra? Too bad.

>>8910735
it is. serious combinatorics is not your shitty
>HOW TO TAKE STUFF MOD P LMAO
freshman class

>>8910839
>literally just counting
>hard

>>8910892
yeahhh it is
any field of math where active research is going on is hard

>>8910900
There is active research in banging your mom but she's easy af.

>>8910829
Idk about that but im using the e-book version and have no problems. No reason to go for physical version desu, just makes navigation harder

Best statistics and probability book lads?
Undergraduate shit, preferably separate books

>>8911509
There is a /sci/-wiki.

>>8910532
>no aops books in pdf

Feels bad man

>>8910532
>>8911589

Yeah I'm having the same problem unfortunately. I think AOPS might be just a brand/meme; otherwise, more people would provide free copies.

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>>8903319
>brusing up
>on precalculus
Bruch up the 12 gauge lobotomizer, my man

>>8912094
i go to stanford you state school brainlet

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>>8912096

>>8912094
>>8912096

That's not me, the OP. I don't go to a state school, nor do I go to the Harvard of the west. I'm a sophomore chemistry major at Harvard.

>>8903319
wanting the best precalculus book in the world is like buying really nice shoes for a babby

>I did fine in high school, but I hardly remember anything because it was an easy class.

Nothing amuses me more than the stupid, foolish, laughable, senseless, naive, ignorant, unintelligent and utter bullshit that brainlets come out with in order to convince themselves that they aren't what they are.

>>8911509
http://4chan-science.wikia.com/wiki/Mathematics#Probability_and_Randomness

>>8911990
Nah, it's actually good

>>8912144

Just telling it as it is. Got easy As, got 5s, etc. Maybe I am a brainlet, but at least I recognize enough to want to improve myself.

>>8903358
Americans need a second class to learn about basic functions

>>8912169

I wish I could trust your word for it. Got a snapshot of a few good pages anywhere?

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>>8913248
Libgen has a few of their books available for download.

The books start each section with a small introduction to the topic. Then it gives you about 5-8 problems that develop the theory. Following that are detailed solutions to the problems. Then about 10 additional exercises, whose solutions are in a separate manual. Each chapter (made up of the aforementioned sections) ends with an easy test and a hard test.

I like it because it has not too many problems. And each problem is medium to slightly hard difficulty. All of the challenging problems have one or more hints in the back of the book. I also like how they often have you prove important theorems, but they break it down into steps for you to make it relatively easy to figure out yourself.

The problem based approach that it offers could be perfect for some types of learners but bad for others. Overall, I'd recommend it as a workbook that goes beyond the average high school courses treatment of the topic.

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>>8913295
Another pic

>>8913295
>>8913301

Thank you for sharing!

>>8913248
>>8914186
hi I'm the anon who brought the subject up, not the one with the above pics but here I'll share 1-2 pics of the e-book version of interm algebra. By the way the e-book system they have has insanely good navigation but yeah the biggest problem is you can't find free PDFs

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>>8914431
And ofc I forgot pic

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>>8914440

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>>8914447
alright that's enough shilling I guess, on their website they have free excerpts if you're interested

>>8903842
This, king Leo is the man. No one else helps develop intuition like him

Check out this ultimate step-by-step list of how to learn the basics of university-level mathematics, depending on how deep into math you need to go.

>CATEGORY 0:
Algebra – Israel M. Gelfand
Functions and Graphs – Israel M. Gelfand
The Method of Coordinates – Israel M. Gelfand
Trigonometry – Israel M. Gelfand
Geometry: Book I. Planimetry – A. P. Kiselev
Book II. Stereometry – A. P. Kiselev

>CATEGORY 1:
Pre-Calculus - Carl Stitz & Jeff Zeager
Statistics - David Freedman
How to Think Like a Mathematician - Kevin Houston
How to Prove It - D. J. Velleman

>CATEGORY 2:
Calculus: A Modern Approach - Jeff Knisley & Kevin Shirley
Linear Algebra and Its Applications - David C. Lay
Ordinary Differential Equations – Morris Tenenbaum
Calculus of Several Variables - Serge Lang
Calculus Vol. I & II - Tom M. Apostol

>Category 3:
An Introduction to Formal Logic - Peter Smith
Introduction to Gödel's Theorems - Peter Smith
Concrete Mathematics - R. Graham, D. E. Knuth, & Oren Patashnik
Introduction to Probability - D. P. Bertsekas & J. N. Tsitsiklis

>Category 4:
Linear Algebra - K. M. Hoffman & Ray Kunze
Introduction to Partial Differential Equations with Applications - E. C. Zachmanoglou & D. W. Thoe
Fourier Series - G. P. Tolstov
Nonlinear Dynamics and Chaos - S. H. Strogatz

>CATEGORY 5:
Analysis I & II - Terrance Tao
Calculus on Manifolds - Michael Spivak
Visual Complex Analysis" - Tristan Needham
A Book of Abstract Algebra - C. C. Pinter

Dunno any good basic topology books, but I'd put them somewhere in either cat 4 or cat 5. Recommend one please.

>>8915656
>Calculus: A Modern Approach - Jeff Knisley & Kevin Shirley
>Calculus of Several Variables - Serge Lang
>Calculus Vol. I & II - Tom M. Apostol

For what reason? Do one set or the other.

>A Book of Abstract Algebra - C. C. Pinter

That's a book for brainlets.

>>8915656
nah mate category 3 has irrelevant stuff, and category 4 should be lin alg, analysis 1 and 2, topology, algebra. category 5 should be ODEs, geometry, higher algebra, complex analysis

>>8915670

>For what reason? Do one set or the other.

Doing Apostol before fully understanding calculus sounds like a fucking mistake for most people. I think Spivak would be doable for bright, motivated people but Apostol? It doesn't seem feasible. Besides, if you're only into, let's say, chemistry, then I don't think you'd really need to do any Apostol/Spivak/Courant.

>That's a book for brainlets.

Recommend a better introduction to abstract algebra that doesn't immediately send you into a world of pain.

This is supposed to be a buffet-style list for those thinking about seriously entering mathematics for the purpose of science, engineering, computer science, applied math, or maybe even pure math (with strong foundations). Not trying to send people off into grad school mathematics.

>>8915692

>nah mate category 3 has irrelevant stuff, and category 4 should be lin alg, analysis 1 and 2, topology, algebra. category 5 should be ODEs, geometry, higher algebra, complex analysis

I think mathematical logic and the "good stuff" from discrete mathematics should be maintained in category 3. It's more of a "discrete mathematics" category without recommending brainlet books like Rosen that don't go into anything with detail. In fact, I would add a good book on algorithms and graph theory if I knew of any that work well for self-studying.

Why wait so long to do linear algebra differential equations? You should have all that you need to at least scratch the surface with single/multivariable calculus out of the way, and doing Strogatz without ODE or doing PDE before ODE sounds like a fuckup to me.

Recommend me some good books or your perfect order with reasons. I'd like to hear it, you sound like you know what you're talking about.

>>8915726
I'm not waiting anymore than you were for linear algebra. I assume Cat0,1,2 are "premath", Cat3 is "preparation" and Cat4,5 are math, so linear algebra is in the math part asap.

you need analysis for ODEs that's why I put it in Cat5. I mean Picard's existence and uniqueness, linear systems and qualitative study of flows up to poincare bendixson (so not the calculus-based ODE methods class for brainlets. I dunno if the brainlet ODE in section 2 is worth it, maybe it is)

I don't know what strogatz does, looks like non-core material and I would defer it for later, or make it optional like other topics including probability, PDEs, linear programming. I don't know what you mean by doing PDEs before ODEs, you can definitely do a brainlet PDE for engineers before the real math part

the ones I said for cat4,5 are what I consider crucial, core material anyone doing math needs to know asap. obviously there's much more. I don't have good recommendations for cat0,1,2, my recommendation for cat3 is maybe a book on set theory like halmos and "how to prove it" or "book of proof".

as for books I like rotman's "A first course ..." for algebra, Tao's for Analysis 1, 2, H&K for Lin Alg, Hirsh&Smale for ODEs, Pressley for Geometry. People recommend Munkres for Topology and Ahlfors for Complex Analysis a lot so they should do (my favorites are not good for a first course / in another language respectively). "higher algebra" still rotman, it's a big book, groups, rings for cat4 and fields for cat5 maybe.

anyway you're not supposed to read all of the books completely, there's plenty of optional stuff in Tao2 and Munkres, it would be good for you to talk to someone when you're doing this

>>8915745

>
I'm not waiting anymore than you were for linear algebra. I assume Cat0,1,2 are "premath", Cat3 is "preparation" and Cat4,5 are math, so linear algebra is in the math part asap.

Yeah, my bad. I did suggest that there's applied linear algebra early on (around calculus) and then there's a much more rigorous study later on.

>Cat0,1,2 are "premath",

I like your categorization. I think I will restructure it in a second post. I would also describe category 2 as "premath/quantitative core".

>Cat3 is "preparation"

And discrete mathematics. A good introduction to discrete mathematics seems to prepare someone well for serious mathematics.

>(so not the calculus-based ODE methods class for brainlets. I dunno if the brainlet ODE in section 2 is worth it, maybe it is)

I'm not interested in creating a "purist" list. Most mathematical degrees start with "brainlet" quantitative mathematics (calculus, linear algebra, differential equations, discrete mathematics) before they advance into the foundations that underlie math used in scientific applications, anyway.

>I don't have good recommendations for cat0,1,2, my recommendation for cat3 is maybe a book on set theory like halmos and "how to prove it" or "book of proof".

Already mentioned "how to prove it" by velleman in category 1. I think I will re-arrange everything, though. Maybe even merge category 0 and 1 at this point.

Revised List

>CATEGORY 0 "Pre-Math: Grade School Review":
Algebra – Israel M. Gelfand
Functions and Graphs – Israel M. Gelfand
The Method of Coordinates – Israel M. Gelfand
Trigonometry – Israel M. Gelfand
Geometry: Book I. Planimetry – A. P. Kiselev
Book II. Stereometry – A. P. Kiselev
Pre-Calculus - Carl Stitz & Jeff Zeager
Statistics - David Freedman

P.S. Reviews should contain only Stitz-Zeager, perhaps Freedman, for the most bang for your buck.

>continued

>>8915980


>CATEGORY 1 "Pre-Math: Entry-Level Quantitative Mathematics":
Calculus: A Modern Approach - Jeff Knisley & Kevin Shirley
Linear Algebra and Its Applications - David C. Lay
Calculus of Several Variables - Serge Lang
Ordinary Differential Equations – Morris Tenenbaum
Introduction to Partial Differential Equations with Applications - E. C. Zachmanoglou & D. W. Thoe

>CATEGORY 1.5: "Honors Calculus & Advanced Quantitative Mathematics"

Calculus Vol. I & II - Tom M. Apostol
Introduction to Probability - D. P. Bertsekas & J. N. Tsitsiklis
Probability and Statistics - M. H. DeGroot, M. J. Schervish
Fourier Series - G. P. Tolstov
Nonlinear Dynamics and Chaos - S. H. Strogatz

Note: Apostol, Courant, and Spivak are all realistically interchangeable. Apostol has the most breadth and aptly “founds” previous studies of calculus and linear algebra. Spivak is more conversational, focused on pure math, easier to read, but has more difficult problems. Courant is also solid, with a heavy focus on applications.

>CATEGORY 2: Discrete Mathematics and "The Transition":
How to Think Like a Mathematician - Kevin Houston
How to Prove It - D. J. Velleman
An Introduction to Formal Logic - Peter Smith
Concrete Mathematics - R. Graham, D. E. Knuth, & Oren Patashnik
Algorithms - Sanjay Dasgupta, C. H. Papadimitriou, & Umesh Vazirani

>CATEGORY 3: Introduction to Analysis and "Pure Math":
Linear Algebra - K. M. Hoffman & Ray Kunze
Calculus on Manifolds - Michael Spivak
Analysis I & II - Terrance Tao
Introduction to Graph Theory - R. J. Trudeau
Topology - J. R. Munkres

>CATEGORY 4: Further Analysis, Pure Math Specialization
Differential Equations, Dynamical Systems, and Linear Algebra – M. W. Hirsch & S. T. Smale
Elementary Differential Geometry – Andrew Pressley
A Book of Abstract Algebra - C. C. Pinter
Visual Complex Analysis - Tristan Needham
Introduction to Gödel's Theorems - Peter Smith
<whatever you want; congrats on reaching mathematical maturity!>

>>8915726
>Recommend a better introduction to abstract algebra that doesn't immediately send you into a world of pain.

Artin, Herstein, D&F.

>>8915980
>Statistics - David Freedman

Why the statistics for polsci majors book? Just do DeGroot, Wackerly after you finish Bertsekas & Tsitsiklis.

>>8915983
>Algorithms - Sanjay Dasgupta, C. H. Papadimitriou, & Umesh Vazirani

That book is utter trash.

>Elementary Differential Geometry – Andrew Pressley

Why so late? After doing calculus on manifolds and topology, you're more than ready to do Lee's Smooth Manifolds. Move Pressley to cat 2.

>>8916038
>DeGroot, Wackerly
*DeGroot or Wackerly

>>8916038

>Why the statistics for polsci majors book?

Basics for those who don't want to go that far. Also not particularly important... only Carl-Stitz is crucial from this list if one wanted to solidify grade school foundations.

I do have DeGroot on the list in Cat 1.5.

>Algorithms - Sanjay Dasgupta, C. H. Papadimitriou, & Umesh Vazirani

>That book is utter trash.

Any particular reason? Seems to be a good introductory text as opposed to "Introduction to Algorithms" by Cormen, Leiserson, Rivest, and Stein, which is more of a reference text than an autodidact resource/introduction IMO. I also don't want to veer too far into the specialization of computer science, just enough that's mathematically interesting (for exposure).

What would you recommend instead?

>Why so late? After doing calculus on manifolds and topology, you're more than ready to do Lee's Smooth Manifolds. Move Pressley to cat 2

I think that placing it in cat 3 (I assume cat 2 is a typo) or cat 4 is a rather arbitrary decision. It's either late in cat 3 or early in cat 4. It is definitely a book that requires "context" in other works, so maybe cat 4 would be best.

Doesn't really matter to me, and I see why that's important.

>>8915980
>CATEGORY 0:
Elementary Algebra for Schools by Hall and Knight
Higher Algebra by Hall and Knight
Plane Trigonometry by Loney

>>8903371
Muh meme books. Gelfand is great for people who are looking for some extra material, but his books are shit for a stand-alone text. Also there is way higher a number of faulty problems in Gelfand's books than there should be which makes going through it a pain.

Basic Mathematics by Lang is great for muh rigor, but it uses some confusing nonstandard/out-of-date terminology in the geometry section if I remember correctly. Precalculus in a nutshell is great for getting a simple overview of material in under a week. After that I'd just go to the library and pick up some mathematics competition books and solve interesting problems until you feel like you "get it". If you're not-retarded and work about an hour or two a day you should be more than ready for Calculus in under a year even assuming you never passed a math course. There's just not that much you really need to know.

>>8915983
This list is shit. Don't do this.
>How to Prove It
>comes after Spivak, Apostol, or Courant
lol
>Algorithms
lol
>Includes Topology in Analysis section
lol

>>8916105

>How to Prove It
>comes after Spivak, Apostol, or Courant
>lol

I didn't add the note recommending to skip ahead to "HTTLAM/How to Prove It" if you were having trouble with Apostol, Spivak, and/or Courant due to a lack of character space and due to the fact that it's kind of an "optional" section (partially or fully, depending on what you want to do).

Good point, though. It will be the first "proofs" text that someone will encounter if they follow this list, and they may need to learn how to prove things first. It's just that "HTTLAM/How to Prove It" is also heavy into a lot of discrete mathematics stuff, so I categorized it in Category 2.

>Algorithms
>lol

A little something for everybody. You don't need to do probability, graph theory, combinatorics, or statistics, but I think that they're fields that are motivated by significant applied concerns, so they're good to include.

>Includes Topology in Analysis section
>lol

Reread the title:

>Introduction to Analysis AND "PURE MATH"

Unless you have a suggestion of where to order it, fuck off.

>>8916095

>
Muh meme books. Gelfand is great for people who are looking for some extra material, but his books are shit for a stand-alone text. Also there is way higher a number of faulty problems in Gelfand's books than there should be which makes going through it a pain.

This is good to know. I'm surprised that they haven't revised the exercises despite being in print for so long.

>Basic Mathematics by Lang is great for muh rigor, but it uses some confusing nonstandard/out-of-date terminology in the geometry section if I remember correctly. Precalculus in a nutshell is great for getting a simple overview of material in under a week.

I think there are 5 solid books that one can use for this purpose, which I will rank in overall quality from best to worst (but still great):

1. Precalculus - Stitz-Zeager
2. Precalculus (AOPS) - Rusczyk (sp?)
3. Basic Mathematics - Serge Lang
4. Precalculus in a Nutshell
5. Precalculus - Cohen (especially for extra problems... otherwise not always the best treatment)

>After that I'd just go to the library and pick up some mathematics competition books and solve interesting problems until you feel like you "get it"

Good point. AOPS books are great for this. There's also "Challenging Problems in [blank]" which I thought were cool, too.

>>8916105

Doesn't seem like a bad list, apart with some minor issues in ordering. Don't be a faggot.

>>8916095
Actually Lang's book is only semi-rigorous. I'm not talking about the liberal assumptions he makes, because those are fine, but rather that there are some unjustified steps in crucial parts. E.g. he doesn't properly define equality, or the famous "equals added to equals result in equals" axiom either.

>>8916127
What do you think of this list of fields?

1. Plane and Solid Geometry
2. Arithmetic
3. Baby Algebra
4. Single Real Variable Calculus
5. Multi Real Variable Calculus
6. Single Complex Variable Calculus
7. Multi Complex Variable Calculus
8. Group Theory
9. Complex analysis or Introductory real analysis
10. Analytic Number Theory
11. Geometric Topology
12. Arithmetic Number Theory
13. Algebraic Topology
14. Homological Algebra
15. Universal Algebra

>>8916483

In order?

>>8916951
Progressive.

File: 1494819152137.jpg (76KB, 505x768px) Image search: [Google]

76KB, 505x768px

>>8912612
>Americans need a second class to learn about basic functions
This is what happens when we nationalize the public education system and apply "affirmative action" quotas.

>>8915980
>>8915983

Any thoughts on this list?

File: Anons list.png (95KB, 1828x897px) Image search: [Google]

95KB, 1828x897px

>>8917329

Brainlet here, I hope this will be useful for me. Thank you anon.

>>8913248
A lot of good books aren't on libgen. For example, Schoen Yau's Lectures on differential geometry is the most famous textbook in geometric analysis and is unavailable on libgen.

>>8915983
this is a really absurd list, if one's goal is to be a pure mathematician, I would streamline it a bit. This list really does not seem to focus on the foundations of what's important to know for a working mathematician. As crazy as it looks, I actually think Misha Verbitsky's curriculum is what you should aim for, albeit slower/with some thing in different places:
http://imperium.lenin.ru/~verbit/MATH/programma.html
The only things I would change here are the speed he does things, the amount of advanced algebra he includes in HS, and the lack of emphasis on more analytic topics. throw in some rigorous versions of fthe categories 1-5-2 stuff and ignore the more complicated geometry stuff (year 3 and beyond, unless you wanna go into these areas) I think you'll have something worth aiming for, at your own pace.

>>8917329
the list is fucking terrible. It's painfully obvious whoever wrote it is not a mathematician.

>>8918570
Fuck, translate it

>>8903371

Have you actually looked through "The Method of Coordinates"? The book is terrible and needs to be rewritten by someone who doesn't have Parkinson's. I will give you "Algebra" is nice, but actually READ the rest of them to see if they are of the same quality.

>>8916965
>>8916483

I made that list and you ruined it by unnecessarily adding complex and real analysis to it. You don't need complex or real analysis. What you need is to spend a lot of time working with complex valued and real valued functions and THINKING about their limits on your own. They are literally parasites of the mathematical world. Same with functional analysis. I see braindead grad students come out of those classes all the time spouting off what they think are "new" concepts, when really they've just learned how to repackage older ones. They are just ways for humanities majors to feel like they are contributing something to math. Whenever you see "theory" and "analysis" attached to something, be skeptical of it.

>>8918570

Is your main qualm that the list focuses on too much "fluff" outside of pure mathematics?

I'm thinking that it would be a good list for those who want a solid grasp of applied mathematics while also having a solid introduction to pure mathematics... with the order and choice of books being easily malleable depending on your interests. Most math curriculums start with some more quantitative, less rigorous stuff anyway, so I made sure to start the reading list in a way that makes sense. If you decide that moving further is not for you, you can hop off any time.

>>8918690

This is something that I was concerned with as well. I really like Stitz-Zeager as an introduction to understanding the details behind pre-calc math in such a way that you aren't clueless when you explore more difficult algebraic manipulations in calculus 1-3.

Maybe I will remove most of the other gelfand books, keep the kiselev, keep Stitz-Zeager, and then potentially keep Freedman (good to have basics, even if it will be explored rigorously later on).

>>8915980

Revised List 2.0:

SCOPE: Basic mathematics, entry-level quantitative mathematics, transitory period between quantitative/applied mathematics and pure mathematics, introduction into pure mathematics. Follow the path as desired.

>CATEGORY 0 "Pre-Math: Grade School Mathematics":
Elementary Algebra – H. S. Hall & S. R. Knight
Higher Algebra – H. S. Hall & S. R. Knight
Geometry: Book I. Planimetry – A. P. Kiselev
Geometry: Book II. Stereometry – A. P. Kiselev
Plane Trigonometry – S. L. Loney
Statistics - David Freedman
Challenging Problems in Algebra – Dover
Challenging Problems in Geometry – Dover
Challenging Problems in Probability – Dover

Note: What an ideal "grade school math" curriculum should look like. Probably best to serve as a reference material for those less skilled in quantitative mathematics.

>CATEGORY 0.5 "Pre-Math: Grade School Review":

Algebra – I. M. Gelfand
Pre-Calculus - Carl Stitz & Jeff Zeager

Note: Best review for the most bang for your buck. Successful completion of both books should adequately prepare you for entry-level mathematics.

>CATEGORY 1 "Pre-Math: Entry-Level Quantitative Mathematics":
Calculus: A Modern Approach - Jeff Knisley & Kevin Shirley
Linear Algebra and Its Applications - David C. Lay
Calculus of Several Variables - Serge Lang
Ordinary Differential Equations – Morris Tenenbaum
Introduction to Partial Differential Equations with Applications - E. C. Zachmanoglou & D. W. Thoe

Note: Standard set of topics that define an early STEM education.

>CATEGORY 1.5 "Additional Topics in Quantitative Mathematics"

>Introduction to Probability - D. P. Bertsekas & J. N. Tsitsiklis
Probability and Statistics - M. H. DeGroot, M. J. Schervish
Fourier Series - G. P. Tolstov
Nonlinear Dynamics and Chaos - S. H. Strogatz

Note: Pure mathematicians can probably skip these books. Applied mathematicians and non-mathematicians will likely find these topics interesting, if not useful or necessary.

>>8915983

>CATEGORY 2 "'The Transition' and Some Discrete Mathematics":
How to Think Like a Mathematician - Kevin Houston
How to Prove It - D. J. Velleman
Calculus Vol. I & II - Tom M. Apostol

Note: This will be the first major introduction into abstract, proofing-based mathematics that defines the field later on. Skilled applied mathematicians should consider continuing in this direction to glean some insight into math foundations, first by challenging oneself to rigorously understand calculus.

>CATEGORY 2.5 "Additional Topics in Discrete Mathematics":

An Introduction to Formal Logic - Peter Smith
Concrete Mathematics - R. Graham, D. E. Knuth, & Oren Patashnik
Algorithms - Sanjay Dasgupta, C. H. Papadimitriou, & Umesh Vazirani

Note: In case HTTLAM & HTPI from the last section didn't provide enough discrete mathematics for your liking.

>CATEGORY 3 "Introduction to Analysis and 'Pure Math'":
Linear Algebra - K. M. Hoffman & Ray Kunze
Calculus on Manifolds - Michael Spivak
Analysis I & II - Terrance Tao
Introduction to Graph Theory - R. J. Trudeau
Topology - J. R. Munkres

Note: This will almost certainly be the domain of pure mathematics concentrators, save for the most difficult applied mathematics found in physics and economics.

>CATEGORY 4: "Further Analysis, Pure Math Specialization"
Differential Equations, Dynamical Systems, and Linear Algebra – M. W. Hirsch & S. T. Smale
Elementary Differential Geometry – Andrew Pressley
A Book of Abstract Algebra - C. C. Pinter
Visual Complex Analysis - Tristan Needham
Introduction to Gödel's Theorems - Peter Smith

Note: At this point, you can consider yourself prepared to continue onwards into any pure mathematical subject in the modern age (of which there are many yet to be covered), and you'll be more than equipped to handle any applied mathematics that you'll encounter, with further specialization of course. Enjoy mathematical maturity!

>>8903319

>Pre-Calculus

There is a copy of Stewart Calculus floating around online. Download that and work through appendices A-E. Also do the review of the algebra, which can be found by googling "Stewart Calculus Review of Algebra". Finally, work through chapter 1 of Stewart Calculus. Make sure to do as many of the exercises as you can for each of these topics. A lot of people here hate Stewart, but it will prepare you well for the standard calculus sequence. The exercises vary well in difficulty, from simple computations to proving results seen in the book such as sine law (that's about as tough as it will get in pre-calculus).

>Vectors/Matrices/Combinatorics/Probability

There is a copy of Discrete and Combinatorial Mathematics by Ralph P. Grimaldi online. Use Appendix B for a review of Linear Algebra. Use Chapter 1 and 3 as a review of combinatorics and probability. You can also use Appendix A for a more refined look at logarithms than you'll get in Stewart.

Up.

>>8920722
>>8920732

Hmm, what is /sci/'s opinion of this booklist?

>>8920722
>>8920732

Seems bretty gud to me

>>8920722
>higher algebra
>limits

Ok this is pretty good shit

>>8920722
no comment on cat 0~2.5.

for cat 3, calculus on manifolds is a bit out of place. if it's calculus then it shouldn't be here, and if it's analysis it should be after geometry (aka something like differential forms by do carmo). graph theory feels out of place because it's non core material. why include that and not include number theory, operations research, or probability?

for cat 4, godel feels out of place and is again non core. you could switch that for something on set theory like jech maybe.


I would:
- take out calculus on manifolds
- take out graph theory
- add first chapters of jech's set theory to cat 3
- move algebra to cat 3, indicating chapters for ring and group theory
- add algebra part 2 to cat 4, indicating chapters for field (galois) theory
- take out godel
- add number theory from ireland & rosen to cat 3 or 4
- add category 5 with algebraic topology, commutative algebra, differential forms, riemannian geometry, functional analysis
- add a section on secondary material

in general lists are of questionable usability so it's not worth it to try to make a great list imo
but it's decent

>>8920722
>Fourier Series - G. P. Tolstov

"The Fourier Transform & Its Applications" by Bracewell makes more sense to include than Fourier Series.

>>8903319
>Depth, Problems, and Understanding

There is only one book you need, Precalculus by Sheldon Axler because it comes with a "Student solutions guide" which means every second exercise is fully worked out so you can see what's going on.

>Basic Mathematics - Serge Lang
Too much errata, plus hard to find. The pirate copies are shit and full of errata on libgen

>Calculus and Analytic Geometry - George B. Thomas 3rd or 4th edition
You want to try something like this after, it's rigorous and proof centric, Don Knuth claims this book made him a mathematician. The first edition is just lecture notes, the second is a cleanup, and the 3rd fills in all the blanks and adds linear algebra with differential eq. The 4th is good too. Anything after that is pure garbage as they watered down the text and removed all the rigor. Spivak's Calculus is good for this too, any motivated highschool student can do either.

>How to think like a Mathematician - Kevin Houston
You may want to get this on libgen, though you will learn all this in your first real math class in university like how to read a proposition, ect.

>The Art and Craft of Problem Solving - Zeitz
Mentioned here already this is a good book but again you'll largely figure this out by yourself the more math you do.

>>8920732
Concrete Math is actually a text any highschool kid can read. They won't get all the very difficult exercises, and it's full of silly tricks to throw them off but generally it's very accessible. I read it in Grade 11 and was generally a shit math student.

>>8922654

>not include number theory or probability?

Already covered basics in earlier books by Bertsekas/Tsitsiklis & Graham/Knuth/Patashnik. I could consider adding more.

>for cat 4, godel feels out of place and is again non core.

It's a different kind of mathematics or mathematical philosophy that would definitely improve one's understanding of how pure mathematics "works", if you get my drift. I will definitely consider adding more set theory, though, since we only have the basics from the "Transition" and "Intro to Discrete Mathematics" section.

>I would:
>- take out calculus on manifolds
>- take out graph theory
>- add first chapters of jech's set theory to cat 3
>- move algebra to cat 3, indicating chapters for ring and group theory
>- add algebra part 2 to cat 4, indicating chapters for field (galois) theory
>- take out godel
>- add number theory from ireland & rosen to cat 3 or 4
>- add category 5 with algebraic topology, commutative algebra, differential forms, riemannian geometry, functional analysis
>- add a section on secondary material

Fantastic additions, but I don't know about subtractions. I think it might be a good idea to remove Spivak here if we already have Apostol Vol II covered earlier. Secondary material could be a useful distinction as well. I don't want to remove graph theory, though maybe I could "shoehorn" it into the Discrete Mathematics "optional" section.

>"The Fourier Transform & Its Applications" by Bracewell makes more sense to include than Fourier Series.

How come?

>Calculus and Analytic Geometry - George B. Thomas 3rd or 4th edition

How does this compare to Apostol or Courant?

>The Art and Craft of Problem Solving - Zeitz

Is this necessary after covering "HTTLAM" or "How to Prove It"? What about the classic "How to Solve It" by Polya?

>Concrete Math is actually a text any highschool kid can read. They won't get all the very difficult exercises, and it's full of silly tricks to throw them off but generally it's very accessible. I read it in Grade 11 and was generally a shit math student.

You're definitely not "shit" compared to the average if you're reading Knuth in 11th grade, but thanks for reaffirming my decision to stand up to any brainlets who are still lingering at this stage and demanding to read a standard discrete math textbook like Rosen.

more bumps

Damn, I just started Pre-Caculus by Carl Stitz & Jeff Zeager and I've got to say this book is amazing. Thanks to all the anons who recommended this book. Makes me want to check out some of the Isreal M. Gelfand and A. P. Kiselev books just to make sure I didn't miss anything in those elementary subjects...

>>8925037

Glad that you've enjoyed it. Took some major research to find a pre-calculus textbook that I'd like after finding the meme Basic Mathematics and Precalculus In A Nutshell to be lacking.

Only worthwhile checking the first Gelfand book. Kiselev is for fun geometry.

>>8925118
What's wrong with the other Gelfand books

They seem to have good problems, all around

>>8925226

Hard to read because they're not written well, which can be a problem for a beginner. Algebra is good, though.

>>8925767

Do you think the other books would suffice to cover the lack of Gelfand's 3 other books?

>>8925226

Do you plan on reading those books? Or only doing the problems?

>>8903351
I didn't graduate high school at all and had to take college algebra my first semester of college, and now I just finished calculus 3 with an A. It took 5 semesters to go from college algebra to calculus 3 because I did not have any high school math past 9th grade, though, but I got an A in all of them except college algebra itself.

>>8903319
What do you guys think of the Larson textbooks? They're the only ones my calculus instructor would use. My professor knows Larson personally and harps on about what a great guy he is all the time.

>>8922752

Never really got an answer for these series of questions. I'm about to finish this list, since it seems nice for a general introduction.

>>8922749
>>8922752

Anybody?

Well?

>>8928429
Both, but mostly just doing the problems.

>>8908844
>For example, engineers be like "multiply both sides by dx and solve the difEQ!" while mathemaricians be like "brainlet. that's not a variable"

If you encounter a mathematician like that, then he's not a good teacher.

A good teacher will not chastise a student for thinking of dx as a variable.

A good teacher will use that as a launching point to mold the student's mind into the right way of thinking about dx.

A good teacher will say "yes, it's true that there are cases where dx can be symbolically manipulated in much the same way that a real variable can be manipulated -- however, it's also important to understand the ways in which dx differs from a real variable".

A good teacher will NEVER miss an opportunity to clarify and expand in a way that builds on the student's intuitions, but that also guides the student toward a more mature understanding.

It's sad that you have never encountered a mathematician who's also a good teacher.

>>8930839

Nobody has ever "chastised" me for doing that, but I have been told that it's not mathematically rigorous to do so, though my calculus background is not strong enough to understand why.

Honestly, I barely even understand why we can manipulate variable notation the way we normally can, let alone dy/dx notation.

>>8903376
Olympiad training is GOAT for problem solving skills

>>8906320
>Lang
The OP posted it you fucking retard

>>8930839
how can an autist be a good teacher?