book-recommendations

[solid bloom]

tysm

[slender cargo]

As an introduction it seems to at bare minimum require a good instructor, or be supplemented by other books.

[remote vortex]

Indeed

[slender cargo]

Great book. That book even teaches enough Calculus in case you want practice on the computational side

[earnest wolf]

i mean, if an instructor fills in the holes that rudin left, will it really be PMA or just a completely different ||(normal)|| course?

[daring lake]

I think I solved all problems in its first 8 chapters, took a very long time though 💀

[slender cargo]

Sounds like a great way to learn Analysis!

I think Bartle's main idiosyncracy is that it primarily teaches integration through Riemann sums instead of the Darboux way.

[daring lake]

I actually disliked that part

The notations were also strange to the point that Rudin's chapter on integrals was easier to read

[slender cargo]

Yeah I can see that chapter as being a reason to use a different book

[vital bane]

that's true, rudin isn't a bad textbook if you're taking a real analysis class. But if you're self-learning it is horrendous it is a good reference book tho

[terse garnet]

whats a good calc textbook

that isnt stewart

or larson

[daring lake]

What do you expect? Since most calc books will be around the same level as stewart

Apostol does things a little differently iirc, would be a little more terse than Stewart

[terse garnet]

hmm

proof based

I think

[daring lake]

Yeah Ig Apostol then (or Spivak but its very terse, I wouldn't really recommend it), or you can pick up an Analysis book like Bartle & Sherbert or Abbot

[neat citrus]

Some book recommendations for differential equations that is a little bit easier to read?

[terse garnet]

Are there any modern books that are proof based

[daring lake]

I don't know any

[remote knoll]

Proof based calculus is just real analysis, pretty much. And there are a bunch of those

[terse garnet]

would I be missing anything by not doing the engineering calculus books

omni

[remote knoll]

Probably, for instance, real analysis focuses mostly on why calculus works, and less on how to compute anti derivatives

So you'll lose out on worked problem practice

[terse garnet]

I guess i'll just do stewart or larson first

then move onto

[neat citrus]

If someone knows, ping me!

[terse garnet]

analysis

[vital bane]

More like proof based calculus + structure of R

[remote vortex]

Structure of R is a necesary part of making the proof-based calculus possible

So it counts as part of that

[vital bane]

I suppose

[earnest wolf]

#real-complex-analysis message

guys, can u summarise in a few words how different LADW is, compared to LADR (or if it's too different, then just the approach it takes)

[elder pendant]

There’s a (fantastic imo) pinned message about linear algebra that includes a comparison of these two. Hth!

[novel hound]

I've had this happen with a specific book - when I mailed customer support about it, they said it was because the book was not available for my region (although they did give me the contact info of the local seller responsible for it's distribution). Try removing books from the cart to identify which book is problematic.

[lusty escarp]

Yeah, you are right. For me, Linear Algebra Done Right by Axler is causing problems.

[earnest wolf]

u mean this review by dami?

tbh did find any comparison btw the two there

ah, ok, I see

balances theory and computation
then I hate that book

[remote vortex]

LADR is much more focused on theory and proofs than computations.

I don't know what LADW does but judging by the title, presumably not that

Also LADR is a title that's hugely full of itself and I'd have expected more from Axler

[earnest wolf]

never judge a book by the title

like, when I heard someone recommend a book to me called '[...] Done Right', my first though was: «it must be smth like 'A guide how to become a surgeon in 24 hours'»

[sudden kindle]

LADR good

[remote vortex]

Axler is generally good at writing books and exposition, so his book is fine at what it does.

I don't do linear algebra much, but Axler's measure theory book is excellent and I recommend it without hesitation

So just in terms of Axler as textbook-writer, I rate him high

[earnest wolf]

I think he also is an editor of most springer undergraduate math series

[sudden kindle]

There are some computations in Axler, but its not the focus for sure

[sage python]

Yeah there's this weird dynamic in linear algebra classes

[sudden kindle]

The problem about using diagonalization to get an explicit formula for Fibbonaci numbers is peak math 👌

[sage python]

Where a lot of people take a class that's devoid of general ideas and is just a lot of RREF

And then they take a second class that does all the theory

[remote vortex]

So sort of like the calculus -> real analysis progression?

[sage python]

The reason for this is that first class is meant not just for future math majors (they don't have manpower or demand to offer an entirely separate track for math majors from the start), but for scientists and engineers

Who would feel the theory is a distraction

So now some linear algebra books are written around "the second course in that progression"

[sudden kindle]

I wanna learn everything there is to know about metallic numbers

[remote vortex]

Sodium numbers, Mercury numbers, Uranium numbers....

[sage python]

Axler is kinda in that category. It's technically self-contained but the reason it has fewer computations was that it has in mind an audience who knows the stuff

[remote vortex]

Yeah, reading Axler if you've never seen a matrix before, might be confusing

[sage python]

I wouldn't say confusing

But you'll come out on the other end missing some stuff

[remote vortex]

Yeah

[sudden kindle]

Is there a book on metallic numbers?

I wanna know connections to rational approximation, continued fractions, geometry and arithmetic

[sage python]

While HK, FIS, LADW are more one stop shopping

[sage python]

Don't even know what those are lol. What's the interest?

[earnest wolf]

I mean.... No?
I took real analysis (1) before calculus and went through LADR (ofc not all chapters) without knowing much computations

[earnest wolf]

and really loved the theory focused approach without «solve this linear system by hand; compute the determinant; does this system has infinitely or only finitely many solution» kind of stuff

maybe the only important thing missing is gaussian elimination

[sage python]

Yeah I'm not saying it isn't doable. I'm saying the reason it ignores computations, as said in the preface

Is that it thinks you already learned them and thus don't need to learn them now

It's not necessary for the presentation but it's less "Axler thinks they're expendable" so much as "Axler thinks you have them covered"

[sudden kindle]

They continue the sequence golden ratio, silver ratio, ...

[trail hemlock]

participation ratio?

[flat marten]

the best linear algebra book is greub 100%

[round shore]

~~so, useless maths ~~

[remote vortex]

That's most maths

[gray gazelle]

should i do discrete math before i do combinatorics, number theory, proofs etc etc

should discrete math be the base of all discontinuous mathematics

or can i learn combinatorics, number theory and all very advanced math easily without starting with discrete

[slender cargo]

Just go with whatever your school's intro to proof class is. Yeah most of the topics in a Discrete Math class will be covered in higher math classes.

[sudden kindle]

Yep

[remote sparrow]

combinatorics is a topic in discrete math

[sudden kindle]

I like useless math

[remote sparrow]

https://en.wikipedia.org/wiki/Discrete_mathematics

discrete math classes are usually a hodgepodge of things CS students need to know

[gray jungle]

its possible but its also important to do some hands on computations

usually a course is either too focused on computations or too focused on theory, its good to know both

with exceptions ofcourse

[earnest wolf]

just to add one more thing: in my uni (EU; discrete math is a compulsory subject) the class differs for different majors

for example, CS students focus more on finite automatons and boolean algebra stuff, while (pure) math students lean more towards ZFC, for example (altho we also have combinatorics and graph theory)

so yeah, it just illustrates how much a course with the same name differs not only across different universities, but even within the same institution

[earnest wolf]

useless math is the best math

[remote sparrow]

CS students need basic combinatorics + graph theory + calculus for algorithms

[earnest wolf]

calculus for algorithms
what's that? like big o stuff?

or u just meant the usual calc: limits, integrals...

[remote sparrow]

analyzing the computational complexity of algorithms is part of algorithms, yes

[molten mason]

As said above. Discrete math means differently in different places.

In my university if you're a math major you take all those as separate full courses with their own textbooks.

If you're a CS major it's just cherry-picked topics from each of these courses all combined in one quick course.

If you're asking about courses at your school, I would ask classmates and advisor.

If you want to learn on your own and get your own textbooks and whatnot, then if you want to just learn some quick nonsense for CS, get a discrete math book. If you want to know each topic and actually learn each subject, get a book in each i.e. a proofs book -> set theory -> algebra, number theory, and combinatorics

[glossy zealot]

Guys what is a source or book to quickly review trigonometry for real analysis?
I forgot most of it, even encountered some inequalities that I did not know about (like |sin(x)|<=|x|)

[misty galleon]

Looking for a physics book of similar quality to something like Stewart or Thomas calculus textbooks with lots of examples and practice problems, intuitive explanations and proofs, also starting my basic, I don’t know much of any physics but I want to the book to contain basics to advanced in it

[violet shuttle]

...i didn't know that inequality, and i passed the written graduate analysis qualy here

don't spoil it for me

clearly we just need to check the interval [0,1]

sin(0)=0. sin(1) can be evaluated to be ≈.84<1. Now I was going to then try to see if I could use convexity but sine is actually concave.

f(x) = sin(x) - x

it is 0 at 0 and ≈.16>0 at 1

we want to show this function is never negative.

it suffices to show it has no roots on the interval (i suspect this is true)

so f'(x) = cos(x) - 1

but this has no roots in the interval (besides 0), QED.

[uncut zealot]

You should not need any trig for real analysis.

[violet shuttle]

The professors here who've I've had opportunity to see do some trig universally don't remember most of them

(anyways you'll rederive anything you need in a real analysis textbook)

[uncut zealot]

If you want to learn trig properly, learn enough complex analysis to understand Euler's identity, then pivot to fourier or harmonic analysis

[violet shuttle]

Yet, all of them can prove the rest easily by just doing algebra on the definition as the real and imaginary part of exp(ix)

Knowing the existence of a double angle formula is more valuable than knowing the actual formula.

[glossy zealot]

I asked about trig because I had to do these proving sin(x) and cos(x) are continuous functions

[violet shuttle]

...?

have you proven exp is?

if so then it's immediate

[glossy zealot]

Yes I did

[violet shuttle]

you know that the difference of two continuous functions is continuous right?

and that if you divide by something that's not 0 you'll still be continuous?

so what was the issue?

[glossy zealot]

Yep I got to use that

[violet shuttle]

like all of the proof is in that exp is continuous

[hallow oriole]

i like the mvt proof of that identity

[glossy zealot]

I did not get to learn anything about relationship between cos/sin and exp

[violet shuttle]

then you don't know what cosine and sine is.

were you using a real analysis textbook or something else?

[glossy zealot]

And the question was forcing me to do epsilon delta proof directly

[violet shuttle]

if this question was asked before you even know what sine and cosine are, it's not good pedagogy.

sine is defined as the imaginary part of exp(ix)

cosine as the real part.

exp(x) is defined as \sum \frac{x^n}{n!}

this definition is well defined because this series converges

specifically, it converges at any point by the ratio test, and on any interval [a,b] you can use M at the boundary to apply the Weierstrass M-test and show uniform convergence.

Thus exp(x) is continuous.

[glossy zealot]

That is very interesting, I will look into it more, thank you!

[violet shuttle]

If you are learning real analysis, you better learn this, and if you're taking a class that doesn't go over this you should revolt.

[glossy zealot]

Ok let me ask the instructor about it

Thank you wiseman

[violet shuttle]

pick up a copy of baby rudin

[glossy zealot]

I intended to go into Rudin after Im done with my intro analysis course

[violet shuttle]

if you finish your intro to analysis course properly than baby rudin won't do much for you

[tribal crow]

Rudin is an intro analysis course

[rich sun]

Not a recommended first intro though

[violet shuttle]

I recommend it to people that happen to be exactly like me when I read it

[charred mason]

Has anyone here heard of or read geometry & imagination by Hilbert and cohn-vossen?

I've heard it's amazing

But it'll be great to get some perspective on the practical side like how hard it exactly is and how much time it requires

[hoary sand]

Any good books to get a better understanding on line integrals?

[uncut zealot]

Functions of One Complex Variable by John Conway

[hoary sand]

Thank you!

[violet shuttle]

...?

[violet shuttle]

what do you actually want

surely you don't literally mean a book dedicated to line integrals

now, I do like Conway's book, but, do note the title.

I can't tell if Pear is memeing or just assuming you wanted a text on complex analysis

[uncut zealot]

Specifically chapter IV, by the way

[uncut zealot]

It's April Fool's Day

[violet shuttle]

(i am likely to call complex analysis line integrals "contour integrals" instead)

[uncut zealot]

I thought contour integrals referred specifically to when the path is closed? Which to be fair is what happens like 99% of the time in CA

[violet shuttle]

yeah uhh actually maybe

i thought "contour" just meant "Cauchy used the word in his stuff"

[uncut zealot]

Valid. You can probably pass the complex qual just by saying "Cauchy contour residue" three times.

[hoary sand]

A book that would help me learn inverse kinematics would be pretty cool for programming.

[violet shuttle]

try a book on robotics

[violet shuttle]

nonono you need to at least write 2\pi n a few times (choosing different n of course)

[uncut zealot]

This is just the value of every integral ever. (where n represents the winding number of the zeros of the bottom of the fraction)

[hoary sand]

Theory of Applied Robotics: Kinematics, Dynamics, and Control by Reza N. Jazar, I'll try this one.

[loud cradle]

why are they called line integrals anyway, when the curve is not generally a line

[uncut zealot]

A line is simply the image of [0,1] under some continuous map

"lines have to be straight" is homophobic propaganda.

[loud cradle]

lines have to be convex, better? 😁

[molten mason]

Euclid homophobic confirmed

Cancel Euclid

[remote vortex]

he's dead

[midnight compass]

Haha polynomial transformation of vector space go brr

[void gazelle]

Whats a good engineering or such book for someone say beginning graduate level math but wants to actually apply what they know into manipulating real world things

[broken meadow]

does Kreyszig advanced engineering mathematics fit the bill?

i havent read it myself but i recall mentions from before

[void gazelle]

I did actually check this out and liked it a lot, however i was thinking more in the lines of a book that is like

This math will help you find this perfect length to saw this block of wood

Or this math is for wiring

Stuff i can actually use my math degree on

[broken meadow]

hmm

[void gazelle]

Thank you btw

[graceful moon]

What does this even mean

[void gazelle]

Just something very practical, something that says for example you need to get this length of wire, here's the formula, get cutting!

Something like that, but for people with a math background

Eh it probably doesn't exist when I put it like /that/

[bright epoch]

Elementary applied topology

[void gazelle]

Thank you i will check it out

[void gazelle]

I checked out Ghrist's book on his website and wow this is great! More of this!!

[vital bane]

Robert Ghrist?

[void gazelle]

I assume that was the author of the Elementary Applied Topology book he meant

[gray gazelle]

its here

[rigid forge]

okay, so I tried to read the books by hung hsi wu, and honestly not a fan. (no, its not about the strong ideology at the starting of the book)

so, can anyone recommend trignometry/precalculus/algebra books that are rigorous and proof-centered? (besides serge lang's book)

[slender cargo]

I think the issue here is that any actual rigorous, proof-based treatment of trigonometry and algebra comes in higher level books. Serge Lang's Basic Mathematics book is the closest you will get, and even that book doesn't really show much besides a few elementary proofs.

[rigid forge]

that's sad, because I find learning math with valid reasoning more fun than just assuming it is true and applying the theorem in problems. i get way more dopamine in the former than the later.

[slender cargo]

That's good. What you should do, is simply find a precalculus book that is easiest for you to learn from, and complete it ASAP.

And then you could get started with something like Spivak's Calculus.

Your goal right now is simply to get good intuition from a precalculus book. The rigorous stuff will come later (and not much later).

[spice sleet]

I hate math so much grrrrrrrrrrrrrrrrrrrrrrrrr

[molten mason]

The problem is the rigorous trig comes in calculus and series (Which would be a typical Calc II class) and then further developed in real and complex analysis. (A couple classes after Calc 3)

Trying to find rigorous proof-centered trig textbook at a pre-calculus level either doesn't exist or if it does is just not commonly known.

[gray gazelle]

I have been collecting every math book I can find a PDF of just by search ( not genlib or anna). Fun to see what gets recommended or used by what universities.

Also I have been going through the internet archive to see what old mathematics books exists ( that can actually be downloaded).

[bright epoch]

Yes

[magic spade]

What didn't you like? I wanted to try using his books

[fervent marten]

what happened to the server's pic?

[hollow peak]

Trans day of visibility

[torn blade]

the torus was removed. thats it

[fervent marten]

dyk if theyre gonna take it off

i hate this

[smoky zephyr]

what's the problem

[violet shuttle]

i mean i do like the torus

[hollow peak]

cringe

[violet shuttle]

given that the trans flag has objectively the best color pallette, one considers transifying the torus and prideifying the background

[gray gazelle]

Why it was removed?

[torn blade]

probably to make the trans flag more visible (haha get it trans day of visibility). clearly its working cause everyone is just starting to notice this server has a trans flag in its icon

[gray gazelle]

Ok, that's fair

Although, let me ask, the trans day of visibility is 04/01?

[granite vector]

Nah, red white and blue

[sage salmon]

03/31

[remote vortex]

The trans flag is fine, and the torus was taken for spring-cleaning and should be returned shortly.

[quaint solar]

Does anyone have some book recommendations for complex analysis

[uncut zealot]

Conway is good.

[remote sparrow]

#book-recommendations message

see pins as well

[crude sage]

What do people think of Lang's Undergraduate Algebra? From what I've seen it's not too chatty, but does some weird organisational stuff

[graceful moon]

James Stewart calculus is ubiquitous, hard to go wrong with that

It’s definitely a calculus book more than an analysis book, it’s not the worlds most rigours text but it does the job of teaching multivar and vector calc imo

[fervent marten]

its not trans visibility day anymore

[glad rampart]

Who cares

It looks awesome

[fervent marten]

ok

[slender cargo]

Trans people are cool so I say we keep it up for some time (although I hope the torus comes back eventually)

[silent edge]

oh its a trans flag

I thought its just corrupted for april fools

[hallow oriole]

we just need to find it again

[slender cargo]

True. It might be lost to the abyss.

[serene raptor]

Yall ever heard of Dune 2?

[molten mason]

It's fine. His graduate book is a standard graduate text. The undergraduate version is a simplified version. If Algebra is too much, Undergraduate Algebra was written to prepare you for Algebra

[crude sage]

In what way is it simplified (apart from, y'know, less details to be verified by the reader)? I know that GTM lang is a reference text, but UG lang seems too short to be encyclopedic/useful as a reference. Also it seems to use a bottom-up approach in some places, like proving theorems in Z before having a later chapter do exactly the same things in more general settings. I thought GTM was much more top-down

[surreal ravine]

What's a good abstract algebra book for beginners, that has a lot of exercises in it?

Does the book mentioned above have a lot?

[remote sparrow]

#book-recommendations message

[surreal ravine]

Thanks!!

[crude sage]

If you're talking about the book I was asking about, then no there aren't as many exercises as in comparable books. I think Artin's Algebra has more, and they have a reputation as good ones

[surreal ravine]

Thank you!

I'll look at them all

[gray gazelle]

Lang’s Algebra

[surreal ravine]

Thanks!

[slender cargo]

Don't listen to eigenpuppet

[surreal ravine]

lmfaoo

[slender cargo]

Along with that list, Abstract Algebra by Fraleigh and another book by Gallian are popular.

If I was self-studying I would probably check out Nathan Carter's Visual Group Theory. That book seems like a neat experiment, but it doesn't cover everything.

[surreal ravine]

ooh thanks! I'll check those all out and see what fits best for me

[gray gazelle]

Herstein’s Topics in Algebra is also good

[uncut zealot]

Probably one of the books that's known for being the most beginner-friendly is "Algebra I: Chapters 1-3" by Bourbaki. Here's a free online pdf that I found: https://math.mit.edu/~hrm/palestine/bourbaki-algebra-1i.pdf

[surreal ravine]

Thank you! I didn't expect so many responses lol

[crude sage]

Keep in mind that it may be April 1 wherever Pear Category Theorem is located...

[surreal ravine]

yup, probably shouldve waited till tmrw to ask lmaoo

[twin dirge]

lol

[violet shuttle]

it's terrible for exercises

I do like the book though

[crude sage]

"Prove all the results in a homological algebra book"

A friend pointed out that you can do this pretty quickly by writing your own homological algebra book with no theorems

[raw bay]

Anyone got good books on combinatorics?? Tryna self teach 🙏🏾

[charred mason]

Is there a good book on galois theory or something that historically motivates abstract algebra? I have some experience with introductory analysis(self study) and some proof based linear algebra

[charred mason]

Interesting wish I could help but I don't have much exposure to combinatorics

[hallow oriole]

stanley's enumerative combinatorics or bona's walk through combinatorics

probably the objectively correct starting points

[tacit crow]

try sedgewick + flajolets analytic combinatorics

[raw bay]

Okay ty guys

[raw bay]

Would those books touch onto these subtopics?
@treesarentreal.com @hallow oriole :
Sets
Functions
Introduction to Counting
Counting with Bijections
Generating Functions
Theory of Generating Functions
Burnside’s Lemma
Polya’s Theorem
Combinatorial Geometry
Graph Theory
Stirling Numbers
Ramsey Numbers
Catalan Numbers
Counting in Two Ways
Recursion
Pigeonhole Principle
Inclusion-Exclusion Principle

[hallow oriole]

yes

probably not going to go fully into catalan numbers, graph theory, or genfs but will give standard-ish treatments still

[raw bay]

Okay thank you sm🙏🏾 on that note would you know good number theory books?

[hallow oriole]

also you'll probably need to know basic set stuff going in anyways, but you can pick that up by osmosis

[olive kernel]

i feel like literally every math book ever has a sets relations functions tutorial at the start of it

[hallow oriole]

for the level you seem to be at i think 'a friendly introduction to number theory' by silverman would be a good fit

[raw bay]

Alright I’ll look into those

Thank u again🙏🏾

[hallow oriole]

nw, gl

[tacit crow]

for graph theory i recommend bondy + murty

[raw bay]

I have the name and author?

You*

[remote sparrow]

you don't typically study galois theory before groups, rings, and fields

[hallow oriole]

galois theory is motivated by abstract algebra, historically, so i have trouble believing you'll find something like this

[uncut zealot]

Proof based LA and analysis are relatively irrelevant for galois theory. A standard UG algebra text, like Pinter or Gallian, should provide the background needed for a modern treatment of galois theory. I learned the topic from Stewart, who has a dedicated book, but most people learn it initially from books like Dummit & Foote or Lang, which are graduate texts just on abstract algebra more generally.

[torn crypt]

Weintraub (dedicated Galois book) I believe takes an LA based approach to galois stuff? (But LA as in including modules, I think). This assumes you know some algebra going in, however, but going into Galois stuff needs you to know a bit of ring & field stuff the way it's usually done anyway

[uncut zealot]

If we're talking solely historically, the motivation is kind of reversed. Galois developed a significant amount of group theory in order to prove the insolubality of the quintic even after he proved the famous correspondence that bears his name. (In particular lots of modern notation about groups, as well as the proof that A_5 is simple). Modern pedagogy kind of does it backwards compared to how it was developed historically - though admittedly in an order that results in students feeling like they're following for most of the semester instead of feeling hopelessly lost until the end.

[hallow oriole]

well, i stand corrected

[remote sparrow]

Galois Theory by david cox is a good standalone book

[sage python]

Wait the linear algebra???

[uncut zealot]

I was thinking like chapters 10-13 of D&F because I'm an idiot. A bit of UG level LA is definitely necessary.

(i.e. I was conflating homological algebra and modules with LA)

[hallow oriole]

galois must have been insanely fucking smart, holy

i knew that academically but it's sinking in now

[uncut zealot]

He did most of it in a single night from a jail cell the day before he died in a duel at the age of 19.

[sage python]

Oh that makes more sense. I was gonna say, stuff like trace, norm, minimal polynomials, Kummer theory...

[hybrid sparrow]

[golden salmon]

I think blitzstein (https://projects.iq.harvard.edu/stat110/home) has a chapter on it that's pretty accessible, starting page 497. Link is sourced directly from the author's website

[hallow oriole]

uh

no links

sorry

unless that's legal?

[golden salmon]

I'll link the site then

[hallow oriole]

which given that it's a google drive link i doubt it

[uncut zealot]

Stop being a narc

[hallow oriole]

im the biggest piracy advocate on this server i just don't want this place to be shut down fr

[golden salmon]

The google drive link is found on the site I found

[uncut zealot]

🏴‍☠️

[hallow oriole]

oh, yeah if it's legal go for it

[golden salmon]

It's by the author and its a preprint, not a pirated copy

arxiv-equivalent

[hallow oriole]

instead of the drive link probably just do https://probabilitybook.net

[uncut zealot]

Fun fact - downloading PDFs is legal in the US, it's just uploading pirated files that's illegal

[hallow oriole]

technically not true in all situations but true enough

[golden salmon]

not everyone is in US :^)

[uncut zealot]

Discord is a company headquartered in the US and thus beholden to US laws.

[golden salmon]

The users themselves are not in the US, which is more what I was angling at

[uncut zealot]

Internet piracy is mainstream enough in Sweden that there's a major political party whose main goal is advocating for it.

[golden salmon]

I see

[hallow oriole]

when are these ever the same thing

anyone got a good book on lattice theory in relation to sphere packing that isnt the one by conway and sloane

[golden salmon]

when you're learning about lattices in some solid state physics class

[heady ember]

Lmao what

Oops wanted to turn off the ping but fat fingered and pressed sent

[uncut zealot]

https://en.wikipedia.org/wiki/Pirate_Party_(Sweden)

[heady ember]

[novel hound]

if you want a light read for motivation and historical notes, Elements of Algebra by Stillwell is nice. Edwards Galois Theory is also notable for focusing on history. However, these aren't the best for comprehensive, well-rounded intro, but you can use them for motivation, history, etc.

[oblique hatch]

What is “towards the mathematics of quantum field theory” by Frederic Paugam like?
(What does it assume, is it any good, etc?)

[lapis sundial]

Source?

[earnest wolf]

for fun, let's do a voting under this message

[remote vortex]

Statement proven already.

[earnest wolf]

that ain't fun

[remote vortex]

What does "fun" have to do with mathematics?

[golden salmon]

why else do it?

[alpine rover]

what are the usual prerequisites for undergrad diff geo

[graceful moon]

Linear algebra, general mathematical maturity

[weak solstice]

Also multivariable calc

[graceful moon]

Some analysis for like the inverse and implicit function theorems but it’s not majorly important you could probably just black box it

[alpine rover]

interesting I thought analysis with multiple variables would be most important

[graceful moon]

It’s definitely helpful, but you can pick up the topology and stuff you need as you go, I’m also assuming for UG DG you mean like curves and surfaces where it’s less important

[alpine rover]

oh I meant more smooth manifolds

sorry

[graceful moon]

In that case yeah some analysis, LA and topology is enough

Take a look at the intro to Lees smooth manifolds and he talks about the prereqs, the appendix also covers basically everything you’d need

[alpine rover]

that's the exact book my course is using

[alpine rover]

ahh thanks I didnt check out the appendix or intro

wow the appendix is so comprehensive

[graceful moon]

Yeah there’s someone here doing ISM in highschool just learning everything as they go which is insane, but shows you it’s possible lol

[tribal crow]

who’s that

[graceful moon]

Marlins

[tribal crow]

isn’t he reading Tu?

[graceful moon]

Possibly, actually

[gray gazelle]

what is the Pinter/Judson equivalent of Linear Algebra?

[haughty cypress]

book recomendation for begining calculus or math self study

[marsh ingot]

Spivak or Apostol

[haughty cypress]

where can i pirate this

[gray gazelle]

Piracy is illegal!

[remote vortex]

I mean, it is

[marsh ingot]

Just google it

Them is up to you

[broken meadow]

Let's not discuss piracy here

it's against tos

to be safe

[gray gazelle]

what happened to the server's logo

[haughty cypress]

facts

[glad rampart]

Math came out as trans so the server icon changed to show support

[earnest wolf]

going by the question, I'd suggest khan academy?

it's great for those kinds of skills i think

[gray gazelle]

Absolutely Agreed this is what should be done. We gotta support everything for a better world and encourage equality!
People like Euler Newton Gauss and All the Geniuses who contributed to mathematics would be so proud!

[glad rampart]

True!

[drowsy thicket]

The torus infront of it was removed because of the April 1 event. The background has always been what you see now.

It will be back to normal in short

[slender cargo]

@glad rampart I remember us talking about math stuff you were learning the other day. Have you considered looking at Nathan Carter's Visual Group Theory? I think if I were in high school, after having gone through Number Theory, Calculus, Combinatorics, and maybe Linear Algebra I would check that book out.

[rustic meadow]

it had the trans colors for a while the torus was just removed

[glad rampart]

Haven’t gone through calc or linear algebra but I’ll keep that in mind

[rustic meadow]

maybe some other changes were made but that's all i noticed

[slender cargo]

It's just a neat book that teaches Group Theory in a unique way that I think can be more easily self-studied.

[night harbor]

is there any book to learn extreme level of math?

[glad rampart]

Define “extreme level”

[night harbor]

i meant hard maths

very hard

[glad rampart]

You mean like competition math?

[gray gazelle]

Hartshorne’s Algebraic Geometry

[night harbor]

kinda

i am not sure

cuz i just wanna increase my knowledge

[glad rampart]

Just read some books on subfields that interest you

[night harbor]

it only contains algebric geometry nice

[night harbor]

r u in uni?

[glad rampart]

No

I’m self teaching myself hs math

[remote vortex]

any particular area? or do you just want something difficult?

[night harbor]

indeed

[night harbor]

great

[slender cargo]

Don't think of it in terms of "hard maths". Sounds like you want to learn more rigorous proof-based math. Start with something like Abbott's Understanding Analysis if you've taken Calculus.

[remote vortex]

good advice

[night harbor]

finished

dude

i saw in on yt

[gray gazelle]

saw what?

[night harbor]

the book

[slender cargo]

As in, you learned the contents of the book through Youtube?

[night harbor]

stephen abbot is damn clever

[night harbor]

i read that

i bought it from amazon

[gray gazelle]

What is your math background?

[night harbor]

nothing

i am a high school student from a creepy country

[gray gazelle]

No, like what have you studied?

Evidently analysis?

[night harbor]

yes

to be honest

[gray gazelle]

Anything else?

[night harbor]

algebric geometry]

[remote vortex]

One natural follow-up to real analysis is measure theory.

Axler's or Folland's book, for example.

Folland is harder

[gray gazelle]

…you’ve studied algebraic geometry?

[drowsy thicket]

Most likely translation error

You must mean analytic geometry?

[remote vortex]

I mean, apparently they're very ambitious

[glad rampart]

Not impossible if he’s terry tao

[drowsy thicket]

I didn't read much of the history, just him saying he did analysis and then algebraic geometry

[slender cargo]

So you have studied group, ring, and field theory?

True. If you want to make your life hard, go with Rudin PMA immediately for self study!

[remote vortex]

I was going to oppose baby Rudin instinctively, but then again, he did specifically ask for a hard time.

And apparently had no problem with Abbott

[peak ledge]

Can anyone recommend where I can from basics to extensive things about complex plane transformations?

[remote vortex]

It's an excellent book, it's just a lousy book from which to learn analysis.

[glad rampart]

No, not a single one exists /j

[slender cargo]

I think it is both a high quality book and a hard book to self study from without an instructor. Just imo

[remote vortex]

Yes, some uncommonly talented and motivated people will be fine self-studying analysis based on PMA.

But that's definitely not the median outcome.

I wasn't accusing you of humblebragging, I'm just saying I've seen a lot of people struggle HARD with PMA

[tribal crow]

definitely not!

[remote vortex]

Yes, and a lot of people don't have your sheer perseverance and grit, is my point.

[novel obsidian]

my ego is so inflated rn

[slender cargo]

List of reasons for why I think Rudin PMA is a difficult book to learn the material from on a first pass:

1. Ton of concepts mixed together that come in very quickly: metric spaces, both complex and real numbers
2. Jumping from Calculus on the real line to considering cases involving both real numbers and complex numbers (and more generally metric spaces) can be confusing
3. Proofs get to the heart of the argument but leave out the more laborious work, which is great as a reference (but maybe not for someone who is still getting accustomed to proofs)

In summary: need a good instructor to go along with it for a first pass, imo

[gray gazelle]

That's a treasure advice

[remote vortex]

I mean, you did end up skipping a lot of it eventually....

[novel obsidian]

This is just not true. I only skipped the multivariable and convex analysis bits

[remote vortex]

Also not enough examples and exposition, in my opinion.

[novel obsidian]

Actual disinformation smh

[remote vortex]

Fair, I'm still impressed you managed as much as you did.

[novel obsidian]

Admittedly I have been using 3 sources so it's not exclusively rudin self study

[remote vortex]

And you had access to people to ask for help coughcough

[sick river]

Idk about telling people to spend as much time as they can on rigorous mathematics at that age is wise if you’re interested look into it but it’s a hobby not something you have to do so don’t see it that way

[novel obsidian]

Yes, make it 4 sources of information haha. You personally have set my maths career forward by about a year icl

[night harbor]

yes

[novel obsidian]

Haven't read them so idk

[night harbor]

sry my english is bad

@drowsy thicket

[remote vortex]

I mean, I think this of all of Rudin, so yes 😛

[fierce hedge]

Ngl this can be a good or a very bad idea depending on how understanding his parents are. In fact okish at best and really bad at worst

I understand, also this is kinda out of topic for this channel

You can try something in between like Spivak and adjust accordingly. Say if you like the rigour then move to a proper analysis book else go for a calc book

[glad rampart]

You’ve gotta learn proofs

Very important

[tribal crow]

Spivak is super readable though (I’m assuming you mean his Calculus)

[molten mason]

Downloading a PDF of copyrighted material without the author's/publisher's expressed or implied consent even for personal use is illegal in the US. It just never gets prosecuted.

[tribal crow]

his exercises can be quite hard though

[fierce hedge]

While you can try proof books, I think Spivak does a fairly good job at introducing you to proofs

If you really wanna go from scratch try Tao's analysis book. He builds almost everything from ground up

[molten mason]

Don't listen to TopDreg, Lang is perfect

[fierce hedge]

If it's Amazon you can cancel/return the order or just keep it in case the books aren't that expensive

I mean 15 usd is expensive by Indian standards

I got a copy of Thomas in trash

Yes

Sure

[slender cargo]

One point in Spivak's favor: he's got a full solutions manual that you can get off Amazon

But yes you could just learn Calculus through Stewart first and then go into Analysis like most people do

[glad rampart]

What book even is Stewart

[slender cargo]

In the U.S. at least it's the most popular Calculus book for classroom use.

[glad rampart]

Yeah but like what is it called

Where do I buy it

[slender cargo]

Uh.. here you go

https://www.amazon.com/Calculus-James-Stewart-dp-1337624187/dp/1337624187/ref=dp_ob_title_bk

Only $185 on sale

[glad rampart]

What the fuck

That’s stupid expensive

[slender cargo]

yes

There are much cheaper alternatives to learn Calculus, but Stewart has been tested and has pretty graphics.

[glad rampart]

Might buy it used when I start learning calc

It’s a book, doesn’t need to be good condition

[molten mason]

Literally that's how it's simplified lol I think most undergrad algebra courses are 2 semesters and his Undergraduate Algebra is like 1, 1.5 semester worth of content, but at the same time he states in preface of Algebra that it's more than enough to prepare for Algebra. Algebra is only a reference text if you use it that way, you can still learn from it, it's just a steeper hill.

As far as approaches, I don't have experience with other algebra books besides Lang, but Lang does his own thing and even different things with different books. His GTM book is older and he wrote it so sections of the book are independent of other sections, so you can skip around and start mostly wherever. His UTM book is newer and chapters 1-7 are designed for a 1 semester course. That might explain his approach in each book.

[molten mason]

It's a mandatory textbook for many colleges and universities across the United States

Supply and demand baby

[molten mason]

It's supposed to have an online access with Cengage (Fun Fact, the Cengage fee is ON TOP of the textbook fee) so instructors can assign and grade homework.

If you self study, Quizlet Plus is $7.99/mo and has not just a solutions guide, but a walkthrough guide for every problem, even and odd, for the entire Stewart 9th edition

[glad rampart]

Couldn’t I just Google the problem solutions? If it’s so popular someone would’ve complied a list

[molten mason]

No

Like

You could Google enough of the solutions, and you'll find them in random YouTube and Quora posts.

But Quizlet literally has them all in one place. And like I said they're not just solutions, but explanations.

Let me find a random screenshot

[glad rampart]

Well I’m 16 and don’t have disposable income

[quiet rock]

Hi does anyone have like a suggested course outline or a course webpage for Apostol's introudction to ant?

[molten mason]

You could just keep creating accounts and use the free trials.

[glad rampart]

True

But that doesn’t change the fact that the textbook is like 250$

I’ll just buy it used

[molten mason]

After college students use them they just toss them, donate them, sell them for cheap. The used market is amazing.

And you don't need the 9th edition, the 6th, 7th, 8th editions work just fine. They're almost identical just different problem sets.

[glad rampart]

I love reading textbooks

[slender cargo]

You would probably really like Spivak's Calculus, although I'm hesitant to recommend it

(and it's cheaper)

[molten mason]

I personally like Howard Anton for calculus and his used books are also cheap.

My physical copy of Howard Anton is the 4th edition from 1994 and it's not that much different from the current edition.

[glad rampart]

I’ve heard good things about spivak

[molten mason]

I'm sure if you were to google "howard anton calculus 10th edition" you could find some good stuff about the book

[slender cargo]

Spivak is very different from Stewart. Spivak will teach you why Calculus works and is much more in-depth. It is also a lot of work

He'll teach you how to do Real Analysis, but it's a major commitment

[glad rampart]

Sounds like my style

[slender cargo]

Be sure to get the solutions book that Spivak wrote so you can check your work while going through the book

since you're self-studying

[molten mason]

Too bad Apostol is outrageously and prohibitively expensive

[glad rampart]

Mhm

It will be fun

Might use both it and AoPS calc

[slender cargo]

I don't know what AoPS calc is like but Spivak is pretty self-contained

If you've already taken Trig though then I would get started on Spivak already. That book will take at least a year to get through

[glad rampart]

I haven’t

Ive still got a ways to go until calc lol

[slender cargo]

I would focus on learning Trig right now then, which shouldn't take long, and then move on to Spivak.

[glad rampart]

I have a plan for getting to calc

[slender cargo]

Sounds like you're ready for Trig if you are currently learning Combinatorics

[glad rampart]

I haven’t learned geometry so i don’t think I’m ready for trig

[slender cargo]

Oh

Get through Geometry and Trig asap then if you want to do Spivak

[glad rampart]

I have a plan

[molten mason]

You're still like a solid year from calc then

[glad rampart]

Yes

I know

[slender cargo]

I'm just offering a suggestion since Spivak takes a serious amount of time, if you want to go that route. That's the main concern

[molten mason]

Basic Mathematics by Lang is $41 on Amazon and would teach you pre-algebra, algebra 1, geometry, algebra 2, and trig all in one book

[slender cargo]

Lang Basic Mathematics to Spivak would be a pretty straightforward path

(or just Khan Academy to Spivak)

[glad rampart]

Yes

It will take time, but I am ready for the commitment when I get there

[shadow dirge]

What would be a great book for linear algebra, I have "Linear algebra done right", are there any good ones? (I am studying computer science so math isn't my strong point)

Thanks!

[molten mason]

#book-recommendations message

Abstract Algebra in college or US class called "College Algebra"

Two completely different subjects.

Stewart, Axler, Blizter, Lang, etc. At that level any textbook will be fine.

516 ratings 4.4 stars, looks good to me

[remote sparrow]

a well-motivated, open-source, free text that's also available as a low cost paperback is hefferon. also consider checking out Coding the Matrix by klein.

[dapper inlet]

it should be illegal to recommended ebooks that cant be navigated

[remote sparrow]

https://tenor.com/view/epic-embed-fail-ryan-gosling-cereal-embed-failure-laugh-at-this-user-gif-20627924

[molten mason]

That website definitely looks 12 years old lmao But material looks good.

[remote sparrow]

older editions are very similar content wise. i got my copy of the 6th edition for $12.93.

[glad rampart]

Damn nice

[remote sparrow]

i didn't need the book by then, i just wanted to have it on my shelf

[glad rampart]

I think im gonna end up buying a used spivak and solutions, looks like I can get both for about 80$ total which isn’t too bad

Yes

[remote sparrow]

it's probably a good idea to get spivak used regardless of cost savings since after his death, the binding quality of his books has gone down

[glad rampart]

You could say he was the thing… holding them together?

[loud cradle]

he published, and then he perished

[remote sparrow]

it's an honors calculus book

[glad rampart]

Sounds like a skill issue to me

[loud cradle]

what forum is offering such dubious advice?

[remote sparrow]

a lot of students that go into honors calculus classes in college have had ap calculus

but not all of them

[slender cargo]

Spivak is 100% suitable for a first course in Calculus. It's just that Honors Calculus courses tend to go fast

[molten mason]

Probably reddit

[loud cradle]

ah yea, math experts over there
about a half step above quora

[molten mason]

Your account is too new

[glad rampart]

Spivak seems very cool I’m excited

[remote sparrow]

you need active, very active, or emeritus roles for image perms

[loud cradle]

well, physicists gonna physics

[remote sparrow]

http://alpha.math.uga.edu/~pete/expositions2012.html

@glad rampart this page has notes for spivak

[glad rampart]

Seems cool

[slender cargo]

Going the Spivak route is, like, an investment in itself. It is its own challenge.

[molten mason]

MSE, PSE, Reddit, and Quora. The four horsemen of the internet math forums lol

Gotta view 30+ threads across all 4 on a topic to get an accurate review on something.

[glad rampart]

Itll be a good while until I actually start spivak though lol

[remote sparrow]

quora fuckin sucks

[molten mason]

I love that half of Quora answers are just "I don't know"

"Hopefully someone else knows"

and other equally-informative answers

[loud cradle]

bro is asking if spivak covers the full content of a real analysis course (answer: no), and the first responder is like wah wah no it's way harder than stewart, talk about a non sequitur

[molten mason]

"What's the difference between A and B and why?"

"Just pick A"

[glad rampart]

Something to look forward to

AoPS intro to counting and probability

Wanted to learn a bit of combinatorics and probability

[molten mason]

Also I just read the rest of this

"It is a rigorous textbook that assumes a strong background in calculus"

It is a calculus book lmao

[loud cradle]

yea this is just way off base
worth every cent it cost to read, i suppose

[molten mason]

Vibes

How that lad views Spivak lmao

[glad rampart]

What approach does spivak take exactly?

[loud cradle]

"i wasn't among the target audience for the book, so it's too hard for anyone to use as an intro"

[loud cradle]

it's just a calculus book that proves the results rigorously, but aside from that it has the same content as any (single variable) calculus book

[glad rampart]

That sounds really really cool

[molten mason]

Also the other half of Quora answers.

"I've never taken this course, or the pre-requisite course, but I think that book is too hard"

[slender cargo]

It teaches how the derivative and integral are actually defined and used rigorously, where as other Calculus texts will explain it to some extent but stop short. Spivak will also teach you how to work with inequalities, limits, continuity, etc. much more rigorously than traditional Calculus texts. It's Analysis + Calculus combined together.

[loud cradle]

exercises are more challenging of course since many of them involve proofs, and you don't have 100 redundant computational drill problems, but it's not like he doesn't teach the computational aspect

[glad rampart]

I have some experience in proofs

[loud cradle]

you will probably be fine

give it a try and see

[glad rampart]

It’ll be a while

[loud cradle]

it's rigorous but the rigor is very well explained in good detail

[glad rampart]

I love rigor

[loud cradle]

you will totally understand epsilon-delta proofs and so on after reading his explanations

[molten mason]

and then Apostol teaches integrals before derivatives

[glad rampart]

I’m so excited to learn integration

[tribal crow]

chapter 19…

so. many. integrals.

lmfao

[loud cradle]

haha, he's channeling most measure theory/analysis books

[tribal crow]

but yes, I agree with this point despite my quip

[loud cradle]

yea the difference is that most of his are challenging and not just a bunch of similar problems with different numbers, like in stewart or his competitors

[remote sparrow]

i don't see much mention of joseph kitchen's Calculus since it was only recently reprinted by dover in 2020, but it's only $40

[tribal crow]

oh I see

[remote sparrow]

it's another honors calculus book

[tribal crow]

yes I 100% agree

[loud cradle]

in fact there were several spivak integration problems that i never could solve haha

[tribal crow]

I still can’t solve some of them

I’m about to be done a course that uses spivak though

it’s been a wild ride for sure

[loud cradle]

ah fun times!

nope, single variable only

[tribal crow]

Check out his Calculus on Manifolds for some of that

[loud cradle]

yea CoM is his multivariable book but it's nothing remotely like his calculus book

[tribal crow]

it’s not quite vector/multivariate calculus though

not many computations at all

[slender cargo]

check out CoM if you want to see a pamphlet of math scribbles

[molten mason]

No and that's my biggest gripe with recommending Spivak

Apostol is the same rigor as Spivak and it covers multivariable calc as well, so basically the whole Calc I, Calc II, and Calc III course series with one rigourous author.

However, Apostol in physical text is extremely expensive.

[tribal crow]

I’m reading CoM very very soon

along with ITM

[loud cradle]

definitely seems too compressed for its supposed target audience
i'd recommend shifrin over spivak any day

[slender cargo]

I mean, just choose a multivariable book for when you get there

[molten mason]

Yeah but I like having a single author to get from A -> B, not using multiple authors.

[loud cradle]

apostol is so boring compared with spivak too, would be great to have apostol's content but spivak's writing

[slender cargo]

I've seen that Shifrin recommends Spivak Calculus over Apostal due to Spivak having much higher exercise quality, supposedly

[loud cradle]

i can't speak much about apostol's exercises, but spivak's are great

[molten mason]

Unironic, Lang has a good multivariable calc book from what I've heard by others.

Ironically it's the only Lang book I haven't gone through lmfao

[loud cradle]

yeah i actually quite like lang's calculus of several variables or whatever it's called

[slender cargo]

Shifrin claims his book is a "successor" to Spivak's book

[remote sparrow]

read hubbard's book for that material

[slender cargo]

Yeah that one

But I mean, there's several books for that stuff

[remote sparrow]

https://www.matrixeditions.com/

[loud cradle]

btw, shifrin has a full year-long course on youtube covering his book, over 100 hours
he's a really good teacher

[remote sparrow]

don't buy hubbard's Vector Calculus, Linear Algebra, and Differential Forms from amazon, it's way more expensive

buy it from his website

[loud cradle]

well it's all relative

rudin is hard for most people's encounter with it

but a large number of people are capable of mastering it and have done so

[remote sparrow]

it is clearly not the one with the most prerequisites

[loud cradle]

compared with something like say hartshorne's algebraic geometry

[molten mason]

Hartshorne

[remote sparrow]

good thing i don't care about algebraic geometry

[molten mason]

@torn crypt

[torn crypt]

Probably linear algebra

[molten mason]

Memes aside, 100% this

[torn crypt]

Used everywhere

And is how you get things done

[molten mason]

It's simple to learn, easy to master, and a lifetime to use. Every field uses it in one way or another.

[remote sparrow]

but does it get done right?

[torn crypt]

Logic is #2 though

[molten mason]

Calculus actually isn't used a whole lot

[remote sparrow]

sheldon axler

you need the very active, active, or emeritus role for image perms

[molten mason]

Calculus is actually used in this though lmao

#book-recommendations message

Although "Comprehensive Explanation" shouldn't be used with Lang

More like... "Concise Explanation"

Or just... "Concise"

Maybe succinct rather than concise

[knotty herald]

There’s a lootbox

Gotta spam it

Till you get every reward

[remote sparrow]

i heard on springer the coupon codes HLT23 and 50off work

[molten mason]

I'm gonna push the Springer sale until June 30th just in case they improve the sale

I think it was only 40% on Mar 1st and now it's 50%

[fervent marten]

what is the springer sale?

[molten mason]

Springer is having a sale on math books, 50% off with 50off until June 1st

[fervent marten]

can you send me the link pls

[remote sparrow]

you just enter the coupon code in checkout

[fervent marten]

yeah idk how to get there lol

oh nvm

[gray gazelle]

book recommendation about research methodology in mathematics please

anyone knows if this exist?
or maybe a book that uses an research approach for learning?

or learning through questions

?

[earnest wolf]

research approach for learning
u mean like scientifically accurate methods for effective studying?

[gray gazelle]

not necessarily scientific proved or something like that

but motivating

instead of presenting the material

making me to enquiry

[torn crypt]

@remote sparrow Have you seen Marker's book on logic (not models)?

[earnest wolf]

I see. Then you may like this one

||/s||

[gray gazelle]

I heard of it , never read it

how is it?

[earnest wolf]

then don't
it was a bad joke

[gray gazelle]

oh

:/

[earnest wolf]

I mean... Find some assignments / exercises and work only thru them?

[remote sparrow]

did he release it already?

[earnest wolf]

or read a normal book and try to prove things urself

[remote sparrow]

i think he is talking about inquiry-based learning

[torn crypt]

No clue, just saw he has one and is supposed to have come out

[remote sparrow]

yes i've known for some time

[gray gazelle]

the problem is :

1. how to not spoiler my self
2. how to despite "researching" , dont deviating too much of the subject

I mean is ok deviating

but , how i state the correct question

?

[remote sparrow]

you're supposed to mimick some proofs when you first start out

[torn crypt]

Gotcha, and seems like it’s not out with that response

[earnest wolf]

how not to spoiler: cover the proof?
hm, idk, it works for me pretty much. try maybe physically removing the book from the room if you can't resist the temptation

[remote sparrow]

don't worry if you don't fully understand the proofs given in the book right away; when you do some exercises using some results, you'll get it

[gray gazelle]

and also , how do I know what prerequisites or tools (mathematical tools) are needed in order to approach correctly (formaly speaking)

some subject

[earnest wolf]

oh, that's a great question!

you don't

trial and error, salted with some intuition

[gray gazelle]

namely, if I am trying to learn calculus with this approach

[earnest wolf]

ah, wait lol

[gray gazelle]

is set theory needed , or just throw in?
how do I state the questions will led my to the right theorems?

[earnest wolf]

I misread ur question. Nvm

[gray gazelle]

how do I find interesting problems?

if a book about this exist (the questions), it will be easier

(calculus was an example )

[gray gazelle]

Read the preface of the book. Prereqs are usually covered there

[gray gazelle]

Look at a result and it’s proof, ask what would happen if you weakened the hypotheses, strengthened the hypotheses, altered the proof, generalized definitions, etc

the major problems/difficulties are:
how to make the right question ?
how many foundation is needed for following for this approach?
that's why I was looking for a book , which I could not find

[gray gazelle]

What do you mean by right question? Like something interesting? Just ask a bunch of questions and see where it leads you. If, instead, you’re looking for interesting textbook problems, any good book should have them

What subject are you trying to study?

not questiong for practice , instead questions that may lead me to a similar theory , maybe not equal , but similar to what I am trying to study

[gray gazelle]

first, logic. but i would like to extend this approach

[gray gazelle]

So IBL?

what means IBL

?

[gray gazelle]

Why logic? It’s basically just a set of axioms and rules of deduction, there’s not a lot of theory to build until you get deep in the weeds

[gray gazelle]

Inquiry-Based Learning

[gray gazelle]

mathematical logic

[gray gazelle]

how is it ?

never heard of it

Depends on what you’re going for. I try to read every math book as if I’m taking an IBL course; that is, treat it as a list of definitions and results (don’t read the proofs) and work out the proofs on my own, using the book as an answer key

oohh , i see .

but dont you try to cheat yourself by , I dont know , sometimes looking at the results?

Like, there’s no way you’re going to sit down with a couple definitions and reproduce modern mathematical logic miraculously. But if you have a list of results, you can work through them

(proofs)

[gray gazelle]

ramanujan style?

[gray gazelle]

Sometimes I give in, sure. My usual approach is to write down the theorems on a separate piece of paper, then put the book away so I’m not tempted to open it

[gray gazelle]

Ramanujan had a summary of results he learned from

[gray gazelle]

why not

I mean , the results is a little artificial , that's the difficultie, I am thinking about questions that lead results

Well yes, you should figure out the motivation for the results and what would lead you to it

not in the sense that a result is intended to be artificial , but the presentation

But it took centuries and hundreds of mathematicians to develop these fields, many of which spent years on wrong turns, so you’re not going to be able to reproduce it without knowing what the key definitions/results are

[earnest wolf]

I mean, @gray gazelle suggested one idea above: use ur theorems as the questions

[gray gazelle]

it was sort of a joke, but the problem is "If some theory was developed on questions (naturally), there is a book with that questions , in order I can look for the questions?"

[gray gazelle]

A good book should be structured to reveal those questions through its development: it should tell a story

[remote vortex]

the problem is that the historical development of most of the theories was a mess

it wasn't a neat sequence of questions and answers

[gray gazelle]

many mathematics courses are taught "upper giant's shoulders" if that means something , so you never get to the real "mathematical" process , and that's what I am loolking for

Sometimes the questions are made explicit, but otherwise, whenever you encounter a result, you should ask:

1. What is it formally saying?
2. What is it intuitively saying?
3. Why ought it to be true?
4. What does it tell us about the broader field?

[gray gazelle]

then I have not read a lot of those. (maybe spivak was an exception, idk)

[gray gazelle]

The “real mathematical process” is asking questions, then throwing stuff at the wall until it sticks. There’s no way of knowing if you questions you ask are good until you try to answer them

but I see the result but no the motivation for that result, what was the person who discover/created that have in mind

[remote vortex]

there usually wasn't the person anyway

[gray gazelle]

You should try to figure that out for yourself; look into the questions that the result solves, or the questions it produces

[gray gazelle]

so just go for it? whithout any preparation or map, simply exploring?

[gray gazelle]

right

Well, if you really want to. I would recommend just reading some books, working through the proofs and exercises, then doing actual research when you’re ready. Ask questions and explore along the way, but it shouldn’t be your exclusive method of learning

where can I find those questions? It seems hard to find

What questions?

[earnest wolf]

u make them urself

[gray gazelle]

that lead the results

[earnest wolf]

oh

[gray gazelle]

Look up “[result name] related problems,”

I looked for some calculus theorems and just find practice problems

[remote vortex]

I think for the motivation your best bet would be to look into the history, e.g with calculus/real analysis

from the exhaustion method of the Greeks, via Newton/Leibniz, then the formalization of Weierstrass

or all the tumult around set theory that led to ZFC becoming a thing

most modern books start with the modern framework and don't discuss the history that led to it, that's true

and I don't know offhand any book that would focus on the motivation and history extensively, but I'm sure they exist

on the phone so I can't search much right now

[gray gazelle]

You can also determine a streamlined motivation/train of thought that leads to a result

Like, for Rank-Nullity, linear transformations are very well-behaved functions, so it’s plausible that their degree of injectivity and degree of surjectivity are related in a fairly precise way. Now, to define “degree,” the notion of dimension gives you a good tool. The image and kernel of a homomorphism are natural constructions, and their interpretations for a linear transformation are related its injectivity and surjectivity. Compute a couple numerical examples to conjecture Rank-Nullity, then prove it

[gray gazelle]

ok , so that's like start studying a book and getting ahead from the beginning , right?

“getting ahead from the beginning,” wdym?

or just making conjectures from basic things?

"wdym" what means that?

what do you mean

you begin with some theory (injectivy and surjectivity)
then conjecture the rank-nullity

so your prerequisite for this approach would be injectivy and surjectivity

As in being familiar with those concepts?

yes, that means this approach needs a base

Yeah, those concepts are kinda essential to… everything

[hallow oriole]

struggling to think of a branch of math that doesn’t need either yeah

how would something like that even look?

[gray gazelle]

Well, odds are you wouldn’t be able to come up with rank-nullity on your own. What I’m saying is that, after reading about the result, working through the proof, and maybe doing some exercises, you should think through how you could have come up with it yourself, what might lead you to it, and what its significance is

[gray gazelle]

I mean, you could do geometry ala euclid and never mention functions

But basically any modern subject is going to need it, yeah

how many base knowledge you think is necessary for begin conjecturing results?

obviously not the more advanced and obscure ones , but a result that is probably well known , but maybe I didn't know

Again, it depends on what you’re studying

And you’re probably not going to conjecture and prove any super important results because they’re often non-obvious

But you should know all of the content that’s typically covered in an intro to proofs/discrete math class

And, more generally, should have a strong foundations in analysis & algebra if you want to study anything past these subjects. Like, you’re not going to pull the idea of a group and a composition series, then conjecture the Jordan-Holder theorem out of the ether

so discrete mathematics/ proofs , real analysis , and algebra are good bases for trying to have a common notion

[gray gazelle]

The more you know, the better

[gray gazelle]

yes , but the goal , is to avoid learning all by jus reading a book and trying to get at least some self ideas. so I dont know where is the correct point to just stop taking the complete theory as truth.
for example , when someone arrives to the post Phd state of mathematics , then it is probably that the person is in the frontier between the mathematics that is known and the mathematics is done.
so , that frontier is relative. but obviuosly as you say more is better , but how much ?

[remote vortex]

basic set theory is the fundamental concepts for most modern mathematics

sets and related concepts are used pretty much everywhere

[gray gazelle]

The frontier isn’t relative, someone with a phd has studied up to the unknown in some specific subfield and is then doing research to discover something new

It takes years to get there and, generally, in someone’s whole research career they might only chip away a tiny bit of important new results

[gray gazelle]

I mean relative , in the sense , there is ever a gauss that al ready knew the unknown,

not relative in a broad sense

Ummm, what?

Gauss didn’t have some mystical eye into the unknown

He studied a lot of mathematics (read a lot of books), and then was a very successful researcher

But it’s not like he read one book and then pulled the rest of mathematics out of his head

He spent years reading, learning, and discussing it

[remote vortex]

also he was a physicist

[gray gazelle]

That too lol

no , I meant that the phd state of the unknown mathematics , is unknown for the most of people , but that does not mean the phd got to the point he is the only one who discover new things

for example gauss knew non euclidean geometry despite , it was considered something unknown , it was not for him

[gray gazelle]

He helped develop it…

Regardless, you can try to produce “original” mathematics with a limited background, but you’re not going to learn real analysis by doing that

so getting to that advanced point or not , does not change the fact that discovering/creating some math theory is sort of personal

[gray gazelle]

ohh, I get it

is sort of a conversation between different mathematicians , what makes mathematics?

I dont know how to approach this correctly , neither if can really be correct

😕

is useful the approach I want to use?
since you said that , seems like no one really uses this

Depends what you want to do

If you want to play around with some set theory for fun, sure

But if you want to learn a bunch of mathematics, maybe eventually do research, then you should read some books

so you are trying to say that big theories and other mathematics are impossible without collaboration?

I’m saying that big theories are too big to produce without a lot of mathematical background/experience to work off of

[remote vortex]

pretty much, yes

[chilly sentinel]

surely all the low hanging fruit has just been picked and it takes a lot of education to find anything new?

[gray gazelle]

and what about
"The Inter-universal Teichmüller theory"
by Shinichi Mochizuki

[hallow oriole]

dont worry abt it

[gray gazelle]

First off, that’s a controversial subject. Second, Mochizuki has decades of experience and training in arithmetic geometry and mathematics as a whole

[chilly sentinel]

what is it about mathematics that makes people who aren't educated in it want to reinvent it?

[hallow oriole]

honestly i doubt it. i genuinely believe there are several results that would be important that a motivated undergrad could find (especially in combinatorics, knot theory, etc) but you’d have to be very very lucky and talented

[remote vortex]

I think it's the allure of the idea of everything following logically from a set of rules

[gray gazelle]

Yeah, but even an undergrad would have been exposed to some analysis, algebra, topology, etc

I dont think that all follows from axioms , but I believe no theory is complete, and every theory can be approached from a diferent point of view

so , basing yourself in what is already constructed limits what you can create

[hallow oriole]

i’m not sure i would explicitly describe this as a “lot of education”, though

[glad rampart]

We’ve known no sufficiently complex axiomatic system is complete for a while now

[gray gazelle]

I know, I’m just saying that no one is going to read 3 pages on set theory and then solve a bunch of outstanding problems

[hallow oriole]

sure, fair

i dont think any undergrad with the capabilities listed above could solve more than one

[gray gazelle]

That’s orthogonal to the problem

still I am having trouble to find the problems that motivates the theory, cause some texts are about the general problems

but not the specific problems

What do you mean by general and specific?

I read elements of the history of Mathematics

and there it talks about the general problems

but does not states what where those problems at all

I mean, if you want specific problems, read a textbook on the subject

[hallow oriole]

can u give an example of a general problem

[gray gazelle]

thats the problem , modern textbooks dont write too much about that

[remote vortex]

the book should cite sources, and those sources should discuss the issue in more detail

[gray gazelle]

I will look at

it

[gray gazelle]

what modern textbooks have you looked at?

[remote vortex]

that's a big part of research, following the sources

[gray gazelle]

right

I was not aware of that

thanks a lot

you are right , I looked at the bibliography and there were cited the papers, is a bibliography of more than 300 citations , I did not look at it maybe because of laziness or fear , or maybe the two

[remote vortex]

well, you found them now, that's what matters

[gray gazelle]

so I think I will do that. That really solved my problem . thanks a lot

[gray gazelle]

do you know a book with this tips , like how to search papers or something I maybe will be missing ?
are courses on this?
once I talked with a professor about this but he wasn't to help too much

There are some articles on advice for research, you can look them up

[remote vortex]

https://academia.stackexchange.com/questions/34038/how-do-mathematicians-conduct-research

here, for example

[quiet rock]

Bumping this question

[violet shuttle]

I agree with you. For most things you should be able to get a book that costs like $30 on average and $60 at most

[sage basin]

Should I press on with Fulton's Algebraic Curves and try to do better with algebraic geometry or set it aside and read Lam's A First Course in Noncommutative Rings?

[gray gazelle]

My algebra is so shit I cannot do most of the assessment problems in Stewart's Calculus. So, I shall be remediating with Intermediate Algebra for Colleges by William L. Hart.

[hallow oriole]

khanacademy works fine

no need to spend money imo

for basic algebra

[gray gazelle]

It is a book I got for probably a dollar at a thrift store. I have tried to do Khan Academy but I really don't like to learn through video, at least not initially. Has a copyright of 1948 and I am really enjoying the explanations.

[glossy zealot]

Sometimes the videos put me to sleep

It’s more about the voice of the instructor and my ability to fall asleep quickly

Reading books or listening to female instructors is less drowsy

[gray gazelle]

is it an easy book? I found this another one by Meckes which seems nice

[junior isle]

What do you guys think about morton curtis' linear algebra book?

[remote sparrow]

meckes is great! i've recommended it before

#book-recommendations message

[gray gazelle]

I even found a course page which is taught by meckes himself

[remote sparrow]

tragically elizabeth meckes died of cancer a couple years ago

[remote sparrow]

elizabeth meckes had a course webpage following the book. not sure if mark meckes had one

[gray gazelle]

may she rest in peace

oh it is by elizabeth?

[remote sparrow]

https://case.edu/artsci/math/mwmeckes/elizabeth/web/math307/

[gray gazelle]

I see

[fleet solstice]

Question, does Blitzer's College Algebra textbook, cover some topics that you learn in Precalc?

Besides the trig stuff

[night zinc]

Hi I want to ask for some books/sources recommendation about Game theory in Auction

[main hill]

Anybody used Swokowski's calculus textbook? Cause I'm looking into it and I see classic edition, alternate edition, the original version with analytic geometry and have hard time choosing the right for me

Don't understand what's the difference

[gray gazelle]

Kindly suggest tough maths books for high school maths
For national Olympiad and competitive exam( JEE)

[glad rampart]

AoPS is great in general and especially for comp math

[gray gazelle]

AoPS?

[glad rampart]

Art of problem solving

[gray gazelle]

Does it have calculas , algebra, conic section, algebra, trigo, probability, permutations and combinations, etc?

[hallow oriole]

dont do combo in aops if you want to learn real combo, pick up a dedicated book

[glad rampart]

AoPS intro to combo has been fine for me so far

[hallow oriole]

sure, it's fine for competition combinatorics

[glad rampart]

What exactly does AoPS not teach that a “real combo” book would

[hallow oriole]

competition combinatorics is just very far away from what people in combinatorics are doing these days and even for a pretty long time

[glad rampart]

that goes for most math that can be taught to grade schoolers reasonably

[hallow oriole]

for reference, my thoughts are based off the book called "AOPS Intermediate Counting And Probability" on archive.org

[glad rampart]

I don’t know about that one, I’m reading the intro one

[hallow oriole]

the intro one should probably be a subset of this one, then

<@&268886789983436800>

[glad rampart]

I don’t think that’s how that works

[night gorge]

Thanks

[hallow oriole]

does the intro to prob one cover more than this?

wait

can u js send the toc

[glad rampart]

It covers different things I’m pretty sure

[hallow oriole]

what's the toc

[glad rampart]

https://s3.amazonaws.com/aops-cdn.artofproblemsolving.com/products/intro-counting/toc.pdf

[hallow oriole]

instantly i see a complete lack of graph theory, basic ramsey theory like R(3,3) computations and such, no sieve anything, no generating functions, no coloring problems, no probabalistic method, no lattices, no complexity theory

[glad rampart]

My guy this is a textbook made for like 8-10th graders

[slender cargo]

I think that’s fine for her case. She’s in high school

Hasn’t taken geometry yet

[hallow oriole]

like i've been saying, i think it's perfectly fine for competition math and such

[glad rampart]

“Being good for competition math” implies it doesn’t actually teach anything about combinatorics which imo is completely false

There is a certain limit to what one can feasibly teach in a textbook with what the target audience is likely to know and I think this book does a good job at covering what it can

[hallow oriole]

fair point!

[hallow oriole]

fwiw i genuinely think this is true to an extent, but this is just my experience from having skimmed comp math textbooks relative to everything i've done in certain other textbooks

[glad rampart]

I think you think they’re much more focused on comp math than they are

[hallow oriole]

aops?

[glad rampart]

Yeah

[gray gazelle]

Yea thanks

[hallow oriole]

fwiw for these purposes aops is fine

[hallow oriole]

am i? they specifically wanted books for national olympiads and the jee

[gray gazelle]

Thanks a lot dude
Do you know any more books?
Need atleast 3

[hallow oriole]

oh

wait

brain fart

ha

[gray gazelle]

@hallow oriole do u?

[hallow oriole]

it's possible, i guess

i'll check out the actual book and come back with a more informed opinion

[glad rampart]

Have fun

[hallow oriole]

for comp math? unsure. having said that if you're taking the jee consider getting a prep book specifically for that exam, there's bound to be something good

[gray gazelle]

Ignore comp math
Just suggest tough maths book for high school

[hallow oriole]

idk pick up rudin or sum

recommendations are dependent on what you need

[gray gazelle]

What I need?
A tough math book that's it for advanced high school

[hallow oriole]

sure but for what subjects

calc, discrete math, nt, etc?

[gray gazelle]

Sets, Relations and Functions, Algebra, Matrices, Probability and Statistics, Trigonometry, Analytical Geometry, Differential Calculus, Integral Calculus, vectors, etc

[fallow creek]

lol

so calculus and an econ class

any standard calc book is pretty good tbh

id shoot for older like 80s

[gray gazelle]

You won’t find just one book covering all of that in detail

Naive Set Theory by Halmos is a good (and brief) intro to set theory

[gray gazelle]

I don't want also
You can name books separately for each topic

By Algebra, do you mean high school algebra or abstract algebra?

[steel patio]

Look up the book “How to Prove It”. This is an introduction to higher level math. It’s an introduction to set theory and notations. It will show you how to prove stuff like why $\sqrt{2}$ is irrational and stuff. Then this can be applied to all types of math like real analysis (calculus), abstract algebra, and other discrete math topics

[hasty eagle]

OGBrownSkin

[gray gazelle]

Thanks man!!

[gray gazelle]

elementary algebra, abstract algebra, advanced algebra, commutative algebra, and linear algebra.

[gray gazelle]

Guys pls help me with this
Am really not getting plus I have to spend money wisely 😞

Ok. well if you’re not familiar with elementary algebra yet, then you’d need to learn that before calculus. Commutative/advanced algebra are subfield of abstract algebra, as is linear algebra(though often you take a linear course before abstract). However, for all of those, you’ll want to be comfortable with proof writing

[fallow creek]

i would probably get a seperate commutative algebra book if you want to learn it specifically

the other stuff is covered in your standard algebra textbook

[gray gazelle]

Atiyah Macdonald is a good first text for comm alg

[fallow creek]

but i would do algebra before calc

[gray gazelle]

you mean elementary, right?

[fallow creek]

as calc and some trig ideas need a lot of algebraic breaking down for lack of better term lol

well... isnt elementary just arithmetic lol

standard stuff quadratic equation

[gray gazelle]

Yeah

I’m just saying that someone probably shouldn’t try to study ring theory before calculus, for mathematical maturity reasons

[fallow creek]

exactly

i tbh wouldnt attempt commutative till after calc

you need a comprehensive idea of space

but thats bc i feel like calc is more rooted in standard algebra than anything else

[gray gazelle]

Well yeah, you shouldn’t try commutative until after a first (or second) course in modern algebra

[steel patio]

I would look into group theory books for a better understanding of abstract algebra especially if you’re still in high school. It will kinda challenge what you’ve learned about commutative algebra and show you what most of math is like

[steel patio]

And you’ll learn about “abelian groups” which are the idea that 1 + 2 = 2 + 1

[molten mason]

Why not just suggest Lang for algebra then

[fallow creek]

i would group trig and analytical geom together tho

but i cant recommend books really

i just self study bouncing around

[gray gazelle]

I feel like I just learned those in precalc & calc

[fallow creek]

well