Current Events > Hey math nerds, where should I go next?

(+) Yeah! (+)

I love math and want to do some self-study, because I'm weird like that. But trying to look up individual stuff on Wikipedia is a never-ending network of prerequisites. As in, I wanted to read up on such and such thing BUUUUUUUT it assumes I know other such thing, and that article assumes I know OTHER such thing, etc.

And dumb dumbs on the Internet answer existing questions that come up in Google results for "what to study next" regurgitate some true but useless crap about how it's not a linear progression thing and yadda yadda when I really just want to know SOME subject, ANY subject I can understand with my existing knowledge. As in, I really don't care which.

Anyway, if anybody has a suggestion for where to go after Cal 3, Diff EQ, Differential Geometry/Vector Calculus, Discrete Source, Prob and Statistics and/or Linear Algebra, that's be great.

Oh and I'm totally gonna go to the library and see if they have any textbooks there since I've finally decided Internet study is just plain terrible.

Go to the Library or internet and look for this book

https://gamefaqs.gamespot.com/a/forum/f/fbef3856.jpg

Kamen_Rider_Blade posted...

Go to the Library or internet and look for this book

https://gamefaqs.gamespot.com/a/forum/f/fbef3856.jpg

Noted. This sounds like a class I can get behind.

RetuenOfDevsman posted...

Anyway, if anybody has a suggestion for where to go after Cal 3, Diff EQ, Differential Geometry/Vector Calculus, Discrete Source, Prob and Statistics and/or Linear Algebra, that's be great.

At this point you might want to dip your toes into some higher level math stuff like Real/Complex Analysis, Topology, and Abstract Algebra.

Will you remember every proof or will any of the stuff in those classes be everyday useful like stats? No. But they are pretty much foundational courses for higher math and once you go through them it makes learning more advanced applied math topics (partial differential equations, stochastic processes, fourier analysis, mathematical statistics, etc.) much easier to get through.

Edit: To put it another way, you were getting frustrated by the fact that trying to learn more math on wiki keeps sending down endless rabbit holes. Basically those rabbit holes will mostly all lead to the subjects I mentioned. Its why they are all required for math phd programs.

Shadow_Don posted...

At this point you might want to dip your toes into some higher level math stuff like Real/Complex Analysis, Topology, and Abstract Algebra.

Will you remember every proof or will any of the stuff in those classes be everyday useful like stats? No. But they are pretty much foundational courses for higher math and once you go through them it makes learning more advanced applied math topics (partial differential equations, stochastic processes, fourier analysis, mathematical statistics, etc.) much easier to get through.

Edit: To put it another way, you were getting frustrated by the fact that trying to learn more math on wiki keeps sending down endless rabbit holes. Basically those rabbit holes will mostly all lead to the subjects I mentioned. Its why they are all required for math phd programs.

Aha! Thanks!

Game Theory was one of my favorite college math courses.

I read the topic title as meth nerds.

Have you tried learning Binary/Octal/Hexadecimal math yet?

Ivany2008 posted...

Have you tried learning Binary/Octal/Hexadecimal math yet?

Yeah, it was part of my degree. While I took almost enough math classes for a math major, that was mostly from electives. Because I'd honestly rather take complicated math classes than, like, art appreciation or whatever normal people take.

My actual degree was electrical/computer engineering. So I'm quite familiar with base 2/8/16 math.

How much proof writing is in your math background? College was a while ago for me, but I don't think there was much in the way of proofs in the courses you listed (IIRC, the math requirements generally needed for an engineering degree). There may have been some in the first Linear Algebra course you encounter as an undergrad, but things get proof heavy once you get past that, like the Analysis, Topology and Abstract Algebra (groups / rings / fields) mentioned.

Set Theory is as well, though I don't think that built on too much that came before it. It kind of brings things full circle, because one of the things you'll do with the concepts there is define numbers and the basic arithmetic operations. Also involved one of the most interesting things I learned in college - you prove that some infinities are bigger than others. (There are just as many integers as there are rational numbers, but there are explicitly more real numbers.)

I second real analysis and then complex analysis.

There's something great about getting beat down by real analysis, only to be brought back by how awesome complex analysis is

MLBloomy posted...

How much proof writing is in your math background? College was a while ago for me, but I don't think there was much in the way of proofs in the courses you listed (IIRC, the math requirements generally needed for an engineering degree). There may have been some in the first Linear Algebra course you encounter as an undergrad, but things get proof heavy once you get past that, like the Analysis, Topology and Abstract Algebra (groups / rings / fields) mentioned.

Set Theory is as well, though I don't think that built on too much that came before it. It kind of brings things full circle, because one of the things you'll do with the concepts there is define numbers and the basic arithmetic operations. Also involved one of the most interesting things I learned in college - you prove that some infinities are bigger than others. (There are just as many integers as there are rational numbers, but there are explicitly more real numbers.)

We wrote proofs a couple times in Geometry in high school. I kinda hated it NGL because if you perform the dastardly evil of just adding some stuff together off panel because, you know, there's a plus sign between them, instead of writing out "I added this shit together because DURRRRRRR" as a formal step in the proof, the teacher counts off.

I'm hoping that's just a teachers gonna teach thing though.

You should probably work though a book that introduces mathematical proofs. I think a new one that I have heard good things about is "The Tools of Mathematical Reasoning" by Lakins. It's expensive but I'm sure pdfs exist.

Another one that's actually open source is "Book of Proof" by Hammack: https://www.people.vcu.edu/~rhammack/BookOfProof/

Proofs in high school geometry are only somewhat representative of proofs you get in the rest of mathematics. However, there are not many more topics you can learn without having to know how to prove things.

lderivedx posted...

You should probably work though a book that introduces mathematical proofs. I think a new one that I have heard good things about is "The Tools of Mathematical Reasoning" by Lakins. It's expensive but I'm sure pdfs exist.

Another one that's actually open source is "Book of Proof" by Hammack: https://www.people.vcu.edu/~rhammack/BookOfProof/

Proofs in high school geometry are only somewhat representative of proofs you get in the rest of mathematics. However, there are not many more topics you can learn without having to know how to prove things.

Proofs are important, but it's also important to get a reasonably accurate intuition, especially for more abstract material. Unfortunately, it's easy for that intuition to get lost/buried in textbooks.

numberphile on wikipedia

Matrices.

EPR-radar posted...

Proofs are important, but it's also important to get a reasonably accurate intuition, especially for more abstract material. Unfortunately, it's easy for that intuition to get lost/buried in textbooks.

I think part of the issue is that it's really easy to read upper level mathematics and convince yourself you understand it if you never actually put it to the test.

Anteaterking posted...

I think part of the issue is that it's really easy to read upper level mathematics and convince yourself you understand it if you never actually put it to the test.

That was something that occurred to me. I think it'll be important to actually do the homework questions.

Actually, I learn a lot better when I start with the homework questions and refer back to the text only when I get stuck. It's one reason I was always so good at math; it's the only "main four" kind of subject that you can generally do this with. Though I also kicked ass at chemistry and physics for exactly this reason.

On the other hand, I seem to remember differential Geometry having relatively few such questions.

Anteaterking posted...

I think part of the issue is that it's really easy to read upper level mathematics and convince yourself you understand it if you never actually put it to the test.

IIRC you've got a math degree?

RetuenOfDevsman posted...

I think it'll be important to actually do the homework questions.

You can't really learn math without working problems. Reading results only gives you a general idea what's going on.

lderivedx posted...

IIRC you've got a math degree?

Yeah, PhD

Anteaterking posted...

Yeah, PhD

Oh nice. What research area?

Not to hijack your topic but the only math I did in HS was intermediate and then geometry. Now I'm signed up for college and my counselor assigned me a pre-calc class. How can I prepare for something like that? I haven't done math for years. I need the class for my major which is computer science.

lderivedx posted...

Oh nice. What research area?

Graph theory and then also semigroup things (I had a weird advising situation).

SomeLikeItHoth posted...

Not to hijack your topic but the only math I did in HS was intermediate and then geometry. Now I'm signed up for college and my counselor assigned me a pre-calc class. How can I prepare for something like that? I haven't done math for years. I need the class for my major which is computer science.

Its mostly just trig + algebra so review those

Anteaterking posted...

Yeah, PhD

Damn thats pretty badass. Didnt know there was a math PhD here.

SomeLikeItHoth posted...

Not to hijack your topic but the only math I did in HS was intermediate and then geometry. Now I'm signed up for college and my counselor assigned me a pre-calc class. How can I prepare for something like that? I haven't done math for years. I need the class for my major which is computer science.

Paul's online math notes (focus on algebra section) for something written with practice problems: https://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx

For lecture-style things, probably Khan Academy or Organic Chemistry Tutor on youtube.

Start working through topics and when you run into concepts you don't remember, identify them and go refresh, then continue.

Anteaterking posted...

Graph theory and then also semigroup things (I had a weird advising situation).

Graph theory is tough. I could never really pick up the intuition for the proofs. One of my committee members was a graph theorist and he decided to tell me there's a better way to do a proof during my defense.

lderivedx posted...

Graph theory is tough. I could never really pick up the intuition for the proofs. One of my committee members was a graph theorist and he decided to tell me there's a better way to do a proof during my defense.

My favorite part about combinatorics/graph theory is that there are a lot of proofs that boil down to random magic tricks. Obviously that can lead to its own frustration but in some ways it was more of a style than a specific discipline compared to e.g. my friends who did commutative algebra.

What was your research in?

Anteaterking posted...

My favorite part about combinatorics/graph theory is that there are a lot of proofs that boil down to random magic tricks. Obviously that can lead to its own frustration but in some ways it was more of a style than a specific discipline compared to e.g. my friends who did commutative algebra.

What was your research in?

I remember one graph theory proof in class used Cauchy-Schwarz out of seemingly nowhere. Although, I think graph theory has some of the neatest tricks/ideas. The discharge method is my favorite thing to tell people who have enough math experience to get the gist.

I do enumerative combinatorics.

lderivedx posted...

I remember one graph theory proof in class used Cauchy-Schwarz out of seemingly nowhere. Although, I think graph theory has some of the neatest tricks/ideas. The discharge method is my favorite thing to tell people who have enough math experience to get the gist.

I do enumerative combinatorics.

This just reminds me of how much I struggled with gfology. Maybe I should take a look at it again.

I Iderivedx do you also have a PhD?

Anteaterking posted...

This just reminds me of how much I struggled with gfology. Maybe I should take a look at it again.

I liked that book. Wilf has a good sense of humor. The footnote for the "x(d/dx)log operation" is something like "the best motivation for the above program is the fact that it works"

Shadow_Don posted...

I Iderivedx do you also have a PhD?

yeah

Anteaterking posted...

I think part of the issue is that it's really easy to read upper level mathematics and convince yourself you understand it if you never actually put it to the test.

Oh yes. Doing the problems, or equivalent levels of engagement with the material, is essential.

I sometime amuse myself thinking about the lies someone would come up with by giving prompts to a chatbot seeking an explanation of Galois theory. The commonest incantations would probably be repeated fairly accurately, but I'm sure the slightest attempt to poke and prod at corner cases etc. would give farcical results.

Shadow_Don posted...

At this point you might want to dip your toes into some higher level math stuff like Real/Complex Analysis, Topology, and Abstract Algebra.

Will you remember every proof or will any of the stuff in those classes be everyday useful like stats? No. But they are pretty much foundational courses for higher math and once you go through them it makes learning more advanced applied math topics (partial differential equations, stochastic processes, fourier analysis, mathematical statistics, etc.) much easier to get through.

Edit: To put it another way, you were getting frustrated by the fact that trying to learn more math on wiki keeps sending down endless rabbit holes. Basically those rabbit holes will mostly all lead to the subjects I mentioned. Its why they are all required for math phd programs.

True but also look into combinatorics, mathematical logic, infinitesimals and cover theory.

[LFAQs-redacted-quote]


Its possible if you already took calc 1 but calc 2 is generally considered as having a leap on difficulty from calc 1.

So you probably really want to do a thorough review of calc 1.

https://m.youtube.com/watch?v=WnDj_wZgrOc&pp=ygUUQ2FsYyAyIG1hdGggc29yY2Vyb3I%3D

Shadow_Don posted...

Its possible if you already took calc 1 but calc 2 is generally considered as having a leap on difficulty from calc 1.

So you probably really want to do a thorough review of calc 1.

https://m.youtube.com/watch?v=WnDj_wZgrOc&pp=ygUUQ2FsYyAyIG1hdGggc29yY2Vyb3I%3D

I dunno if I'm just weird or what but Cal 2 to me was much harder than Cal 3.

More than anything, I think it was because in Cal 2, you study stuff like series that have a bunch of formulas that you're better off just memorizing than trying to think about analytically, while in Cal 3 you just put two or three integral symbols in front of everything. Sure, figuring out what to integrate in each nesting is the trick, but you can at least reason it out.

RetuenOfDevsman posted...

I dunno if I'm just weird or what but Cal 2 to me was much harder than Cal 3.

That's pretty normal. I've not watched the posted video but IMO one thing that makes Cal 2 much more difficult for students is that you've got to choose between different techniques (either convergence tests or integration methods). In calculus 1 and earlier classes you usually don't have such decisions (optimization in calc 1 an exception here).

Isn't differential geometry a doctorate level class?

Anyways, here's today's college level math problem:

2024^p - 2023^p is divisible by 2027 for some prime number p. Deduce the value of p.

RetuenOfDevsman posted...

Anyway, if anybody has a suggestion for where to go after Cal 3, Diff EQ, Differential Geometry/Vector Calculus, Discrete Source, Prob and Statistics and/or Linear Algebra, that's be great.

Complex Analysis

FLOUR posted...

Isn't differential geometry a doctorate level class?

Anyways, here's today's college level math problem:

2024^p - 2023^p is divisible by 2027 for some prime number p. Deduce the value of p.

It was certainly the highest numbered class I took. A jump from 3600 for Prob and Stars to 5260 for Diff Geo. But I had all the prereqs so why the hell not. It was a much more enjoyable class then the fine arts elective I didn't manage to weasel my way out of with Professor Nutcase.

Then again, I never found that how high the number is made any difference in terms of how hard the class was.

...

Actually, now that I think about it, they were all pretty easy. The one that I came closest to failing was Prob and Stats, and that was because the prof graded homework. Funny story, I actually went from looking at an F to an A in seconds in that class because the prof said in class one day "I noticed some of your averages on exams were higher than homework, so since the purpose of homework is to learn, I decided..."

Here he makes direct eye contact with me.

"If your grade would be lowered by your homework scores, I won't count them."

Id highly recommend Linear Algebra before Differential Equations FWIW as Linear Algebra teaches matrix math youll need for it.

As for after, I recommend Abstract Algenra and studying Abelian groups etc. I hated Analysis / Advanced Calculus,

FLOUR posted...

Anyways, here's today's college level math problem:

2024^p - 2023^p is divisible by 2027 for some prime number p. Deduce the value of p.

I did actually do a tiny bit of number theory self study a few months ago, just based on a short tutorial I found on the Internet. If I remember correctly...

2024^p mod 2027 should be equivalent (I forget the real term) to 2023^p mod 2027...

There was something about exponents... I remember they used this to prove the thing about adding the digits of a number divisible by three. 2024 mod 2027 is equivalent to -3 mod 2027, 2023 mod 2027 is equivalent to -4 mod 2027... I think that theorem only applied when the mod of the base was 1. A couple quick examples in my head confirms at least that it doesn't work in general in the way I think I remember it.

Eh, there's gotta be a different thing with exponents. Let's see, I think you can multiply as follows:

ab mod n = ((a mod n)(b mod n)) mod n, I believe. So that would mean a^b mod n = ((a mod n)^b) mod n

(-3)^n mod 2027 = (-4)^n mod 2027. Since n is always the same, we can ignore the negative:

3^n mod 2027 = 4^n mod 2027. Since 4^n is always going to be greater than 3^n when n is positive, we know 3^n has to be larger than 2027.

ceil(log3(2027)) = 7

...

Wait a minute... Is it 13? Have I been overthinking this? Lol, 4^13 mod 2027 won't even Google.

Iunno, that was my best shot at it.

Pretty sure p=1013.

Since 2027 is prime, the multiplicative group of integers modulo 2027 has order 2026, which has prime factorization 2*1013. If such a p exists, this means the cyclic subgroup generated by the element 2024^p must divide 2026, and so p must be either 2 or 1013. We can check that 2 doesn't work: 2024^2 - 2023^2 = 2024+2023 = 4047, which is not a multiple of 2027. Thus if p exists it must be 1013.

Not sure how I'd verify this p by hand but WolframAlpha confirms it.