/math/ general Comrade 21-12-20 05:19:20 No. 338 [Last 50 Posts]
All good communists study math.
Comrade 21-12-20 05:19:22 No. 347
>>345 Yeah I could say a lot about that. Discrete or Intro to Proofs is where math starts to get serious at the undergraduate level, you could say the same about analysis, but the techniques used in analysis are what you learn in discreet. What's really cool about that class is that it opens up all sorts of areas of mathematics to you, where as before discrete the track is pretty linear: pre-algebra, then algebra, then pre-calculus, then calculus and so on. That track continues through Vector Calculus, Ordinary Differential Equations, Nonlinear Dynamical Systems, Partial Differential Equations and so on, but there are a ton of other really interesting classes that tend to only have Discrete as a prerequisite (Linear Algebra, Abstract Algebra, Higher Geometry and so on). If you know some basic calculus, discrete also equips you with everything you really need to start digging into analysis, which is where math really gets interesting imo.
It will probably be pretty different from math classes you've taken in the past, and you may find it particularly challenging even if you are generally good at math. The inverse may be true as well though, I know students who had a really hard time in math generally and then began to excel once they got to proofs. I can't say if this will be the case for your class, but in the experience the workload tends to be quite a bit lighter in terms of homework, the tests don't have so many problems on them, and there isn't nearly as much computation involved as in something like calculus or algebra.
Unless you had a really good geometry teacher or have a background in computer science or philosophy it is likely that some of the ideas presented in the class will be pretty novel to you. In earlier math classes there are shortcuts to doing well in the class if you don't understand the content: memorizing formula, grinding problems etc. There is no way to fake proofs. My first advice is to have a system for taking good notes. If you have a hard time taking notes, look into finding a way to record lectures or get notes provided. Look over the notes regularly, use a highlighter to mark certain definitions that come up regularly, and rewrite new notes as you go that takes the most important concepts from your previous notes.
Next is to read the textbook, and utilize office hours if they are available. Between the book and your professor, all the answers to questions you might have are yours for the taking. This really goes for all math class at the college level, and is something I wish I had taken to heart when I was in school.
Third is to write things out in plain English rather than only in mathematical notation! People don't tend to think in mathematical notation, so it's hard to understand the math you are doing if that's all you're writing down. This also goes for math in general, but people tend to look at me funny when I recommend this for classes before discrete or proofs.
In terms of preparation the best thing you can do is to start reading the book ahead of time. The hardest concept is going to be different for different people, but when I took the class I remember a lot of people struggling with proof by contradiction, so you might want to look into that ahead of time. I've attached a textbook (second attatchment) which is actually a book on analysis, but it also contains an introduction to proofs which is the part of the class I am familiar with (I think discrete also includes some other content but I could be wrong about that).
Good luck and if you run into challenges feel free to post questions in this thread and I'll do my best to help you out.
(The first attachment is a book on real analysis, just for anyone interested)
Comrade 21-12-20 05:19:22 No. 350
>>347 Aww, thank you very much for your helpful and thorough response! I appreciate it a lot. I'll take a look at proof by contradiction. I have looked at the textbook we're using (125967651X) and it really does look quite interesting to me even though I'm a person who usually struggles with math. (My biggest problem is just remembering all the little rules and gotchyas about computation, which as you said, I can usually get better at by just grinding problems)
But I'm somewhat excited to start this class because it looks very different than Calculus and I am already familiar with a lot of the logic operations/concepts at the start of the book. And as you said, it's a prereq to Linear Algebra which I very much want to learn for my programming endeavors. It's just a bummer I have to take this class online cause of the virus… I learn much better in person.
That's a really good tip that I hadn't thought of doing. Thank you again for your help anon, I'll ask you if I run into any troubles with the course.
Comrade 21-12-20 05:19:22 No. 352
>>345 >>347 Discrete math is completely different from "math" that's taught in grade school, aka continuous math. It's not particularly difficult, it's just not taught as much as it should be. It's thought of as challenging mostly because we're taught to think about math according to continuous math and we don't get the fundamentals of discrete math the same way (lots of practice at a young age). There's a number of reasons the discrete side of math isn't taught to kids, and they have more to do with the system education exists within than the nature of the math itself. It's harder for a teacher to grade proofs, for one. There's less investment into understanding how to effectively teach the content for another.
And then of course discrete math is actually much more broadly applicable than (continuous) math and (bourgeois) primary education is kind of designed to make you hate learning and think it's useless to your life… Some of the fields of discrete math are in use by everyone constantly, without realizing it (or realizing that the formal math could be very helpful), such as:
<Logic
<Probability
<Set Theory (Venn diagrams etc)
<Game Theory (strategy)
<Graph Theory (series of two-way relationships)
>>350 Good luck. Hopefully you have a good teacher. That can make a huge difference with this material.
Comrade 21-12-20 05:19:23 No. 359
>>350 If you have a hard time remembering gotchas then I think you might find discrete to be a relief! There is hardly any of that, there's just a lot of principles and definitions you have to understand. The rest of this post is kind of theoretical. I had hoped this thread would get into some of the questions
>>352 brought up, but don't feel like you have to know all the stuff I'm about to launch into to do well in or understand discrete, and also keep in mind I might be wrong about some of it.
>>352 I want to add something to what you pointed and my wording might sound like I'm contradicting you but I don't think that anything you said is wrong.
The math that you learn as a little kid is discrete math, and this is by necessity. Integers is the obvious example, when you learn to count "1, 2, 3… " you are learning a discrete system of numbers. When you learn basic operations as a kid you don't learn them over the real numbers, but rather the integers. Once you get to division though you are usually introduced to fractions which is really where everything goes awry in terms of there being any sense to contemporary western math pedagogy. I always suspected there was something very spooky about them as a kid and there absolutely is. Fractions are great, don't get me wrong, but they are introduced WAYYYYY before the student has the context to understand what the hell their teacher is on about. You need to know how operators can be defined and how they work in the abstract before it makes any sense to use them to construct a new number system. By new number system I don't mean fractions, I mean the real numbers. And by that I don't mean that Pythagoras was actually right about irrational numbers, but rather that we teach fractions AS THOUGH Pythagoras was spot on all along.
We introduce integers, then operations, then once we get to division we introduce fractions, and then we say "oh so here's another way to write fractions: decimals! and btw all these new numbers in between the integers are real numbers" When really they are not equivalent. To show how they aren't equivalent though you need square roots which aren't usually introduced until later.
All this is to say that fractions should really be introduced alongside square roots. My bias is towards a geometric explanation for both of them, partially because it is visually elegant and intuitive, but also because this way the actual weight of what is being taught would be made apparent. This would be a radical shift though since it would mean delaying teaching fractions quite significantly which would create all sorts of problems if it wasn't adopted universally. I'm not sure if we can even make the changes to the way math is taught that we need to under capitalism because of how important it is to capital that new workers know a very specific set of operations. Anyway, I'm off on a tangent which I guess is fine. I attached a pdf that I think is the vague beginnings of a book I want to write on the topic of math pedagogy. Let me know what you thing if you're interested.
We tend to equate logic and proofs with discrete math because at the college level they are introduced around the same time, but they aren't really the same thing (even though the actual classes are pretty much the same). Proofs are important to continuous math as well and if you study analysis you'll see what I mean. It's just that for some reason we delay proofs until grade what, 14? They really ought to be being taught to elementary schoolers.
Man I'm tired. I dunno if this post is even going to make sense when I read it tomorrow but I hope someone gets something out of it.
Comrade 21-12-20 05:19:23 No. 360
>>350 >programming endeavors Cool! I am really bad at programming. I've tried to read the SICP but I always get stuck on like the third chapter or something.
What kind of stuff do you want to make computers do?
Comrade 21-12-20 05:19:25 No. 378
>>371 Yeah people tend to have an easier time with one or the other. I think part of it has to do with different types of intelligence: verbal reasoning vs processing speed and working memory. I think there's ways to improve in either area though.
I am sorta skeptical of psychometrics but I think it's more relevant for mathematical ability than for a lot of other fields. I have slow processing speed and poor working memory to the point that it constitutes a learning disability. I grew up in a very working class intellectual type household (no TV but lots of books) and I've read every day for most of my life. I always thought I wanted to be a journalist so math was the last thing I was expecting to major in.
When I got to proofs I had a much easier time. Discrete was an easy A, Linear Algebra was hard but I seemed to get it more quickly than most. Vector calculus was brutal for me though and I do envy people with strong computation skills. I think both are necessary in advanced math, so if you only have one you gotta try to develop the other, or find someone you can work closely with who's has the inverse strengths and weaknesses. That's sorta what I did. My best friend when I was in college was a math physics double major. He is wicked smart and can do computations way faster than I can, he struggled a bit with proofs though which I thought where a breeze. The times that I was doing really well in school was when we took the same classes together and would do all the homework together.
I'm glad that there are some people in this this thread with both areas of strength. With our powers combined we shall crush the enemy!
Comrade 21-12-20 05:19:25 No. 379
>>375 Tell me something about combinatorics! I don't know very much about it, although one of the professors I worked closely with had done her PhD in that field I think.
>>363 Tell me something about Topology! I was really excited about taking that class but I dropped out before I got to it. I know that there's some trippy stuff related to set theory that you learn in topology, clopen sets and so on.
Comrade 21-12-20 05:19:27 No. 390
>>382 Libgen is your friend when it comes to "buying" yourself a textbook hehe. I would say that you understand a theorem when you can both apply and prove it.
So take the Pythagorean theorem for example. Being able to apply it means that you can find the length of the hypotenuse of any right triangle given the lengths of the legs. It also means that you can find the lengths of one of the legs given the length of the leg and the hypotenuse.
If you can do those things you are half way to understanding it. The other half is being able to show that the theorem is true for ALL right triangles. There are dozens of proofs of this theorem but my favorite goes like this:
With any right triangle you can take 4 copies and arrange them into a square like in pic related.
There are two ways to find the total area of the construction. So for any right triangle the following equation will hold true…
The left side is the obvious way to find the total area, and the right side you get by using the formula for the area of a triangle, and then adding the yellow part in the middle (c)^2
I'll leave the rest of the proof to you, but it's pretty simple from here. Just simplify the equation and you'll derive the Pythagorean theorem.
>>381 I think intro topology is a 300 level class at most schools so about as advanced as NODEs or Higher Geometry or maybe Abstract Algebra, but less advanced then analysis which is what you usually have to take to finish and undergraduate degree in math.
I don't think there's any reason you couldn't start learning topology once you know discrete / proofs, which really isn't all that much content. I didn't take it but I had friends that did and it sounded challenging. Lots of set theory involved.
Comrade 21-12-20 05:19:28 No. 399
>>396 Absolutely disgusting
Kill it with fire
Comrade 21-12-20 05:19:29 No. 402
>>396 I mean, math pedagogy has been fucked since the burning of the library of Alexandria, but the common core is particularly trash.
>>399 Based.
Comrade 21-12-20 05:19:29 No. 404
>>390 A lot of people have been sleeping on z-lib, it's what I use for a lot of my books, though they can usually also be found on libgen.
>>344 >Euclidean geometry please no
Personally I like combinatorics and I've been trying to learn topology and group theory, which are pretty cool. Can anyone explain what the fuck category theory is? I can't seem to get a grasp on it.
Comrade 21-12-20 05:19:37 No. 458
>>408 I've been trying to get through this pdf but it just feel really dense to me.
>>423 Oh, really? Then it's better than the impression I had of the American school system. It seemed like unless you were rich or talented they wouldn't even make the slightest effort to help you catch up. Depends on the school and teachers as well of course. Aggressive standardization however isn't good for learning.
Comrade 21-12-20 05:19:40 No. 477
>>459 Thank you anon! I will check that out.
>>461 Nice! I dropped out before taking analysis or starting my thesis. I did get to do a teeny bit of original research which was fun. It was in geometry but such a undeveloped area of geometry that I can't really say more without doxing myself.
Please share pdfs, the more the better! An annotated index would be cool too, but if you just want to do a dump I can try to index them.
>>462 I'm not a good person to ask about that really, I tend to get overzealous, go all in, get burnt out quickly, take time to lick my wounds, and then jump back in. This pattern doesn't get you very far in academic settings which is why I dropped out. I am interested in research mathematics rather than something more lucrative like actuarial science so it didn't seem like there was much point staying in school and racking up debt. I would rather just work on my research independently so I have complete autonomy.
Comrade 21-12-20 05:19:40 No. 478
>>404 >category theory it's the cool kid of mathematics that arose from the study of algebraic topology and geometry, which supplants the notion of sets with objects called "categories" containing _collections_ of objects (without set structure, so no russel's paradox etc.) and morphisms (basically arrows between objects). the morphisms represent different things based on what the category is, but as an example: the category *Set* of sets has all sets a objects, and functions as morphisms. This generalizes set theoretic mathematics in a trivial way, but also exposes more structure when we think about set-theoretic properties instead as being relations on the morphisms and stuff like that.
This was a loose summary, but I strongly recommend checking out one of the many books available on the subject at an introductory level (you should probably have abstract algebra to understand examples, depending on the book choice). I learned from Mac Lane, but it leans heavily on mathematical maturity. I've heard good things about awodey at an introductory level, especially if you are interested in learning about category theory's deep relations to type theory.
Comrade 21-12-20 05:19:41 No. 482
>>480 >>481 This conversation is a little over my head, but if you're patient I think I can wrap my head around it. I learned about groups in the context of geometry, specifically in defining "geometries". I haven't studied groups much in the abstract.
Is a group a type of category?
Comrade 21-12-20 05:19:41 No. 485
>>482 a group is both a type of category and the object of another. We may talk about *Grp*, the category of groups as objects and homomorphisms as morphisms. But a group can also be described as a category with a single object and morphisms corresponding to group automorphisms.
>>483 In the category *Grp*, the objects are groups and defined on sets. But one can take the properties expressed in this category and _categorify_ them to talk about more abstract objects. Ultimately, however, a thing with all the properties of a group is a group.
Comrade 21-12-20 05:19:41 No. 487
>>484 That makes a lot more sense, thank you anon!
>>485 I thought I knew what morphisms are but I'm realizing I kinda don't… book suggestion?
Comrade 21-12-20 05:19:41 No. 488
>>487 You're probably confusing morphisms with group homomorphisms. In *Grp*, the morphisms are group homomorphisms.
I mentioned Awodey earlier, but another good one is Fong & Spivak's "Seven Sketches in Compositionality", for which Baez has produced an online course with accompanying videos
https://www.azimuthproject.org/azimuth/show/Applied+Category+Theory+Course Comrade 21-12-20 05:19:42 No. 491
>>490 From my biased perspective as a geometer and classicist I'd say the best place to start is pdf related.
Another place you could start is by reading Godel, Escher, Bach: an Eternal Golden Braid by Douglas R Hofstadter. It's not exactly a math book, but it takes you on a sort of idiosyncratic tour of some questions in math, logic, computer science and art. It may give you an idea of what exactly you are interested in.
There's some math related YouTube channels that are pretty fun too. My favorites are Vihart and 3Blue1Brown. There's also Numberphile which is kinda hit or miss imo.
https://www.youtube.com/watch?v=X1E7I7_r3Cw https://www.youtube.com/watch?v=1SMmc9gQmHQ Comrade 21-12-20 05:19:42 No. 493
>>478 Thanks for the recommendation, I'll try to check it out. Can you explain how its structure differs from sets to avoid Russel's paradox?
>>490 If you want an intro to some undergrad topics I can recommend
https://venhance.github.io/napkin/Napkin.pdf It requires some mathematical maturity and understanding of proof-based mathematics, but it motivates its stuff a lot better than most DTP textbooks. It also doesn't require too much highschool knowledge afaik, I haven't read all of it. I'm afraid I don't know any good highschool level materials to recommend.
>>491 Is Elements good? I feel like classical geometry isn't the greatest intro to math, because it's so narrow, but then again I'm biased against classical geometry.
Piggybacking on the 3B1B recommendation I can recommend mathologer.
https://www.youtube.com/channel/UC1_uAIS3r8Vu6JjXWvastJg They don't have as much highschool stuff as 3B1B, but it's pretty interesting.
Comrade 21-12-20 05:19:42 No. 494
>>493 >Is Elements good? It's great in terms of demonstrating what can be accomplished with logic. It's also good in the sense that you don't need any outside knowledge to understand it, although some parts can be challenging. It's important to note that you need to be highly motivated to get through it because there is zero motivation provided by the text.
What do you find narrow about classical geometry?
Comrade 21-12-20 05:19:43 No. 496
>>477 Here have an analysis book and a cheat sheet I made. Don't enjoy it that much but one of my best courses grade-wise.
>>477 In my mind you would have to stay in school to get qualifications to then become a researcher, are you doing it in your free time or do you make any money off it?
Comrade 21-12-20 05:19:43 No. 499
>>495 Gotcha. I guess I kinda like that it is arbitrary, it feels pure in a way.
>>496 Thanks for the pdf! Analysis is pretty cool I would have liked to take that class. I posted the elementary analysis book that I use here
>>347 but I should really get around to studying real (which is what that textbook looks like at first glance)
I don't know how one would make money off of research unless you mean like a stipend from an institution. To me research just means making novel discoveries in the field. Anything you could submit to a journal is research in my eyes.
Comrade 21-12-20 05:19:47 No. 526
>>525 Gotcha. I currently work as a math tutor, mostly for college students but some high school / middle school students as well. Picking up some other kinds of freelance work would be cool though.
What modelling strategies / books / software do you recommend? I did a bit of modeling in school. Took part in the COMAP MCM which was fun, although the problem my team ended up doing kinda sucked. We where given a spreadsheet with several million data points so we spent most of the time just trying to get MATlab to parse it correctly.
Comrade 21-12-20 05:19:47 No. 529
>>526 Nice, I don't think I'll ever get my eyes back to their pre-hundreds-of-hours-starting-at-MATLAB's-retina-burning-design state.
I'd recommend learning all you can about numerical ODEs and PDEs. MATLAB is quite useful useful I won't lie, although if you do anything big you'd probably want to learn C/C++. This book is amazingly comprehensive in its field of finite difference schemes in a musical context, a masterpiece. You could also look at finite element models too.
Whoops file too large, here you go
https://ccrma.stanford.edu/~bilbao/booktop/ the format sucks but couldn't find it quickly
Comrade 21-12-20 05:19:47 No. 530
>>529 I took ODEs in school as well as a class on Nonlinear Dynamical Systems and Chaos. We used mathematica rather than MATlab in that class.
Nowadays I use GNUoctave for everything. I think I should probably learn R as well though.
Thanks for the link, I will check that out :D
Comrade 21-12-20 05:19:47 No. 531
>>529 Ah and yeah C++ was my first programming language. Learned it when I was 13. I haven't used it in a while but I still have a compiler and a pretty good reference book on it (C++ Primer Plus)
Why do you recommend C++ for math modeling?
Comrade 21-12-20 05:19:47 No. 533
>>532 > Matlab in terminal say sike right now…
oh god why did no one tell me???
Comrade 21-12-20 05:19:48 No. 536
GNU Octave is more or less a MATlab clone and it runs in the terminal by default. It is a lot easier to do
>>535 sort of thing with which is why it's the main Mathematical Programming Language I use.
Out of curiosity, what operating system do you use? This might be a little off topic but it would be cool to hear a few perspectives on how people configure their workspaces for mathematical modeling.
Comrade 21-12-20 05:19:48 No. 543
>>535 >>536 This is life changing, thanks a lot. Mathematicians aren't usually very tech savvy so I guess this never came up with my colleagues.
I currently use
macOS for a few reasons. There's no reason to use any software for modelling in particular. The most important stuff is done on paper.
My workspace configuration though is minimal with a tiling wm (yabai), usually need a vim w/latex up, a pdf of something I'm looking at, probably a web browser too and another desktop with code running. So clean and organised is a necessity for me (also because autism).
>>531 >>542 I recommend it simply because when you get to a certain level on complexity you need the fastest thing you can get. If you are a hobbyist then C/C++ is the only way you're going to get this kind of speed (I've heard of some models taking days on super-computing university grade machines) especially within musical applications, you want to get as close to real time as possible. There are easier ways to convert from MATLAB to C if I remember correctly though.
Comrade 21-12-20 05:19:49 No. 547
Thoughts on Wade's vs. Tao's analysis books? I'm trying to pick one to self-study with.
>>536 >Out of curiosity, what operating system do you use? Archtard here. Might switch to gentoo later.
>>543 >Mathematicians aren't usually very tech savvy It could just be a demographics thing, but most of the people I know who are going into pure math are good at programming. Though of course, they aren't mathematicians yet.
Out of curiosity, what reasons do you have for using macOs? I don't want to come off as elitist or anything, but the best one I can think of is ease of use.
Comrade 21-12-20 05:19:49 No. 549
>>547 I also use Arch. Never tried Gentoo, but I'm considering trying either that or LFS.
Interesting. Could be that I went to a liberal arts school but not many of the other students in my department where into computers. I think I met like a total of 2 other math majors who knew their way around a terminal, but it's possible people where hiding their power levels idk…
I'm not very good at programming, and I think of that as being a different skill from being "tech savvy"
Comrade 21-12-20 05:19:49 No. 551
>>547 I used Wade's, so I can recommend that. The cheat sheet I posted
>>498 is based off Wade.
>>547 Students around me either were pure math, and that's where their interests lie solely, or were very applied and did a lot of programming. Of course pure mathematicians would be naturally better at programming than say, a painting student, but I haven't encountered many that actively enjoyed it. If they did they wouldn't usually be 'into' technology, so would be like 'what is this hackerman terminal shit', similar to what
>>549 said.
Regarding MacOS, I got the laptop before I was really into free/open source software, which is unfortunate, the reason I don't change is because currently I have a good workflow and while still in the academic year would be dangerous to swap. Also I make music, the propitiatory software for it is just so much better. When summer comes I think I'll make a partition and have a music-boot, would help with focus as well.
Comrade 21-12-20 05:19:57 No. 618
>>606 I don't either.
I know: BASH, C++, Python, Lisp, Mathematica, and MATlab/Octave
I want to learn: R and that's really the only one
Is there anything I am missing?
Comrade 21-12-20 05:20:12 No. 749
>>748 Looks interesting! I have never used one but perhaps I will try one out. I have always just written proofs by hand but just reading about them online they look like they could be useful as software.
In terms of the question "is there any value in this level of formalization of proofs" I think the answer is yes. Formalization, while sometimes annoying, generally helps eliminate ambiguity. I don't want to establish conventions about how proofs need to be expressed, because that would be limiting, but specificity in logic and notation is always good.
Have you used this software before?
Comrade 21-12-20 05:20:31 No. 922
>>905 I don't see how that's a bad thing.
>>886 Based.
Comrade 21-12-20 05:20:31 No. 927
I've always been more of an Arts guy, but I recall only one instance in my life where I enjoyed maths.
Comrade 21-12-20 05:20:34 No. 955
>>928 Thanks! Like I said it is very rough and not even a fully formed idea. I wrote it in one sitting a couple months ago and haven't gone back to it. I might start working on it again since I am hitting a wall with my current project (critique of cockshott) and I try to rotate between works to stay interested.
Let me know what you think!
Comrade 21-12-20 05:20:34 No. 956
>>927 >I've always been more of an Arts guy, but I recall only one instance in my life where I enjoyed maths. In school we are encouraged to categorize ourselves this way. I am an arts guy. I am a math guy. I am a stem guy. It is easy to see why this is encouraged. It serves the interests of capital, by categorizing students by aptitude, and shoehorning them down tracks where they will end up with specialized educations that allow them to form a particular skill set that will allow them to perform a particular function in the capitalist sphere of production. The renaissance man or woman is of little use to capital, and they may even constitute a threat to the status quo because they are more likely to grasp the totality of our situation.
>It was awesome and so quickly I began to "get" it. Sounds like a good teacher. For me the "getting it" moment came in college when I took proofs. I had already chosen math as a major, but I chose it in particular because it was the class I had always struggled the most in, and I had this bizarre idea that this meant I should pursue it. I've always been motivated by challenge and turned off when things come easily, which is kind of a double edged sword because it makes it hard to stick with things once they start to click. As soon as I had contributed to a novel discovery I dropped out. I don't think the paper was ever published. There where financial reasons to, and I was getting more involved in organizing, but that was definitly part of it. I realized that I could be a research mathematician if I wanted (which up until that point had been my goal) and I realized that is not what I wanted to do. I needed to work on a less achievable goal like ending capitalism. This might sound defeatist, and it absolutely is, but it's how I operate.
So many math teachers come across as straight up sadists. I didn't find this to be the case in college, but I hated most of my math teachers through grade school. My high school geometry teacher was really cool though.
I also can't stand the way that students are encouraged to think that getting it = being smart = having value. It's not just schools that do this, parents who want their kids to get good grades all the time carry just as much culpability.
Synesthesia? It's a real thing and it can have some really interesting implications for mathematical ability. I knew a woman with synesthesia who could recite digits of pi for like half an hour straight. Not slowly either. I don't know exactly what colors had to do with it, but she said it was everything.
Comrade 21-12-20 05:20:34 No. 957
I'll just semi hijack this thread…
Comrade 21-12-20 05:20:34 No. 963
>>957 Sure anon. I normally do that for work anyway but it's kinda dried up since we entered this recession and all. I've actually got a discord server that I set up as a sort of curated library of math related content, but I never got around to finishing it and I've barely given anyone the invite link.
[email protected](dot)ch
Comrade 21-12-20 05:20:43 No. 1052
>>1051 Huh
>bc 3+2 returns
<File 3+2 is unavailable
I'm on arch.
Comrade 21-12-20 05:21:12 No. 1307
>>341 Linear Algebra
Probability
Calculus
Linear Algebra General /LA/ Comrade 21-12-20 05:24:01 No. 2952
-Linear Algebra General-
Welcome to /LA/ comrades. In this thread we will work together more or less in line with the MIT OCW Linear Algebra syllabus.
The OCW page can be found here:
https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/index.htm On the OCW page you can find the calendar, recommended readings, lectures, and problem sets and exams. The lectures are done by Gilbert Strang who also wrote the recommended textbook. I think he is a very good instructor and I believe you should certainly give his lectures a watch if you are interested in learning more.
The Calendar is divided into 40 sessions which correspond to 40 assigned readings and lectures. There are 10 problem sets and 4 exams with all the solutions online. This thread will serve as a place to discuss lectures, readings, and, probably most usefully, ask other anons for help on problem sets or exams.
Comrade 21-12-20 05:24:01 No. 2953
-Useful Resources-
Instead of using Matlab (proprietary and therefore costly software for non students) you may choose to use GNU Octave, an open source (and free as in gratis) alternative.
https://www.gnu.org/software/octave/ The video lectures can be found here:
https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/ The problem sets can be found here:
https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/assignments/ And the exams can be found here:
https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/exams/ Comrade 21-12-20 05:24:03 No. 2967
Here's a question I have for the reading assigned for session 1. Strang introduces the Schwarz inequality which is the proposition that, "Whatever the angle, this dot product of v/||v|| with w/||w|| never exceeds one." Could other Anons explain the proof behind this? Specifically Proof 1 on the related wikipedia page (
https://en.wikipedia.org/wiki/Cauchy–Schwarz_inequality ) I'm a bit hung up on the notation. For instance what does u subscript v indicate?
Comrade 21-12-20 05:24:03 No. 2968
>>2967 So I will just go through the proof from basics.
We seek to prove that
|<u,v>| =< |u||v|
If we have either vector=0 then this holds trivially.
Now to prove the case where v is non-zero we set
z=u-u_v
Where u_v = <u,v>/<v,v> * v
It is then showed that z is such that it is orthogonal to v (by the property of dot products).
So we now have 3 vectors. u and v, which are any nontrivial vector, and z which is orthogonal to v. From our definition of z, we have from a simple rearrangement
u = u_v + z
Basically we can then square the norms of everything on both sides from the Pythagorean theorem which is pic related. Basically we technically have a sum on both sides. So we take the norm and square it (left hand side of the equation), then from the right hand side (usually we call it RHS) we can just expand it out.
Using various properties of dot products we can rearrange this to get Cauchy-Schwarz. Do let me know if you'd like me to clarify any of this. :)
So we can see that u subscript v is just a vector that is useful for us to get u in a form that is nice to play with. In a more intuitive sense u_v is the magnitude of the dot product between u and v, in the direction of v.
Comrade 21-12-20 05:24:18 No. 3074
>>2922 many many courses on youtube.
if you want something more structured and gamified start with KhanAcademy.org
The calculus playlist is great.
Comrade 21-12-20 05:26:55 No. 4742
>>4741 You do it every day and at some unknown point it just becomes an extension of your mind. The essence can only come from solving problems, usually with other people to get different perspectives, and creating a set of skills which you can call upon to solve problems.
I remember in middle school trying to desperately remember which axis is the x and the y, and how linear equations work. However these attempts are actually counterproductive IMO. At a certain points of maths you let go of trying to 'get' it, and just move on, then after solving future problems it may come to you.
Anonymous 18-04-21 08:58:59 No. 5466
>>4741 You just do it. You want to develop a certain set of skills which are useful in the sense that, when solving problems you always ask yourself
– why am I doing this?
– what happens without this assumption?
– what happens with this assumption?
– why does method work for a finite case but an infinite?
– where have I seen similar structures as this?
and so on. It's a skill really, took me 3 years to get somewhat good at it.
Anonymous 23-04-21 10:01:47 No. 5516
>>338 It worries how different countries teach maths and how different their methods are
IN my country we had BODMAS but apparently everybody else learn PEDMAS
Anonymous 04-09-21 17:16:07 No. 6980
>>338 Good thread OP.
How do you anons stop making basic blunders in exams? I fuck up the basic math and make stupid mistakes, and then I end up ruining the whole question. My grades are suffering because of it, even though I have a good grasp of the advanced stuff.
Anonymous 10-10-21 14:07:06 No. 8052
>>8050 yeah I screwed up. he didnt develop calculous but differential calculous.
I was thinking about Leibniz, mb
Anonymous 10-10-21 18:31:07 No. 8058
>>6980 >I fuck up the basic math and make stupid mistakes, and then I end up ruining the whole question Iktfb
And to be honest I never really figured out how to deal with it, I just lucked out and managed to not make enough blunders to pass. But something that kind of helped was really slowing down and doing nearly every calculation by hand, writing down all the steps for the question instead of just diving in, etc.
Anonymous 25-10-21 03:58:41 No. 8506
Number theory, counting, comparisons, ratios, addition subtraction multiplication division, exponentiation, negative exponents, rooting, factoring, GCF, LCF, primes, prime factoring, distributive property, adding like terms, fractions, order of operations, variables, isolating variables, polynomial equations, graphs, discrete and continuous values, difference, rise over run, y = mx + b, quadrants, points, lines, polygons, circles, higher dimensional topology, angles, soh cah toa, algebra with sohcahtoa, trig functions-1, perimeter, area, surface area, volume, units, unit conversion, si notation, significant digits, measurements, estimation, rounding, patterns, circumference, diameter, radius, y = x^2, y = a(x - p)2 + q, y = ax^2 + bx + c, x = 0, x int, y int, x = (-b +- sqr(b^2 - 4ac)) / 2a, a^2 + b^2 + c, itg(x -> 1/0)*dif(x), transformation, translation/reflection, rotation, radians, unit circle sin = y val & cos = x val, Pi * rad = 180deg, analysis, sets, coordinates, groups, mapping, matrix operations are about the operations on coordinates from vectors
Anonymous 19-12-21 15:14:18 No. 8985
>>390 pink = all - 4 green = red + blue
>>404 The rocket pigs version:
https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/ >>493 >differs from sets to avoid Russel's paradox? The standard solution is based on von Neumann's work and predates category theory. Cantor's naive sets are renamed to classes. Classes are partitioned into sets and proper classes. Sets are those classes that can be built up using ZFC, which removes unrestricted comprehensions. You can no longer take "all X [with P]", you have to take "all X from S [with P]" where S is already a set. The question becomes whether the class of all sets is a set. The resolution to Russell's paradox is to provide the negative answer by becoming the proof that the class of all sets is a proper class rather than a set.
Anonymous 31-12-21 19:38:37 No. 9138
Anyone here reading more "historical" mathematical texts? I've started reading through Bourbaki's theory of sets, just to kill some time until I go to university. Its a little incomprehensible to a brainlet like me, but its very nice to read about mathematical concepts that don't get much or any use these days.
>>7466 Amateur mathematicians aren't taken very seriously by academics, because a lot of quacks come up with "solutions" that are blatantly incorrect. All of the open problems require a lot of study to even understand, or exist in highly specific fields. If you want to study math for the hell of it, just pick up some books on the fields that interest you and read through them.
>>9033 Not sure if any website with exercises beyond high school level math exists. Getting a textbook in the topic you'd like to brush up on, and looking at the exercises in there should work. Skimming through the chapters might also be a good way to see if you've forgotten anything too.
Anonymous 31-12-21 20:23:50 No. 9141
>>9138 >a lot of quacks come up with "solutions" that are blatantly incorrect. All of the open problems require a lot of study to even understand Counter-example to your second claim: Collatz conjecture (also a great example
for your first claim).
Anonymous 02-01-22 14:48:05 No. 9168
>>9155 ?
First claim in the quote you've addressed is 'a lot of quacks come up with solutions that are blatantly incorrect'
Second claim is 'all of the open problems require a lot of study to understand'
Anonymous 23-01-22 00:16:46 No. 9480
>>9155 cute and aesthetically pleasing reaction image
i am monke
Anonymous 11-02-22 11:56:33 No. 9731
https://www.youtube.com/watch?v=v68zYyaEmEA does this video have to do with mathematics
is maths the way to solve wordle the best?
Anonymous 23-03-22 05:46:46 No. 10113
can we continue with you responding to
>>>/leftypol/873853 here?
Anonymous 23-03-22 05:50:04 No. 10114
>>10113 >https://www.wolframalpha.com/input?i=%7C%7B%7D%7C%3D Wolfram is interpreting it as the absolute value and not cardinality of sets. (See pic 1)
>you can't have a set without an empty set, no? Can you clarify this question? Are you referring to the construction of natural numbers starting with the empty set?
Anonymous 14-09-22 02:39:38 No. 11648
>This makes me feel mentally disabled You're a namefag, of course you feel that way. Drop that junk.
That's set theory, with a couple of complex numbers. I'm assuming you know what those are. If not, look it up.
I'm not great with (nor generally interested in) mathematics, but if I'm reading it right (Q = the set of rational numbers, which is normal for the blackboard-bold symbol Q,
https://en.wikipedia.org/wiki/Rational_number , and the final question being what is the intersection between the set S and Q) then it's just asking which of the 6 elements of the set S are rational.
So for example, 1/3 and 22/7 are obviously rational, [pi]/3 is obviously not, and so you need to figure out if the other 3 are rational. I forget all my trig and odd/even powers of those complex fractions so someone else needs to sub in.
Anonymous 14-09-22 14:04:19 No. 11651
>>11646 > a step by step https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic https://en.wikipedia.org/wiki/Roots_of_unity https://en.wikipedia.org/wiki/Polar_coordinate_system#Complex_numbers https://en.wikipedia.org/wiki/De_Moivre%27s_formula You are asked to count the rational numbers in S. Two of them are evidently rational. π/3 is irrational, but in most contexts this will be accepted as known since the proof is non-trivial. If you can't remember the sine of π/3 take a right-angled triangle with an angle of 60 degrees and take the ratio of the opposite leg and the hypotenuse. Complete your right-angled triangle by reflection to an equilateral triangle and you will easily find the ratio to be sqrt(3)/2. For the irrationality of sqrt(3) take a^2==3*b^2 with a and b coprime, take the unique prime factorization of both sides and simply count the parity of the number of times 3 appears on each side.
For the first two values, identify them as roots of unity, a cube root and an eighth root. Recall that when raising roots of unity to natural powers you may discard multiples of the root order. Reduce 2019 modulo 3 and 8. This resolves the first value, while for the second you are left with the cube of an eighth root. Since you only need the rationality of the imaginary part you can avoid doing any computation by recalling that exponentiation by natural powers on the unit circle amounts to rotation by multiples of the base angle. Since the base angle is π/4, first quadrant, cubing takes you to 3π/4, second quadrant. This has the effect of flipping the real part sign and leaving everything else untouched, which resolves the second item in S.
Anonymous 18-01-23 04:53:53 No. 12227
>>9475 Because
e = lim{x to infinity} (1 + 1/x)^x
Anonymous 19-06-23 07:03:35 No. 18223
>>18222 I read these two books for similar reasons. Although in my case I can't blame it on the teacher. There might be better books, I chose these two because they are relatively short compared to other calculus textbooks
and because they were written in emacs .
I am not sure about programming itself, but if you are interested in actual computer science, like the theoretical stuff, logic is the calculus of computer science.
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