book-recommendations

[marble rock]

jacobson

pinter if beginner

gaillan

[gusty smelt]

lol well dont do dummit foote

but depending on what you want i always recc artin or jacobson

[marble rock]

bro every1 says its the classic

on like stackmath

[gusty smelt]

artin is for more LA stuff (doesnt assume you have LA background)

dummit and foote is a shit book

and no one should study from it unless they want to get turned off from algebra altogether

[marble rock]

its just dry ig

but isnt all math

srs

[gusty smelt]

no

math is extremely fun

dummit and foote is not.

[prime oak]

got it, I'll check em out thanks

[sage python]

I have the definitive guide to algebra books

[sage python]

There are some books that are more introductory than these, such as "Gallian", "Pinter", "Fraleigh", etc. Honestly I feel like if you're struggling with D&F and Artin... you might not be ready for algebra yet, better to revisit foundations or smth.

Dummit and Foote: the default, nice coverage. Very wordy, imo to the point that it becomes boring, but though that can be a plus if you're looking for something gentle. Pretty decent problems. This is where I learned most of my ring and field/Galois theory (though I mostly went off lectures). In principle could be done without serious LA but the jury is out on whether that's a good idea.

Artin: The objectively correct entry point for most people. Does a good job at showing you algebra is cool/situating it within other areas of math, and doesn't assume any background in anything (defines a matrix). Doesn't cover as much as some of the others, but that can be deferred to a second/graduate algebra class.

Jacobson: Extremely clean writing, my personal favorite. Prefers to explain things in English rather than symbols. Covers an interesting/non-standard set of topics. For this you want some LA going in.

Herstein: Clean writing, good for group theory in particular but doesn't cover enough for the other subjects. Uses x(f) instead of f(x), so you'll have to unlearn it which is a pain in the ass (though I get the point). This is where I learned GT.

Hungerford: I've seen it described as a "watered down rewrite" (presumably of Lang?). Seems clean. Probably a book for which you want LA going in

Lang: The king, good writing (based on reading a bit of his field/Galois theory) but probably a bit too efficient to be a viable first pass

Aluffi: Category memes (good to know but at this level it can be a distraction), kinda slow/way too wordy, exercises that I've seen were not the best.

Knapp: Artin but with a lot more coverage. Is a fair bit harder as a result

People also seem to like books by Isaacs and by Rotman. May glance at them at a future date.

[valid moth]

i recommend EGA for an intro alg book. there's even some geometry thrown in as a bonus

[marble rock]

@sage python what about the beginner texts like pinter or gaillan

fraleigh

AA theory and applications the free one

and what about like

[sage python]

Ehhhhhh I feel like Artin's kinda one of those books where like

If it's too hard you probably should hold off on doing algebra you know?

[marble rock]

yea but maybe artin is defficint

in like GT

[sage python]

You heard me say that because it doesn't cover semidirect products lol. I'm mostly tongue in cheek, like you can patch the holes anyway

[hollow kindle]

oh I started with Hernstein, it's pretty concise but does make some assumptions

but mostly good for complete beginners

i'll check out artin and lang, thanks for the recs

[sage python]

I feel like Artin mostly makes other intro books obsolete, except for like, the best applications oriented one (stuff like coding theory)

[hollow kindle]

that's a bold claim, will def look into it :p

does it have good problems?

[sage python]

Iirc yeah

[hollow kindle]

nice

[gusty smelt]

oh nice you should pin that dami

[marble rock]

well i think iwas the one who told you about that

lol

what about the books specifically for like

like a group theory text

is that bad

[sage python]

So I did a Rotman speedrun and found it real nice (I feel like Rotman is just an S tier writer). I've heard of like, Rose, Robinson, and Alperin-Bell

But I feel like all of them (including probably Rotman) assume you know some algebra already

(them = the ones I know, I won't exclude the possibility that some first pass group theory books exist lol)

[marble rock]

yea col

[hollow kindle]

this is a math text speedrun?

never heard of this lmao

[marble rock]

yea i wonder what that is

[gusty smelt]

yeah dami grinds the same textbook over and over in hopes of world record

ofc

[marble rock]

really

[cold lagoon]

a quick pass through instead of working through it

prolly

[calm crane]

also jacobson book 2 basically covers much further than any intro alg book

[marble rock]

jacobson old

the real algebra text that every1 should read https://www.amazon.com/Abstract-Algebra-Beginners-Substructures-Homomorphisms/dp/0999811789?creativeASIN=0999811789&linkCode=w61&imprToken=YTKortC08DWP8f65NHrn.Q&slotNum=4&tag=uuid10-20

[flint forge]

I still like pinter

i think its a nice read

[valid moth]

nii-tan* @tranquil ocean

wait wrong channel

[sage python]

Yeah I was just like

Trying to learn a bunch of group theory in second year

Because I was thinking of taking a class called "Algorithms in finite groups"

So I just did like, 4 chapters of Rotman in 3 days lol

[civic carbon]

I find DF to be wordy in a good way, and it has the best/most thorough examples

though I will say, I think DF is a textbook to learn from, not the best as a reference

[sage python]

True but I also think he like, drags out explanations too much

D&F is good to follow along lectures with where the main learning comes from the lectures. But to just read through it is like

Boring really

[civic carbon]

I still end up reading it when I teach the course just cause I find the writing compelling, but obviously tastes vary.

[civic carbon]

but there is no algebra book I'd look things up in. I'd look it up in a specialty book depending on what area

[sage python]

Example 3 here is a good demonstration of what I'm talking about. Like to show (2,x) isn't principal in Z[x] should be, 3 lines if you're taking it real easy

[civic carbon]

it's easy if you know a lot more stuff for sure

but no doubt, it is an advanced undergrad early grad text

but one with an enormous amount of content, like their little intro chapters to e.g. representation theory I like

but, of course, if you are a representation theorist, their chapter won't be helpful

a lot of people complain the exercises are too easy

sadly it is too advanced for the undergrad algebra course I teach, but it is my favorite text to teach out of full stop.

I'm actually surprised anyone still talks about Lang's books

[flint forge]

D&F is fine accompanied w lecture

But its a dry read

[civic carbon]

I was definitely fortunate to have good algebra teachers

[sage python]

Lang I feel is a good grad algebra book tbh, like I read through his field theory stuff to learn the proofs of existence of algebraic closure and all

And it was extremely pleasant

Actually he had some annoying terminology

[civic carbon]

I really don't like the "allergic to examples" writing style, which Lang is famous for.

but if it's a textbook for a course, and the professor provides all the examples for you, I can see it clicking nicely

[quick hornet]

i have a very "cool example"-oriented lecture style

when i think i should probably do more "gritty/inelegant examples" lmao

[civic carbon]

you do the coo lexamples in lecture, and put the gritty inelegant exampels on the HW that the TA will grade.

[sage python]

Hmm, I guess I haven't seen enough of it to tell about the examples. Especially because in raw field theory (like before Galois) there's not as many examples as like, finite group theory

[quick hornet]

well yeah, but i think it makes students think that everything should be as clean/slick as what happens in lecture

i try to tell students "I know we get through this stuff in 30 seconds in lecture, but be prepared to spend 20 minutes on it during homework"

but im not sure thats as effective as just demonstrating it?

like i was acting as a TA filling in for a lecturer who was off on some conference

for a group theory lecture on quotient groups

and i gave "slick" examples like

[civic carbon]

I think "put the grittier examples on worksheets and have students do them in groups while you walk around" is a good midway

[quick hornet]

R/Z is the unit circle

but a lot of the isomorphisms students are expected to show arent necessarily as "nice"

[sage python]

Maybe, but at least once it'd be good for a lecturer to demonstrate getting in the weeds

Like this is what it looks like when there's no trick

Also we prob should keep it to books I think

[civic carbon]

I wish more books would do the gritty ugly examples 😛

especially in algebraic geometry 😛

[quick hornet]

imagine examples in an algebraic geometry course

what is this, geometry?

wait

[pulsar aurora]

"PTSD for Mathematicians" The series that makes all Mathematicians cry.

[echo kiln]

lol

[hearty steppe]

Huh there is a series called PTSD for mathematicians? Bloody hell

[calm crane]

it is a book where every example is find a counterexample to xxx

also in really bad notation

and written in old cursive english

[hearty steppe]

World of Mathematics seems like a good series

I checked out the contents of it and I have the volumes on my cloud drive

[pulsar aurora]

There are compilations of essays by mathematicians usually regarding topics all stringed together under a theme. I don't remember what the book series ia called though. I would read them on campous when nothing else to do

[frosty pulsar]

Hey guys, I recently started reading a book in which the author talks about Dirichlet's theorem, but I can not find any info on it online, the only theorems I find are about Fourier series. Do any of you know?

[wooden sparrow]

World of Mathematics seems like a good series
@hearty steppe
What level is it at?

Ohh it's like 'the story behind the equation' book kinda thing?

[gray gazelle]

@frosty pulsar https://www.google.com/url?sa=t&source=web&rct=j&url=https://en.m.wikipedia.org/wiki/Absolute_convergence%23Rearrangements_and_unconditional_convergence&ved=2ahUKEwitlqe8iPDpAhXS4HMBHXMZDKAQygQwAnoECAMQBQ&usg=AOvVaw30iQoG_uz60e0EwZ6OVoNF&cshid=1591545404586

[true veldt]

the closest theorem i can think of that has a name

is Tonelli's theorem

https://en.wikipedia.org/wiki/Fubini's_theorem#Tonelli's_theorem_for_non-negative_measurable_function

which basically says that you can do any kind of swapping you want as long as all the terms are nonnegative

[frosty pulsar]

thanks

[neat basin]

@frosty pulsar Maybe you could search the theorems involving commutative convergent series

[frosty pulsar]

Thanks again, I just found out it's sometimes called Steinitz's replacement theorem. Edit : Nope, totally different thing.

[neat basin]

Oh cool, glad to help 🙂

[upbeat vine]

Anybody who have read books by Haim Brezis (Especially his FA book)? What are the general thoughts on his presentation and writing style?

[radiant crown]

I'm having a quick look at his FA book right now and you may want to attentively read the four warnings at the beginning as they announce a few things that may or may not dissatisfy you

the first and fourth especially

[gray gazelle]

https://www.amazon.com/Introduction-Probability-Theory-Applications-Vol/dp/0471257087

what do you need to know to read this

[limpid gazelle]

da fuk

that theorem is called dirichlet's theorem???

[frosty pulsar]

Apparently so

[hearty steppe]

Some intro set theory book recs would be nice. Halmos glosses over things too quickly

It's good but I am finding there are some weird tricks not glossed over in Halmos as he just brushes over concepts

[flint forge]

You should not really need a set theory book

But instead should pick up any set theory you need outside the basics as you go

I think munkres has a decent “set theory” intro

But set theory normally describes a subfield of logic

Rather than “how to work w sets”

My point is i dont think he means logical set theory given his other interests

I think he just means an introduction to working w sets

I could he wrong

[upbeat vine]

@radiant crown Were you talking about the brief user guide?

[radiant crown]

idk how the translators managed the thing

1. basically says some results will not be proved in the book, but are proved in another book (an exercises compilation book)

but there's a catch

because of some beef between Brezis and the editor, that exercises compilation book was never made and will probably be never made

however, it is possible to find on the internet the original document the exercises compilation should've been adapted from

here it is, though I'm not sure this will be of any use to you

and warning 4) says the book deals with real vector spaces only

maybe you don't care about warning 4) but the situation with warning 1) seems more difficult...

but since I haven't read the book personally, I don't know how this will affect your reading

• digging old forum threads, people seem to be satisfied with this book

[north spire]

what do you need to know to read this
@gray gazelle usually, the preface of the book will give information on prerequisites. Regardless, starting off with probability theory just requires you to know how to work with sets. Such books will usually go through combinatorics too. You’ll need calculus later, though.

(when you deal with continuous distributions)

[upbeat vine]

@radiant crown Oh that's the french version. You can find the updated version here.The 1 is already addressed in this one and there is a big big collection of exercises with solutions at the back. Apologies for the inconvenience but it will be a lot helpful if you can go through this one.
https://link.springer.com/book/10.1007/978-0-387-70914-7

[radiant crown]

o that's nice

so I did all that archeology for nothing

[upbeat vine]

xD

[radiant crown]

but it wasn't wasted time because I had fun !

[upbeat vine]

I am sure you did 0.0

So thoughts about this book? @radiant crown

[radiant crown]

idk it's almost 3 am for me now and I'm going to go to sleep

[upbeat vine]

Oh ohkay! G.N.!

[neat basin]

Guys, what do you think of Knuth's Concrete Mathematics?

I just finished the first charpter and so far i'm liking it

[sage python]

It's huge, like trying to lift a physical coy of that book... is like lifting concrete

Lol jk idk it just wanna make that pun

[limpid gazelle]

I really liked the finite calculus part

[neat basin]

Ok nice, i'm gonna push trhough it 😄

[limpid gazelle]

It's super interesting and cool that you can find an expression for sums using manipulations similar to those on integrals

[neat basin]

Yes, one of my most favorite things! Sequences and Series is the part i most love in Analysis

[wise vine]

Wow the exercises on Knapp's Basic Algebra book are pretty good. I'm thinking of reading it with Artin since they cover the same topics

[gray gazelle]

@wise vine is it introductory?

[wise vine]

For me, it's pretty readable since there's lots of examples. And some exercises are quite like those guided exercises in Baby Rudin.

[warped wave]

Hey y'all I'm high school student and I'm very proficient in algebra and stuff, I'm really struggling to get geometry though.

Do you have any books for geometry?

I don't want basic ones, I need an intermediate one but not so much like topology or something

[calm crane]

nahhh screw high school geometry lol

Try like the first 3/4 chapters of munkres topology that should be enough point set topology, the other chapters could be learnt from better books or are quite boring

[warped wave]

I'll definitely check that out. Thanks a bunch.

Also @calm crane do you mean first chapters until 3-4 or 3/4 of total chapters?

[calm crane]

until like chapter 3 or 4

cant rmb which was uryssohn lemma lemme check

like first half of chapter 4 is kinda useful

then uryssohn tietze and tchynoff are like if you want to read then sure but you could just accept they are true lol

chapter 6 makes me sleep and i stopped XD

[gray gazelle]

Lang: Basic Mathematics. Worth the price?

[warped wave]

Alright, thanks for the help!

[gray gazelle]

Or overrated?

[calm crane]

l i b g e n
honestly my brain checks if it is cheaper to print than to buy, if so not worth

[north spire]

Do you have any books for geometry?
@warped wave Since you asked geometry, I liked Coordinate Geometry by Luther Eisenhart. I also liked Kiselev's two translated volumes of geometry. I read through a decent amount and they were good.

[gray gazelle]

I don't mind paying for books that I will use heavily, or will refer back to in the future (hence the question)

(I like physical editions)

[north spire]

Richard Silverman's calculus text also has over 400+ pages of geometry and all of it is very well written. So, you can get that too. It's a dover book so it's rather cheap too.

[calm crane]

Ah ic, maybe check it out on libgen and also see what others say

honestly trying to do things with a euclidian geometry point of view sucks

[warped wave]

@north spire Looks cool. I'll definitely check em out 🙂

Thanks !

[gray gazelle]

https://www.amazon.com/Modern-Calculus-Analytic-Geometry-Mathematics/dp/0486421007 this one?

[north spire]

Yea

[calm crane]

(btw if you want to go into topology directly munkres is super chill not much prereq, just know how to write proofs)

maybe better to do some real analysis first so things are motivated

yea

[gusty smelt]

i agree, do analysis before top

[gray gazelle]

Thanks this looks good.

[calm crane]

i think it's better to at least like be comfortable with first few chaps of rudin

[warped wave]

i downloaded bartle's analysis once and holy mother of god

it's like reading chinese

[calm crane]

i need a chinese math textbook

my chinese math sucks

[gusty smelt]

write one kek

[calm crane]

^(if anyone has good suggestions for some fun chinese math stuff do ping me)

im wondering, would you give rudin to someone who only done like high school calculus or is my math pedagogy completely messed up

[gusty smelt]

i think you need proofs

but yeah calc is enough

(a lot of hs calc ppl dk proofs rip)

[calm crane]

wait right high school doesnt teach people how to proof damnit

[north spire]

Thanks this looks good.
If you're familiar with the more popular calculus texts like spivak or apostol, then just know silverman is written on the same level as they are but with a good amount of emphasis on geometry too.

[warped wave]

I own Thomas'

I've been trying to finish the first 5 chapters but man procrastination is bad.

[calm crane]

same im reading like 10 books at once

[gusty smelt]

lol im reading like 3 at once

bad idea but w/e kek

[calm crane]

im doing like AM, occasionally poking at the random bits of jacobson i didnt read mainly second volume, lie groups stuff, miranda, silverman ec, some alg top and i decided to poke at hartshrone chap 2 exercises today

[warped wave]

wait right high school doesnt teach people how to proof damnit
@calm crane I'm actually very interested in this. Been wanting to start Velleman's but never had the discipline to actually finish it.

[gusty smelt]

im doing cat memes in reihl, comp anal in gamelin, ANT from marcus and AT from hatcher rn kek

[calm crane]

i do what i feel like doing on the day itself

[gusty smelt]

same

[gray gazelle]

@north spire I'm re-taking Calc 1 (it's been a long time) and then doing Calc 2, so by the semester after that it's a multivariable class with some analysis I think (Zorich). I want to get my feet wet a bit, while also addressing my geometry (our HS geometry is really rando and disconnected and that was like 15 years ago). I wish everything was Dover prices lol.

[gusty smelt]

hs geometry is literally useless

in like

[calm crane]

nah dont bother addressing hs geometry

actually

[gusty smelt]

everything

[calm crane]

dont

[gray gazelle]

okay

is there a better way to go about it?

[gusty smelt]

you wont ever need anything beyond "this is how triangles work"

[wooden sparrow]

hs geometry is literally useless
@gusty smelt
What really?

[gusty smelt]

in any of higher stem

[north spire]

@north spire I'm re-taking Calc 1 (it's been a long time) and then doing Calc 2, so by the semester after that it's a multivariable class with some analysis I think (Zorich). I want to get my feet wet a bit, while also addressing my geometry (our HS geometry is really rando and disconnected and that was like 15 years ago). I wish everything was Dover prices lol.
Zorich is just a bit too hard. Do silverman first, then zorich.

[calm crane]

heck all you need to know for constructible numbers which is the most useless concept is that circle and lines are equations of degree 2 and 1

hs geometry is just like a fun intro to hs students on how to proof things ig?

[gusty smelt]

no

no

its not a fun intro

[wooden sparrow]

???

[gusty smelt]

to proofs

[wooden sparrow]

@gusty smelt really?

[gray gazelle]

not fun, gosh, taught so poorly.

[gusty smelt]

its how to turn ppl off of proofs

[gray gazelle]

^

[calm crane]

rip lol

ic

[wooden sparrow]

Wait what??

I'm listening, please explain!

[gusty smelt]

"apply similarity to prove this routine thing that is obvious in 10 lines"

[calm crane]

oh

ok nvm

[wooden sparrow]

Ohh

[calm crane]

my thinking is super hs olympiad math line lol

[gusty smelt]

yeah lol i mean same i never actually took hs geo

my friends told me

and i was like wtf

[gray gazelle]

@north spire Man that scares me. Multi doesn't scare me in and of itself, but the analysis part does.

[gusty smelt]

they showed me their proofs and it was actually so bad

[north spire]

not fun, gosh, taught so poorly.
silverman is a nice introduction to proofs. Like I said, equivalent to spivak or apostol or courant. I didn't personally like apostol or spivak. I liked courant. You can have a look at all of them and see what you like.

[calm crane]

nah dont worry, unless the book is terrible it is quite easily understandable with doing enough exercises

[wooden sparrow]

@gusty smelt bro then what is a good geometry book?

[gusty smelt]

a good geometry book

[wooden sparrow]

Or course?

[gusty smelt]

is not doing geometry

[wooden sparrow]

@gusty smelt what?????

I didn't understand

[north spire]

haiz

[calm crane]

Is doing actually interesting geometry

im quite liking miranda

[gusty smelt]

hartshorne is a good geo book

[calm crane]

no it isnt lol its quite trash content

[wooden sparrow]

And what is interesting geometry?

[gusty smelt]

yeah true lol

[calm crane]

not introducing abelian cats in sheaves are a scam

[gusty smelt]

i returned hartshorne

after getting angry at it kek

[calm crane]

differential geometry, riemannian surfaces, alg/arith geo

lol

exercises are nice tho

supplement to AM

[wooden sparrow]

So Euclidean geometry is just feces?

[gusty smelt]

true, i will just pdf the exercises if i ever want lol

yes

[calm crane]

yup

Differential geometry is basically calculus analysis on nice looking shapes (with a bunch of added teasers)

[wooden sparrow]

Why? Wasn't it very logical though?

[calm crane]

as in?

tbh i learnt differential geometry for general relativity

[gray gazelle]

god damn geographic retail differences

[wooden sparrow]

I like how area of parallelogram is base× height because you can cut a triangle from one side and attach it on other to make a rectangle

[gusty smelt]

lol most ppl do

[calm crane]

ah

ic

[wooden sparrow]

Isn't it good though?

[gusty smelt]

i mean sure but like

[calm crane]

like yea it's great to be interested in math

[gusty smelt]

doesnt excuse all the other attrocities of the subject

[calm crane]

Like sometimes you just got to move on

and study more thingd

[wooden sparrow]

Attrocities?

[calm crane]

instead of trying to draw lines

and circles

[wooden sparrow]

Ohh....

[gusty smelt]

@gray gazelle pls show examples

of attrocities of euclidean geo

just pull out one of ur problems

[calm crane]

copies OMO problem

[gusty smelt]

kek

[wooden sparrow]

I'm unable to get you...

I'm not mathematically literate tbh

[calm crane]

so basically what we mean is that, it is better to move on, leave hs geometry behind

[gusty smelt]

yeah the topic is a. not useful and b. doesnt have meaningful depth

[calm crane]

Doing hs geometry doesn't make you more mathematically literate tbh, learning more math makes you like
know more math

[wooden sparrow]

Ohh

Well I liked the triangle inequality property applying in vectors though..

[calm crane]

Try out some proof books first
maybe also can play with a bit of elementary number theory too
then like idk just go to analysis, algebra, uh topology

[wooden sparrow]

Ohh

[calm crane]

you'll see that reappear don't worry

[wooden sparrow]

Okayy

[calm crane]

the reverse also appears in general relativity

[gray gazelle]

Book on Amazon US: $28 - Book on Amazon AU: $50 plus $20 shipping; Book on other retailers that ships: $80

[hasty eagle]

Klossowski:

[wooden sparrow]

Ohh, wait...
So in the end, you guys are saying " HS geometry is as trivial as numbers itself and doesn't do justice to the depth of mathematics" right?

[calm crane]

@ mods allow external emotes pls

eh

numbers have a lot more interesting property

but yea

number theory is literally the most uwuest field

[wooden sparrow]

Lol uwu?

[calm crane]

idk how solving polynomials with solutions being whole numbers is so amazing but it is

[warped wave]

There was like a 1-2 month period when I really loved number theory and I brought plain papers and my phone to do number theory problems everywhere

[wooden sparrow]

Like those oddly satisfying videos on YouTube?

[gusty smelt]

yes nt is uwu af

[calm crane]

Number theory just

appears everywhere

or that it is just what i choose to read

but anyways the math goes pretttty deep

[gray gazelle]

$85 on eBay for a $28 Dover edition. I give up on reading or even having eyes.

[hasty eagle]

Klossowski:

[north spire]

try book depository

but honestly, you can also just download the material off of libgen

[gray gazelle]

The worst. $80.

[wooden sparrow]

Okay so what I take from this conversation is " properties of numbers and their relationship with each other is more interesting than some uniform shapes that appear to us in our 3d world"

[gray gazelle]

This is true, however, I really dislike reading technical books as PDF.

[calm crane]

some shapes have nice properties, they just aren’t the ones you study in hs geometry

[wooden sparrow]

Ohh

[north spire]

The worst. $80.
haha it's a tradeoff you gotta make. It was $20 when i ordered it

[hasty eagle]

Abhijeet Vats:

[north spire]

but that was a long while back

[wooden sparrow]

So should I skip HS geometry then?

[calm crane]

yes

[wooden sparrow]

I'm at prealg tbh

[calm crane]

oh

uh

[gray gazelle]

Yeah that's true - so like a 30-40 trade off is a fair price imo. But Australia is like, geo-cucked into paying 2x RRP for everything.

[calm crane]

maybe just like

[north spire]

So should I skip HS geometry then?
Here's a suggestion

[calm crane]

learn a bit lol

dont skip so far

[north spire]

ask that same question on math stackexchange

[wooden sparrow]

Lol okayy

[north spire]

see what you get

[wooden sparrow]

I'm giggling

Okay I'm laughing now

[calm crane]

if you're at prealg a bit of hs geometry will help

than like completely ditching it

[north spire]

Yeah that's true - so like a 30-40 trade off is a fair price imo. But Australia is like, geo-cucked into paying 2x RRP for everything.
Tbh silverman's book is very very comprehensive. It covers basically everything

[wooden sparrow]

Guys posting any book reviews?

[north spire]

It's not gonna be enough for my own analysis course but, well, my course is different from the cookie cutters offered in other places

[calm crane]

jacobson - uwu
am - ikea
hartshrone - crappy intro

sure

[north spire]

So should I skip HS geometry then?
I was serious by the way

ask this question on stackexchange

[wooden sparrow]

Like writing an essay about why the book you specified is recommended.

Others can upvote it and ask the owner to pin that message :)

[gusty smelt]

artin better fight me

[gray gazelle]

Pung

[north spire]

see what you get as a response from others

[gray gazelle]

@gusty smelt

[wooden sparrow]

So it's easy to find

[gray gazelle]

@gusty smelt

@gusty smelt

[tranquil ocean]

There are literally thousands of relevant books lmao

[calm crane]

It may be better to learn a bit of hs geometry

[wooden sparrow]

Ohh

Okay okay

[gray gazelle]

What was your questiob

[calm crane]

like genuinely at least know like what similar triangles stuff are what circles are etc and some nice properties

[gray gazelle]

Just a random geo problem?

[gusty smelt]

yes i wanted an example of attrocities of geo

[gray gazelle]

Oh I found an edition on Abebooks, $9 used plus $9 shipping. Might need to wait 2 months though.

[hasty eagle]

Klossowski:

[calm crane]

once you're toying with multivariable calc you can screw hs geometry

[wooden sparrow]

@gray gazelle it's a different question, asking if you guys are reviewing books here and asking the admin to pin the reviews to find out what the book is about,

Because different people like different writing styles, they can probably find theirs here without going through a lot of books? Maybe?

[north spire]

Oh I found an edition on Abebooks, $9 used plus $9 shipping. Might need to wait 2 months though.
@gray gazelle I'd say it's worth the wait

[hasty eagle]

Abhijeet Vats:

[north spire]

But read the pdf and see if it's what you want to get

[main flax]

multivar calc toyed me

[gusty smelt]

ill toy you

[calm crane]

there was a great algebra books review somewhere

[main flax]

go ahead john

[calm crane]

someone pin dani's algebra books review list?

it's quite nice imo

[gusty smelt]

yeah i agree

someone give me honorable so i can pin

obv

[calm crane]

same

[gusty smelt]

jk @steel viper

[calm crane]

give me admin to pin

[steel viper]

nyanpfp

[gusty smelt]

pin damis algebra book review

like yesterday or smth

lol

https://discordapp.com/channels/268882317391429632/716264872018706443/718931426346795099

@steel viper

[steel viper]

lang uwu

[gray gazelle]

@north spire cheers, I looked through it and it looked ... good? But really I don't know where to go to prep for that class so I'm willing to take the gamble. I'm slowly etching away at Hammack's Book of Proof; and Alcock's How to Think About Analysis was recommended to me as prep material to get more comfortable with proofs.

John I gave you a geo problem, now do it

@gusty smelt

[north spire]

There are multiple books available at that level. There are people on math stackexchange who've reviewed all of them. Search those reviews and see what you want.

[gusty smelt]

stop spamming the books channel frucht

[tranquil ocean]

@gray gazelle stop

[gray gazelle]

@gusty smelt you literally asked me here

[calm crane]

serious question here - where to find books in non - (english/french)
and please no manga intro to xxx they hurt my eye

[scarlet crescent]

France

[north spire]

Like I said, I liked silverman and courant. Nothing wrong with spivak or apostol. It's just that i didn't particularly like their writing style @gray gazelle

[gray gazelle]

non english = french

[gusty smelt]

nonfrench she means

i assume lol

[steel viper]

hebrew translation of EGA

[scarlet crescent]

Japan

[steel viper]

[gray gazelle]

Is there one?

Probably not I guess

[steel viper]

no

afaik no lol

[scarlet crescent]

Prolly, I actually have no clue... Im just waiting until ya'll are done to ask a question...

[steel viper]

unless its not a book question

in which case dont ask

[scarlet crescent]

Ok then...

[gray gazelle]

אשנב למתמטיקה: קבוצות; פעולות בינאריות; פונקציות

Best book

[steel viper]

this is like the 3rd time you've sent that

[gray gazelle]

It's the only hebrew I know

[steel viper]

no one cares

[gray gazelle]

[scarlet crescent]

What would be a good post-undergrad book to start off with? I finished college, but dont want to stop learning math... I also dont want to get my masters yet with all this COVID business. I want to find some books that would be good to read while I am between college and grad school.

[tranquil ocean]

What things have you learned already?

[calm crane]

and what are you interested in

[scarlet crescent]

Everything that my college taught, I ended with 180+ credits. Real analysis, Abstract algebra, Complex variable, and topology are the highest-level I have learned.... but I did a study in Measure theory as well. I am intrested in pretty much everything I just listed, and i'm sure there is more that I would like but dont know yet.

[tranquil ocean]

Well, that's good. Those are the baseline topics that you need to learn anything else. You could go basically where ever you want

If you want to learn more abstract algebra, you can go on to learn galois theory or commutative algebra

Or if you want more topology, you can learn algebraic topology, or differential topology

If you want to learn more analysis things, you can learn some functional analysis

[scarlet crescent]

Looking for something that would prepare me for my first few years in a graduate program.... not just stuff I like though. Galois would be good, I heard that is pretty hard so that might be good to review before school.

[tranquil ocean]

You could also learn number theory things

[calm crane]

basic galois isnt too bad until fancy stuff appear tbh

[scarlet crescent]

Yeah, I took NT in college too... it wasnt my favorite. Mostly just playing with integers, which isnt that exciting to me.

[tranquil ocean]

I'm going to teach a galois theory course to some people here that you could join in on

[sudden kindle]

Really? @tranquil ocean

[tranquil ocean]

NT starts to not really look like that as you get more advanced

[scarlet crescent]

Im guessing in english? that would be cool.

[tranquil ocean]

@sudden kindle yes bc poco said he was bored

[gray gazelle]

Any reccs on absolute beginner NT? 😉

[scarlet crescent]

Depending on the format and what times you do it, I would love that.

[sudden kindle]

This is not for the highschool program?

[gray gazelle]

I've heard a friendly introduction to NT is good

By Silverman

But if you know algebra I guess Ireland Rosen is good too

[tranquil ocean]

its vaguely related to it, but not for the high school program no

[sudden kindle]

@tranquil ocean I wanna teach something too

[gusty smelt]

wait zoph how much do you plan to teach in it btw

[tranquil ocean]

what does that even mean

[gray gazelle]

@tranquil ocean is it over Zoom?

[gusty smelt]

like topics wise

[tranquil ocean]

#chill

[main flax]

oh hey zoph papa is bacc

[upbeat vine]

Oh that's the french version. You can find the updated version here.The 1 is already addressed in this one and there is a big big collection of exercises with solutions at the back. Apologies for the inconvenience but it will be a lot helpful if you can go through this one.
https://link.springer.com/book/10.1007/978-0-387-70914-7
Would be very helpful if someone can review this FA book.

[sage python]

My functional analysis class used it. I didn't read it too much but what I've read of it seemed pretty good

(We deviated a fair amount because it kinda presents functional analysis as a prelude to PDEs, and doesn't talk about certain things our prof felt was important, like spectral measures)

[worldly gale]

Hello, I'm new to the server but I'm looking for book recommendations, I live in the US, I completed HighSchool but did not care for mathematics during then but now I do. I wish to learn high school math again, after I've gotten that down and back into my head I'm planning to learn pre-calc and calc. Right now it is algebra-1 so any books, videos or resources for those things I would much appreciate. I'm not too smart but I just got out of a program for software engineering and self study Computer science so anything there I am happy to assist with.

[flint forge]

khan academy

is my universal rec

for these situations

[upbeat vine]

My functional analysis class used it. I didn't read it too much but what I've read of it seemed pretty good
@sage python Oh I see so for the presentation of topics that are already present in the book, I suppose it is a good resource.

[ionic wren]

Also 3b1b obviously

[radiant crown]

I had a quick look at it and it seems alright @upbeat vine

[sage python]

Yeah the exposition feels clean

[upbeat vine]

Thank you all!

[radiant crown]

and the exercise are those which should've been their own book

[frigid comet]

Brezis is a great book. I wish I learned of its existence earlier than I actually did.

[worldly gale]

Thank you Max

[willow relic]

Is Landau’s foundations of analysis a good place to start to get ahead for a university course?

in analysis

[midnight skiff]

I used baby Rudin for that. Is it for real analysis or calculus?

[willow relic]

“An introduction to rigorous analytic proofs involving properties of real numbers, continuity, differentiation, integration, and infinite sequences and series.”

Looks like it gets at part of it but doesn’t cover some of the stuff that’s more like the calculus I know and love

[flint forge]

Rudin is the most common book

[willow relic]

Isn’t baby rudin not that great for self study?

[flint forge]

for that level

Uh

I guess people feel this way yes

[willow relic]

I was taking a look at Tao’s real analysis too

Presently I have no assistance but I figured since my job is closed until further notice and I don’t go back until august I may as well get something done

Regardless landau’s book has been interesting so far. So I’ll read it either way

[gray gazelle]

taos analysis is nice imo, it covers a lot of important stuff before starting what people usually call analysis

[midnight skiff]

I felt that baby rubin was great because it was my first introduction to ”rigorous” mathematics and gave me a lot of maturity. I didn’t have to re read anything about the subject later thus saving time. Baby rudin dictated the path/topics but I used other supplemental books also, of course. I guess it depends on what your goal is.

[warped wave]

what is baby rudi

is there adult rudin or something?

are they different versions?

[marble rock]

yea

baby rudin is principles of mathematical analysis

papa rudin is real and complex analysis

and i think uncle or like god ruddin is functional analysis

not sure

[gray gazelle]

(grandpa rudin)

[gray gazelle]

What should a reader be familiar and comfortable with before starting Tao's book on analysis?

Assuming said reader hasn't been doing math comps since they were 11

[hearty steppe]

@flint forge your on point with the idea of better working with sets, yea

@flint forge which book by munkres exactly?

[calm crane]

munkres topology probably

[flint forge]

point set topology

[digital pier]

Hey does anyone know a good book to learn about Duality and Tensor products (Linear Algebra)? I cannot seem to wrap my head around it, and would love some suggestions.

[valid moth]

@digital pier math3ma's blog

just look it up

[digital pier]

@valid moth Bless you 🙂 Truly helpful. That really cleared things up!
Link to what I read if anyone would like to read about it: https://www.math3ma.com/blog/the-tensor-product-demystified

[valid moth]

np

[weak fossil]

I'm a returning student and I need to learn algebra 1 until precalc. Khan Academy seems to explain well but I feel like the practice problems are really doing nothing for me and I just go through concepts without a deep understanding. Would you be able to recommend a book? I'm currently reading Lang's Basic Mathematics but it's insanely hard and I'd like to find something in between KA and LBM to feel like I'm stretching but not far enough to make me give up

[gray gazelle]

@digital pier When was tensor products mysterious in the first place?

@weak fossil What about Lang's BM is hard?

[weak fossil]

I'm not a native english speaker and i sometimes have to reread sentences a few times. It might be just because I took a week long break

and my brain was just feeling uncomfortable about studying

but compared to KA it's really really raw

[gray gazelle]

ya its kinda rigorous tbh

there is that one jewish math book that is easy algebra 1 and 2 i will look for it

[weak fossil]

thank you!

[valid moth]

jacobson?

[gray gazelle]

gelfand

https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773

[valid moth]

wait you mean hs algebra

i have no idea then

[weak fossil]

Thank you so much!

[gray gazelle]

Also there is this mammoth https://www.stitz-zeager.com/

Too much material, but if you can create your own list of needed topics you can get through it

[weak fossil]

Thank you thank you

[gray gazelle]

https://www.stitz-zeager.com/ch_0_links.pdf

And if you need the ultra basics, from integers all the way to complex numbers it's this

Is Abstract Algebra by G. Lee good?

Or should I read Serge Lang's?

[tranquil ocean]

#book-recommendations message

[gray gazelle]

Thanks zoph

[hearty steppe]

damn it Namington i liked your old pfp more ;\

[weary nymph]

I'm looking for books with lots of exercises about verifying trig identities and simplifying trig expressions, and maybe some insight into some of the approaches used.

Does anyone know about a book like that?

[steel viper]

huh

what is jewish math

what's gentile math??

[quick hornet]

sloth you've been living a lie

[steel viper]

??? have I been doing jew math??

[quick hornet]

look deep inside yourself

and youll see

[steel viper]

[ionic wren]

@steel viper Jewish maths?

Is that in the Zohar

[ionic wren]

Nathan Jacobson

Iirc he’s Jewish

Sloth

Do you think that’s an accurate thought

[hearty steppe]

@flint forge is there any pre-reqs for Munkre's Point-Set Topology book?

I was under the assumption that I should learn a bit of analysis and abstract algebra before going into topology?

[civic carbon]

I don't think Munkres requires much

there is some algebraic topology near the end, but you probably won't make it that far on a first read.

[calm crane]

I think you should only just care about the first 3-5 chapters of munkres lol

[lone flower]

For the first few chapters, having some analysis background can be helpful for motivating certain concepts (continuity, the reals as an important example of a topological/metric space, etc), as well as the mathematical maturity that comes with it, but neither analysis or algebra are technically required.

When you'd need algebra would be when you'd move to something like Hatcher's Algebraic Topology anyway.

[calm crane]

imo you should have some analysis backgrounr

[lone flower]

I agree. Having the background from analysis will help motivate a lot of (otherwise) seemingly arbitrary definitions.

[civic carbon]

My analysis class certainly defined these notions the way Munkres does, but I can see how that could be helpful. It's all connected closely.

[lone flower]

For self-study, I highly recommend Abbott's Understanding Analysis to get that analysis background.

[hearty steppe]

yea im working on the analysis background haha

i am pure math newb still at this point

[valid moth]

oh no

[civic carbon]

That book is excellent, and certainly my favorite to teach analysis from. But I think real analysis' importance in mathematics is greatly overemphasized.

[valid moth]

we have to read some "successful" college essays for my english class

but this essay

i cannot read it

i am in pure pain

http://s3.documentcloud.org/documents/691589/describe-something.pdf

@steel viper amazing essay here

[civic carbon]

(but it is beautiful, and of rich historical interest)

[hearty steppe]

yea I have Abbot and a few others to check out. Actually. I am enjoying Rudin rn. After I finish Rudin first chapter, I am gona go over Tao and Pugh, possibly also Schroder to go over Reals again

and probly re-read that Rudin chapter

[valid moth]

oops this is #book-recommendations
will move to #discussion

[lone flower]

Baby Rudin is an impressively terse textbook.

[sage python]

Baby Rudin is why I became a math major lol

[gusty smelt]

honestly rudin is great, i love the book

[sage python]

Metric spaces from chapter 2 was just like

Yo this shit is dank af

[gusty smelt]

i dont think it is unfair at all

[slender sphinx]

I shall start reading it soon; all this hype is exciting

[gusty smelt]

and the pacing was eprfect for me

[frigid comet]

Baby rudin is why I became an analyst, although perhaps that would have happened regardless.

[slender sphinx]

[sage python]

Half that and half my summer linear algebra/graph theory class

[civic carbon]

I became a geometer because someone tricked me by talking about the Weil conjectures

[lone flower]

At least your Riemann hypothesis is proved.

😄

[civic carbon]

haha yes, I even usually work with curves where BSD is proved, I'm spoiled.

[frigid comet]

lol

[lone flower]

I became a number theorist because I liked how you could (with a bit of a stretch of terminology) say that an integer is squarefree (or, equivalently, two integers are coprime) with a probability of 6/pi^2

[gusty smelt]

oh i like

that

[frigid comet]

haha yep I can get that.

number theory was my first love

[civic carbon]

That is a beautiful fact. The video currently in my editing pipeline is about Euler's proof that zeta(2) = pi^2/6

[lone flower]

That one problem (and its generalizations) kept me hooked for a long time. 😄

[gusty smelt]

i know you can handwave that a bit by considering the product of probability at each prime

and then euler product

but theres also a nice way to do it rigorously

[lone flower]

Basically, if you take a uniform distribution on the integers in [-N, N], you get a probability that is 6/pi^2 + o(1) as N->infinity

[hearty steppe]

yea baby Rudin is actually really great

[limpid gazelle]

What's a good book on projective geometry?

[lone flower]

Actually, I wrote a blog post with a fun generalization to the Binomial distribution a couple of years ago. I don't know if links are allowed.

[limpid gazelle]

You can send it, it's fine

[gusty smelt]

oh interesting

yeah i would be interested to read it

[lone flower]

https://www.anothermathblog.com/?p=355

Don't judge the terrible header too much lol

[limpid gazelle]

I'm sorry for the ping again, but do you know a good intro book about projective geometry? @civic carbon

[civic carbon]

Shafarevich is my recommendation

[limpid gazelle]

Basic Algebraic Geometry?

[gusty smelt]

i liked that a number theorist

[valid moth]

@civic carbon What are the prereqs for arithmetic dynamics

[gusty smelt]

(lol i just noticed ANT)

[hearty steppe]

I def need to see what this number theory hype is all about at some point

lol

[civic carbon]

Arithmetic dynamics requires a lot of algebraic number theory and probably also some algebraic geometry?

You should not assume because I have an article about it that I know anything about it haha

I really, really recommend the book Fearless Symmetry as in introduction to modern number theory. It gives a really good lay of the land

[hearty steppe]

thats a pretty badass title

[slender sphinx]

Wait, whoever why are you learning proj geo

I thought you were gonna take a class on it next year

[hearty steppe]

whoever seems to learn whatever

xD

[sage python]

Lol my journey through number theory was interesting. Originally worked with friends through a few chapters of this book called "Number Theory for Beginners" by Weil

Do not confuse with "Basic Number Theory" lol

[lone flower]

hahaha

[civic carbon]

hahaha that book

[lone flower]

I love that title

[civic carbon]

isn't page one local class field theory?

[sage python]

Not quite but yeah overall it's a CFT book

But yeah that book was real nice, then in algebra we did Lagrange's sum of two squares and I was like 😍

[lone flower]

I remember seeing the Haar measure mentioned and thinking "Yeah, I don't know about that title"

[civic carbon]

the modular forms proof of the four square's theorem is beautiful.

[quick hornet]

the higher level of math you get, the more the title of textbooks demeans the reader

"elementary basics of arithmetic"

[sage python]

Silverman-Tate's "Rational Points on Elliptic Curves" was also real nice

Though I mostly looked at chapters 1/2/6

[civic carbon]

I think literally every book by Silverman is somewhere between great and essential

(at the graduate level, I should say)

(I'm not a huge fan of a Friendly Introduction to Number Theory or the crypto book)

[lone flower]

I was seen carrying Jacobson's Basic Algebra 1 to class one day by a physics major (they were on the floors below the math department). They asked me what I needed it for if I was a math major. 😄

[sage python]

Neukirch has notation that makes me wanna choke him but his writing is clean af

[quick hornet]

well he wrote that one mathematical cryptography book

oh you mentioned it

[sage python]

Though few exercises so I didn't really absorb it as well as I should've. Got some off Buncho but should've done more

Serre Course in Arithmetic is also super nice

Tbh I have an overall positive opinion of number theory books I've read

[lone flower]

The first time I had a "favorite" math textbook was when I read Apostol's Introduction to Analytic Number Theory.

[frigid comet]

I forgot how cute/small Serre was.

[civic carbon]

I think it is Andrews that proves all the elementary number theory theorems with combinatorics and I really like that.

[sage python]

Obv now I'm liking Goldfeld-Hundley

Like a lot

But yeah funny how different areas have really different like, overall levels of literature quality

[frigid comet]

We should have a GOAT textbook knockout bracket.

[sage python]

Like in analysis it's overall positive as well but not quite as good

Rudin I like chapters 1-7 but the multi is meh and measure theory is pointless. I saw one book by Igor Kriz and someone else which seems to me like the best possible replacement for Rudin

[valid moth]

i propose the monster book

also ega

[sage python]

Complex Analysis, I don't like anything I've seen completely, though I haven't looked enough at Gamelin or Narasimhan

[limpid gazelle]

Oh I've actually read the book fearless symmetry, and I also really like it

[sage python]

Measure theory, Bass honestly feels good but his later topics are kinda cursory. Functional analysis has great stuff

Linear algebra though, doesn't really have a great book out there that I know of

Topology same modulo possibly tom Dieck

Etc

[civic carbon]

I think Hocking and Young is remarkably good for topology

[gray gazelle]

@sage python were you trolling when you recommended spivak?

[lone flower]

The book that was used when I took Measure Theory was "Measure Theory and Integration" by Michael Taylor. It was okay.

[civic carbon]

but Munkres is my go to nowadays for topology

[sage python]

Oh when I said topology I meant algebraic lol

I don't believe point-set deserves a full book lol

@gray gazelle not at all

[civic carbon]

I'm trying to remember who even wrote my AT text. I know it is on my shelf somewhere haha

[lone flower]

Spivak's Calculus is good. Calculus on Manifolds is also decent if you're okay with relearning some notation later.

[civic carbon]

Yeah Calculus on Manifolds achieves S-tier status

[gray gazelle]

I'm doing spivak calc now cause ppl here recommended it

but

it seems like it was a troll suggestion

[sage python]

How? That book is amazing wtf

[pulsar aurora]

I prefer apostol's calculus myself. :p

[gray gazelle]

did you see my questions in the channel i pinged you in?

[lone flower]

Not really a troll suggestion, but definitely something that'll require more than Stewart.

[sage python]

Not really I mostly ignore pings in the questions channels because I assume it's gomez

[gray gazelle]

he makes terrible, vague, definitions like
consider a > 0, then a in P instead of just a is positive or something

[frigid comet]

what an outrageous assumption @sage python

[gray gazelle]

his section on inequalities

is absolutely fucking incomprehensible

i dont know how anyone can read this and be like "ye good shit"

or worse

recommend it to someone else

[frigid comet]

the terrible vague defn is yours

"just a is positive or something"

[sage python]

I remember his inequalities being fine

He just says okay there's a set P

Which in your mind in positive numbers but because we're doing things axiomatically just stick with P

[frigid comet]

his is precisely the way you formalise the notion of positivity and order in fields.

[sage python]

The idea is that you define a set which will be the positive numbers

And you say what properties it has

And now a>b if a-b is in P

You can very easily prove that a\in P iff a-0=a is in P

So a > 0

Boom

[lone flower]

@gray gazelle Essentially, Spivak is formally covering the foundations in order to formally cover single variable calculus.

[pulsar aurora]

I always felt he was brief myself, but I went into spivak without ever touching calculus or proof based math. Felt like he expected me to know more than I did when using the book.

[quick hornet]

that seems like a weird thing to claim about

"a is positive" and "a is in the set of positive numbers" are synonyms

and writing "a in P" is the cleanest, shortest, least ugly way to express this

[gray gazelle]

I think you missed my point @sage python

[quick hornet]

what's your point, that it defines everything formally?

you need formal definitions to prove things, and spivak is a proof-based text

if you don't want to prove things, use another textbook

[vestal basalt]

^

[sudden kindle]

@frigid comet chess?

[vestal basalt]

Yes spivak and apostol has been discussed extensively @gray gazelle they are seen as the heavy hitters. The proof is what discourage people

Keep going you will get there I would partner up with you in completing spivak but I’m retaking some math classes I’ve signed up for I’m thinking of pursuing my masters in math

[sage python]

If you're upset that it's saying P instead of the phrase "positive numbers"

[quick hornet]

just sharpie it in

tbh

[gray gazelle]

It's nt that sorry im very busy atm

[sage python]

The point is that people are inclined when they see positive to use what comes to mind. The idea of that chapter is just

[civic carbon]

oh is there really an Apostol calculus book? I thought that was a joke about Apostol's Analytic Number Theory book

[gray gazelle]

but the way he explains stuff is so bad I can't do any proofs

[pulsar aurora]

@civic carbon. Yes

[gray gazelle]

I don't feel like i understand inequalities at all

[sage python]

You train yourself to just reference the axioms

[vestal basalt]

@civic carbon apostol calculus book is his most famous

Well I thought it was

[slender sphinx]

Apostol does have a calculus book (2 volumes)

[vestal basalt]

^

[sage python]

Just every step you do you say "Oh yeah this holds because blah is in P"

[quick hornet]

apostol's calculus textbook is in the same vein as spivak but takes an integrals-first approach

pedagogically im not sure whether this is better or worse

[sage python]

With proof-based calc you gotta put most of your existing intuition to the back

[sudden kindle]

Integrals first is good

I think apostol does integrals first

[quick hornet]

that's... what i just said

[lone flower]

It's chronologically accurate, I believe.

[pulsar aurora]

Photo proof

[quick hornet]

ehhhhhhhh

im not sure i buy the "historically accurate" argument

[civic carbon]

it still blwos my mind that the ancient Greek's knew how to find the area of a parabola

[pulsar aurora]

The difference between Apostol and Spivak, from what I have noted... Apostol really expands on the details starting with historic context regarding Archemedies Method of Exhaustion.

[quick hornet]

like yes the series definition of the riemann integral was frmalized before the limit definition of the derivative except

[vestal basalt]

@pulsar aurora I purchased a copy as well I wanted a hardcover but would’ve had to shell out more bucks

[quick hornet]

infinite series werent really formalized back then

since we didnt have a limit lmao

¯_(ツ)_/¯

liek there was a gap in the first definition that was just smoothed over

[sage python]

The way I see it with integration vs differentiation first

[civic carbon]

The fact that no one defined continuity until the 1800s makes the history really weird

[pulsar aurora]

Apostol doesn't even bring up Riemann and only sticks with Archimedes.

[vestal basalt]

@pulsar aurora yes I did the method of exhaustion I felt so badass afterwards

[quick hornet]

uh

what do you mean

he doesnt mention riemann sums?

the fuck

[pulsar aurora]

As far as I have seen, no

[sage python]

It's a trade-off of do you get your fancier functions right away vs do you get to compute with FTC right off the bat

[quick hornet]

how do you define an integral without riemann sums

[sage python]

Also I think differentiation is simpler to set up the apparatus for

[quick hornet]

like i doubt he's doing another definition since

no derivatives lol

so how is he defining it

[sage python]

And like it's fine to get stuff like exp and log as you go

[sudden kindle]

You define it as the lim sup of integrals of simple functions (step functions)

[civic carbon]

I mean, how do you define the area of a circle? Clearly you can define area without Riemann sums

[pulsar aurora]

@quick hornet He defines it the same way, from what I can tell, but he refers back to the Method of Exhaustion/summations. He just never brings up Reimann.

[sage python]

Hence I prefer differentiation first

[quick hornet]

ah okay

he just never uses the phrase

"riemann integral"?

[pulsar aurora]

Yep

[vestal basalt]

@sage python does spivak do differentiation first?

[pulsar aurora]

Never comes up

[quick hornet]

¯_(ツ)_/¯

sure

[sage python]

Yeah

[vestal basalt]

Interesting

[quick hornet]

well spivak spends like a hundred chapters waffling around with field axioms, set theory, how functions work, limits of sequences, limits of functions

[sage python]

Spivak does differentiation chapters 9-12 and integration 13

[quick hornet]

but yeah it does derivaatives before integrals

[sudden kindle]

I like the definition of integral using supremum, reinforces the supremum axiom of the reals, which tells you why were doing this over R

[quick hornet]

once you have derivatives, the theory for integrals basically falls from the sky

like its not "obvious" but

once its presented

its fairly clear

[sage python]

So sup of step functions is halfway to Lebesgue

[quick hornet]

the converse probably holds too but

[sage python]

Because you just replace step with simple

[quick hornet]

i dont have experience

with that

[sage python]

The question basically boils down to how it changes proving theorems in the Riemann setting

[civic carbon]

I think it is really difficult to appreciate how miraculous the fundamental theorem of calculus is until the second time you see it

[pulsar aurora]

As for my decision to stick to Apostol than Spivak is that I couldn't visualize a whole hell a lot of what Spivak was trying to explain. I preferred Apostol for that.

[sage python]

E.g. lower vs upper sums and uniform continuity on compact sets basically makes the proof that continuous => integrable fairly nice iirc

[civic carbon]

(I will say, though, I'm not a fan of proofs in calculus I courses)

[sage python]

Might be harder for step functions. Might not be idk

[vestal basalt]

@pulsar aurora what other books have you enjoyed

[sudden kindle]

Why

[vestal basalt]

@civic carbon I’m not either but I would like to learn about them more

[pulsar aurora]

@vestal basalt I'm early in my math exploration. I've tackled Concrete Mathematics, and got my ass handed to me. Honestly, Apostol is my real major math text I've studied as deeply as I have.

[civic carbon]

I'm a big fan of people learning about them after they understand Calc I. That's why analysis is a cool course.

[vestal basalt]

That is still impressive @pulsar aurora that book is hard

[pulsar aurora]

Which book? Apostol?

[vestal basalt]

Yes

[pulsar aurora]

I don't think it's that hard. Concrete mathematics is much harder, in my experience. 😛

[sage python]

I think the way my undergrad did it was optimal

You have a calc with proofs and without proofs

[vestal basalt]

^

[lone flower]

@pulsar aurora Concrete mathematics by Knuth et al?

[pulsar aurora]

Yes

[vestal basalt]

@pulsar aurora I’m frightened to look at concrete mathematics then

[sage python]

If you know you're gonna be a math major take calc with proofs. If you take calc without proofs and decide oh I like math, you do intro to proofs before you do analysis and linear algebra

And then analysis jumps right to R^n

[lone flower]

If you got through that, try Art of Computer Programming. Concrete mathematics is basically Chapter 0.

It's a lot of fun.

[vestal basalt]

Omg @lone flower are you serious

[civic carbon]

I think a flavor with and without makes sense.

[pulsar aurora]

@lone flower I'm well aware. I'm unsure how accessible his series is. I've not looked. Just had a friend who had a CS degree, and said if I could manage Concrete Mathematics, I'd be set for any math-stuff I had to do for CS.

[lone flower]

That's true. Art of Computer Programming is basically a mathematician's CS book.

Knuth makes his own computer architecture with assembly language (MMIX)

[civic carbon]

(Calc II is a very nice introduction to proofs via convergence tests, so I usually let that be the first place it comes up)

[pulsar aurora]

The ironic bit though... Concrete Mathematics made me more interested in Mathematics than CS. 😄

[sudden kindle]

Ravioli

[lone flower]

And you basically learn to program in a fake architecture. 😄

[vestal basalt]

@civic carbon it really is that is a very good point

[sage python]

I guess it depends a bit on how you organize it. Tbh I'm not sure if there's really a good way to teach a class that serves both as "Get the people who use math but don't need the theory to learn their tools efficiently" and "Get math majors set to go"

And here Calc II is extremely the former

[pulsar aurora]

That said, the more I learn analysis regarding calculus, the easier Concrete Mathematics becomes each time I tackle it.

[steel viper]

Idk I found calc without any proofs a bit boring

[vestal basalt]

I remember the amount of people struggling with series/sequences. I aced that section in calculus 2.

[steel viper]

The routineness of computation set in very quickly for me

and then dami made me do spivak and i died

and was reborn from the ashes as a mathcel

or I guess the beginnings of one

[sage python]

That's how it goes

[sudden kindle]

Sloth are you a mathcel

[steel viper]

probably at this point

[vestal basalt]

Mathcel?

[sage python]

Forged by the furnace is the way to go

[sudden kindle]

I'm sorry to hear that

[steel viper]

if i wasnt doing math maybe id have enough extra time to get a bf

[civic carbon]

I think Calc I is kind of like learning arithmetic. It is not going to be the most exciting. But then Calc II is beautiful and exciting and is when you see that lightbulb really start to shine in student's minds

[steel viper]

but ngl idk if id date any of the bxsci boys pty my standards arent that low

[sudden kindle]

Lul

[sage python]

But yeah Hegel that's because of your preferences, if someone doesn't care for math in itself but needs it for like, science/engineering/business or something

[steel viper]

yes my true sexuality is math

[lone flower]

Calculus 2 is my favorite in the sequence, but then again, I'm a sucker for infinite series. 😄

[vestal basalt]

@civic carbon I saw students upset they had to take it twice I remember a couple of guys told me they had to and I freaked out and thought I would fail. Somehow I got an A first try

[gray gazelle]

@civic carbon makes me want to learn it. Are you a teacher?

[sage python]

Then it's like alright, this delta epsilon shit is great but I will never use it in physics

[steel viper]

pty thats the nlab

[sage python]

And physics kinda quickly draws upon math I feel. Like if you never needed calculus until the last n weeks then like

[vestal basalt]

What is a mathcel? I want to be one.

[steel viper]

a haven of degeneracy

[sage python]

I'd just say okay it's fine to teach everyone proofs

[civic carbon]

Infinite series are great. Integral approximation is great. Integrating sec(x) is great. Taylor series and applications are great. It's just all awesome

[sage python]

But I feel like the physics department is gonna be like

[vestal basalt]

@civic carbon are you studying any books at the moment?

[steel viper]

there is a notion of negative degree htpy groups in top spec apparently

which is awful

[sage python]

uh fam why you giving these kids delta epsilon? We're trying to get them to integrate transcendental functions and you're taking years to define them because of this petty stuff

[lone flower]

Integrating sec(x) is awesome because you multiply by 1, but 1 came dressed up all fancy.

[sage python]

And then they start a "Calc for Physics Students" class and your department probably loses a bunch of funding and is sad

[civic carbon]

@vestal basalt I'm making a youtube series about all the math that goes into the Riemann Hypothesis and figuring out how to chop it all up to make sense. So I've been studying a lot of books looking at perspectives and then doing my own writing/programming/animation/etc.

@lone flower That ways is a lot of fun, but I prefer the partial fractions proof. I always do both.

[sage python]

Obv being a bit dramatic but that's kinda the problem, subjects that use math kinda use calculus right from the get go so the earlier you can teach them tools the better. And in particular Calc II in my school is like

[pulsar aurora]

I have a physics book that slowly adds calculus in each chapter where you slowly build up your mathematics knowledge as far as what the physics needs. Gives me a different perspective of calculus that allows me to fine-tune how I understand calculus

[sage python]

Alright now we're gonna teach you the tools to do things

[lone flower]

@civic carbon Please do expound on that partial fractions version, I'm intrigued.

[sage python]

While Calc I is more conceptual

Also if we're switching from talking about books to talking about math or pedagogy let's move it outta here

[civic carbon]

sec(x) = 1/cos(x) = cos(x)/cos^2(x) = cos(x)/(1-sin^2(x)) now do u-sub

[lone flower]

Oh yeah!

[gray gazelle]

@quick hornet @vestal basalt @sage python
The proofs are not what discourages me, he is extremely vage and lacks any ambition in explaining things
I mean you'd think 4 pages of inequalities would be enough but no, he goes on with simply doing showing properties and rarely any examples
And then when you get to exercises everything is wrong. He never says "distance from 0" or something either on absolute value

Because even though if he said a > 0, if a-b is in P
When you get to an exercise that's like (x+1)(x-2) > 0, how do you derive from the shitty definitions he gave you which X's are allowed?

[sage python]

So what's the worst math book you've ever read?

[gray gazelle]

spivak calculus so far

[frigid comet]

lmao

[lone flower]

I don't know about worst, but the book that came to mind was Hatcher's Algebraic Topology. I don't know why, I just didn't like it.

[quick hornet]

He never says "distance from 0" or something either on absolute value

is this a criticism

"he never uses a concept he hasn't defined"

gasp

[gray gazelle]

@quick hornet true, but then he expects you to know them in exercises

[quick hornet]

does he?

[vestal basalt]

@gray gazelle I’ve heard that many times about his book and apostol’s book when it comes to proofs they assume you know a certain amount of information

[civic carbon]

I've found all the homological algebra books to be thoroughly demoralizing.

[gray gazelle]

Up until now the problems i've had in math have been my mistakes cause I generally get the idea behind things

but spivak?

Nah

[lone flower]

@civic carbon What about Lang's section on Homological algebra? 🙂

[gray gazelle]

I feel like i have to memorize all the definitions

[sage python]

Uh

[pulsar aurora]

@vestal basalt I disagree on Apostol. He really gets into the grit of details, I think. At least, enough for me to keep up

[sage python]

Yes

That's how proof based math is now

[quick hornet]

(x+1)(x-2) > 0
i mean you'd use some of the properties he's presented to you

[civic carbon]

Lang is very anti my taste. But I definitely respect his work, it's just not for me.

[gray gazelle]

Like how kids keep a multiplication table in their pockets

[quick hornet]

surely you've seen properties like

ab is in P if a is in P and b is in P

[lone flower]

@civic carbon It was a joke, because Lang's Homological algebra section is just “Pick up a homological algebra book and prove all of the theorems yourself”

[sudden kindle]

You dont have to memorize definitions, you just need to understand them

[quick hornet]

so that gives you one thing you can check

when are both (x+1) and (x-2) in P

[gray gazelle]

@sudden kindle That's right, but how can I understand them if he never explains them?

[sage python]

Thinking

[quick hornet]

like okay

all of mathematical proof comes down to definitions

[sage python]

The thing about that part of Spivak is that he limits what you can use really hard

[quick hornet]

often these definitions are fairly formal and can seem a bit unmotivated

[frigid comet]

You have to think about them yourselves. Simply use the tools and concepts introduced alone, and be totally precise.

[sage python]

That's both a constraint and a hint

[quick hornet]

in order to get motivations

you need to think about them

[sage python]

Like I only have 3 axioms and what I've proven up to now

[quick hornet]

you need to prove properties about them

this is how all of mathematics works

its not unique to spivak

if you approach this like you approach you're high school math classes, then that's like approaching essay writing like your first grade short story assignments

"The cat is in a box."

"Okay that's great but I wanted a 4 paragraph essay on the causes of the American Civil War"

im sorry, that probably sounded a bit condescending in hindsight

my point is that

highe rmathematics is structured in a very problem-oriented format

you're not spoonfed everything; rather, you're expected to figure out the important properties yourself

now, you're heavily guided along

as this happens

like the book will straight-up tell you what you need to prove or whatever

but you're not just going to be spoonfed problems in a

predictable format

you cant solve this all by memorizing SOH CAH TOA

you dont memorize "when i deal with a problem where the car is going around the corner, i have to plug in the coefficient of friction here and the radius here and..."

[vestal basalt]

I completely agree no more hand holding when it comes to higher levels

[gray gazelle]

I guess I can't disagree that was an interesting take

But how do I tackle it then? I feel completely lost

Re-read the chapter over and over?

100 times until I understand?

[pulsar aurora]

Try another book. 😛

[quick hornet]

look at properties that seem relevant to the problem in hand

like youre given the question

(x+1)(x-2) > 0

okay so let's look what's happening here

we have two numbers

[pulsar aurora]

Sometimes, it may be too deep of a water, honestly. There are lower level books that help with the transition

[quick hornet]

multiplied together

and we're asking when it's positive

[vestal basalt]

I went to my professors for help in math but I stopped going to them because I didn’t want to continually bother them

[quick hornet]

so if we wanna figure out this behaviour

we want to ask ourselves

"how does 'multiplication' interact with 'positiveness'?"

focus in on results/statements/definitions related to that

I guarantee somewhere that there'll be a proposition that ab > 0 iff either (a > 0 and b > 0) or (a < 0 and b < 0)