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 No.30554[Last 50 Posts]

I liked the first math thread, but that hit the bump limit so I'm making another one.

Here is a neat tool posted in the previous thread that shows you how to do geometry the way the greeks did.


Here are a series of MIT OCW courses that will help you learn calculus:



Full MIT OCW Mathematics catalog:


Attached is the a Numberphile video about the seven bridges of Königsberg because I dunno what else to attach to this OP.


Why would someone learn math if they're unlikely to accomplish anything with it? Entertainment?



Most humans learn things they are interested in without any real end goal. I mean, people devote a lot of time learning video game mechanics that have no real world applications. So yea, learning about math can be entertainment.

In the attached video in OP, a problem that anyone can think of is discussed. Mathematics help you analyze and properly solve these problems, which is something one could consider an accomplishment.


Has anyone here learned graph theory? Did you like it? Any good textbooks you recommend?


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Another wizard warned me to study for my next exam in the last thread. Guess how that went down over the course of an entire fucking week. Just looked at last year's paper. Might go and rest my head on the train tracks now.


I'm taking a linear algebra course and the guy constantly got us finding inverses of 5x5 matrices and solving systems of equations with 6 variables that have infinite solutions (I got one of those questions wrong because he was looking for discrete solutions and my notation used solutions in the R domain, I thought the discrete steps were implied…). It is an online class so I can understand him making shit this hard, but man… fuck this shit.



Any computer science major should know quite a bit about graph theory. I am a CS major and math minor and all graph theory I know comes from my CS classes. I guess any math major wouldn't get into graph theory until grad school.

Thinking back on all my CS classes, I guess the best one that goes into graph theory the best is the discrete math class, which was mentioned in OP's video (just about any textbook in dicrete math would make a mention of the bridges of konisberg).

Keeping with the spirit of the OP, here is a youtube playist that has some MIT lectures on graph theory. I don't think that subject of math is that complex and can be explained with a few lectures but I dunno, I'm not a mathematician. Of course these videos come from the MIT "mathematics for computer science" course.




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>I am a CS major
Did you study parsing theory? Currently I'm analysing a language using derivations and all that stuff in order to improve its grammar and then, if possible, programming a compiler.

>Graph theory
I didn't like it because its proofs seemed too informal to me, they were like a silly talk… but there are books transforming graphs into matrices and proofs are formal.

>Any good textbooks you recommend?

What kind of book are you looking for? Do you wanna proof or only results (like an engineer) or algorithms?


>What kind of book are you looking for? Do you wanna proof or only results (like an engineer) or algorithms?

Maybe just pure maths, but I'm not sure. Any book that is good will do the job, I guess.


>Did you study parsing theory? Currently I'm analysing a language using derivations and all that stuff in order to improve its grammar and then, if possible, programming a compiler.

Yea, that was part of my advanced programming class.


>pure maths
>I'm not sure
Do you know the proof methods? If not, you probably wont understand a pure maths book.


>Pure maths.
What are you guys on about.
Every maths in books is pure maths..
Applied maths just means it's useful.
The distinction is engineer majors (and other) and maths majors books.
Only read the maths majors books.


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I'll play. The usual view is that applied maths means maths currently useful and applied to another field (e.g. physics, economics). If maths is not applied, then it is pure. So applied maths is normally not considered a subset of pure maths: when something becomes applied it ceases to be pure.
The only cause for confusion here is when something is initially pure but later applied and universities choose to call it pure for historical reasons (e.g. group theory has been useful in particle physics for decades but some call it pure).

I know I posted this one or two thread ago, but for anyone interested in applied maths (for physics in particular, but also other things), you owe it to yourself to read this book. You can probably pirate a .pdf somewhere.


My point was that it's nonsensical to talk about pure maths/ applied maths when you're still a student.
What students mean by pure/applied is is it for maths majors or not. Is it just calculation and applying methods (shitty engineering stuff) or is theory heavily involved (formal proofs).
Not one book that is for maths majors will be "applied" in that sense. Otherwise it's not a maths major (or a good rigorous worthwhile one at least).

But I did already hear about that pure/applied definition of yours. It's nice.
It shows how nonsensical it is to try and categorize pure/applied.

Eventually (maybe) evrything gets applied.


Do we really need someone to make a youtube video of bridges of konnigsburg whatever.
A simple 1 paragraph description + 1 paragraph proof suffices.
I realize if there's a niche someoen will try to fill it to spam ads / make money, but this is ridiculous.


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Well… The point here is not whether or not graph theory or other mathematical theory is pure (probably the correct word should be formal), but the books you study. For instance, engineering books only show results and give no single proof (and in case they do, it's a very informal one) and are fille up with hundreds of examples and thousands of exercises, they both numerical. There is no abstraction at all.

Pure… I mean, formal books on mathematics, on the other hand, give you an axiomatic development on the theory they are about, and due to that you need to know the methods of proof because proofs are the only thing you will be studying; most of these books have no single numerical example, things like these are "left to the reader as exercises" because you must be able to figure out how to "translate" numerical examples into abstract entities, even if you have never seen one.

>I am a […] math minor
>I'm not a mathematician.
Wait. What? Could you explain what you must study in order to be a mathematician? what is the difference among minor, bachelor, career, and other terms? I thought having a minor was enough, or do am i confusing terms?


>Wait. What? Could you explain what you must study in order to be a mathematician? what is the difference among minor, bachelor, career, and other terms? I thought having a minor was enough, or do am i confusing terms?
I don't think someone can call themselves a mathematician or physicist until they are actually a professional working in the field as their career.



Numberphile isn't for people that want mathematical rigor. It is to get normal people interested in math so they spend forever explaining the most simple of junk. The channel also makes a few ridiculous video, such as the sum of all natural numbers equaling -1/12 to get people talking about math.


I have read a book on proofs called "Book of Proof" by Hammack. Is that enough?


I don't understand this talk about proofs/pure.
There is no distinction if you're buying the books aimed at maths students.


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I really want to master this shit but I am stumped. How do I find the second order derivatives? I tried continuing the chain rule that I used to calculate the first orders but I must be doing something wrong because I can't get the right answers.

Specifically I want to know d^2z/dx^2, d^2z/dy^2, d^2/dxdy, dz^2/dydx



The answers are:

d^2z/dx^2 = -(16z^2+x^2)/(16^2z^3)
d^2z/dy^2 = -(4z^2+y^2)/16z^3
d^2z/dxdy = -(xy)/(64z^3)

No idea where that junk came from.



If figured it out. I was being dumb and didn't try the easier method of using implicit differentiation. Implicit differentiation makes it easy as fuck.


This is the most complicated simple explanation of e^(pi*i) I have ever scene.


Anyone here went to grad school for mathematics? I'm wondering if it is worth it.



I'm doing a doctorate in applied maths in the UK.

It's pretty good, my whole course of study is funded so I get enough money to live comfortably and no additional debt. It's much more interesting doing research than studying, and it isn't too much work as long as you keep on top of it. I basically work 3 days a week. There's very little socialization required, and though you have to work in teams on some things and talk to supervisors, it's always about work stuff. Most mathematicians are normalfags but there's enough weirdos around that a wizard won't stand out. Despite what some on here claim, there are plenty of non-academic jobs available for people with advanced degrees in maths/physics, most of them well-paid and reasonably tolerable.

I hear terrible things about grad school in the US though, how people are used as slave labour, get into massive debt and still don't have a PhD after 10 years or more. Better to come study in Europe if you're a US wizzy.

Hope this helps.


I like to learn maths as much as I can do, but it's been ages since I've last studied maths. So, I need to take "mathematics 1". Do you know any good source?



Thanks. I'll look at it tomorrow.




I hear terrible things about grad school in the US though, how people are used as slave labour, get into massive debt and still don't have a PhD after 10 years or more. Better to come study in Europe if you're a US wizzy.

Anything related to to education in the US is complete shit compared to the rest of the world. The only thing the US has over any other country is that if you are able to become ridiculously rich if you are lucky and dedicated enough.


PBS now has a math channel, like space-time was for astrophysics. The host is a succubus. Doesn't bother me, but I know that stuff triggers some guys here.


Could someone recommend a good pre-calc textbook?


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Who else?
I just finished the first chapter, and it's pretty good, feels like it's exactly the kind of book I was looking for. I heard somewhere else that it's really really hard past this chapter though, I wonder how true that is. But first I gotta get the excercises done, and some of them are pretty weird actually, I don't really know how to approach them (yet).


Welp, right after writing this I tried solving the "warmups", and they are HARD.
Maybe I'm just stupid



Looks like a discrete math textbook. What are some examples that you are having problems with?



Pre-calc is just the basis of trigonometry, geometry, algebra etc. Basically a culmination of everything you are supposed to learn in highschool, excluding calculus. Khan academy is very good for those subjects.


I took an undergraduate course in graph theory and I absolutely loved it. My professor has a low Erdos number.

We used Graphs & Digraphs by Chartrand, Zhang, and Lesniak. I thought it was a very good book.


I'm afraid to ask this, but is there anything I should read or look up to brush up on math? As in, "I failed all my math classes in high school" levels of rusty. I'm not expecting to unNEET myself, just something to past the time instead of just refreshing Wizchan.


see >>34091
>Khan academy


>be me
>retaking real analysis, which is god awful
>doing undergrad research
>motivation a shit this semester
>if I pass/finish, I'll graduate with a BS in Math.


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So, there is this course about discrete mathematics in Coursera. They made it sound like "hard enough, but not too hard", which seems to be a bit of an understatement when it comes to someone with a humanistic educational background: I know jack shit about anything beyond college freshman arithmetic,and though formal logic is a friend of mine, I can barely prove that every odd number has the form k+1.

Trying not to give up too easily, I took a look at textbooks recommended on that course. All right, junk in those books seems definitely less hieroglyphical than I presumed, but one big question remains. How the hell am I going to figure out whether or not I've gotten anywhere near the right answers to exercises introduced in the books?

To give an example: "We form the union of two sets. We know that one of them has n elements and the other has m elements. What can we infer about the cardinality of their union?" The question comes from "Discrete Mathematics Elementary and Beyond", 2003, Lovász, Pelikán, and Vesztergombi (page 9, exercise 1.2.13 for those possibly interested).

Well, this doesn't look too daunting; so, naming our sets N and M, having n and m elements respectively, I can infer that |N U M| <= n+m because |N| = n, |M| = m, and N U M has at most n+m elements. Neat, I now have a candidate answer. But is this right? Probably. Is it the correct and exhaustive answer to the question? Most likely not.

Of course the book does not offer correct answers, so I am sitting my ass exposed to the burning sun while having my head deep in a pit full of shit. Taking a class in the nearby university might be an answer to this, but I am not up to anything that requires moving my rear end from this sofa it's currently parked on. Should the universe ask me, I'd glad to die on this sofa.

Now there might be someone willing to play a personal tutor in this thread, and that'd be brilliant, but my actual question is: Where do I look for a supporting community in basic shit like this? Any other ideas or advice about how to know when I've arrived to the right answer instead of concocting an answer which has wrongly convinced me about it's truth? There must be other people not naturally talented with this and yet willing or trying to get some grasp of it on their own time without the help of professional educators.


Okay first
>I can barely prove that every odd number has the form k+1.
lol, I think you mean 2k+1, but on second thought, every odd number DOES have the form k+1.
But you have the right idea of mathematics (most laymen seem to think it means doing computation).

The question you struggle with is "What can we infer about the cardinality of their union?", what can you know about N and M when |N U M| = k hint: compare with m+n
It's hard at first when you are comfronted by math books because of the nature of the questions. Don't worry though, the questions are there to have you think, with the correctness of your answer being a side effect.
In fact, the vast majority of the books use the excercises to introduce results that aren't stated in the book. The point is that you find these ideas for yourself, which is the best way to learn.
Don't beat yourself up, do your best effort. Math books are not easy.
I can also suggest you read a little book called "How to Solve It" by G. Polya.



>every odd number DOES have the form k+1.

True, but you can find an even number that is k+1 (e.g. k = 1), therefore every odd number does not take the form of k+1, which is a proof by contradiction that shows that all numbers do not take the form k+1, which makes it an insufficient definition of odd numbers.

But yea, if you follow this line of thinking with similar shit, real analysis becomes a breeze. Keyword is analysis, which in other words can mean you thinking of an initial solution and challenging that solution and proving your results with logic.



It is not entirely out of the question how I might have indulged myself with intoxicating substances during the writing process of my post. All of which I now present as my excuse for failing to type 2k+1 instead of k+1.

But anyway, thanks for the encouragement.


I've been looking at the channel introduced in this post >>33247

Her latest video is supposed to be about proofs, and starts strong, but it devolves into a series of mathematics mumbo jumbo with visual explanations that ware probably obvious to people who understand the math mumbo jumbo, but confusing to those who don't. I think this is the problem with moth math textbooks. A lot of things seem "trivial" when you understand what is being explained, but if you don't you are completely confused and feel like an idiot for not understanding. I can see how it is easy for any author to fall into this pattern, which is why it is a problem to begin with in math education.

The channel i very promising, though. I think the target audience of the channel is people have an interest in math, but do not have any formal background in it. I think she just needs to upload a series of videos that goes over the foundations of mathematics and its axioms. Things such as what the triangle equality actually means, and perhaps things explaining things such as convergence, continuity, and the basics of set theory should be some foundation videos for the channel, I think.


I don't know what 2+2 is but I like math music, so I'll let this be my contribution.


I hate to be a pedant, but I think you guys meant integers not numbers since x=2k+1 is trivially true for any numbers, odd or even.

>therefore every odd number does not take the form of k+1
I actually don't follow your conclusion in there. k+1 isn't a valid definition to begin with, so the negation of the statement doesn't really affect the truthfulness of the other.


Disregard what I said about integers, I am an idiot. It's actually standard to say odd or even number.



It might be standard, but you're right. A number may or may not be an integer, and the banter was about integers, not numbers in a looser sense.


Can't disagree with you. Personally, my way to get into mathematics is by reading books. Right now I'm trying to go through Concrete Mathematics, it pretty much introduces to you the methodologies of doing math.
I think those videos are more suitable to spark the interest in mathematics, rather than to actually /teach/ anything.


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If I've understood this correctly, mathematics is something a game of exploring a world, P, while avoiding both the meadow F and the forest T ∖ W, and hopefully find within P the sacred spring of W:

P = { p : p is a mathematical proposition }
F = { f : f is an element of the set of all false mathematical propositions }
T = { t : t is an element of the set of all true mathematical propositions }
W = { w : w is an element of T and w is non-trivial }

|P| = ∞
|P| = |T| + |F|
|W| < |T| <= |F|

The game may only be played in a closed semantic system, which is non-complete ie. it is not expressive enough to prove all propositions in itself. All moves in the game must be deductive, though one may use a special flavour of induction as a method of proof.

If it's anything like the game I think it is, why most, if not all, mathematical education I've received in the grammar school, high school, and even in the university I went was about mechanical calculation and mindless application of formulas? Nobody told me what the purpose was and I can now clearly see that quite a bit of that mechanical arithmetic shit was not very useful when it games to the actual game.

PS: I might be slightly tipsy, so I blame all and every mistake above on the drinks I may have had.


The same reason that all other classes in school usually only cover the most shallow, banal, and basic repetitious aspects of almost any subject. The majority of all schooling in the modern world is not there to help you actually develop a deep and abiding understanding of the art you are being taught about, instead it's essentially job skills training, out of date job skills training at that.


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> out of date job skills training at that

Unless, of course, you're going to be an instructor in an institution producing out of date job skills training …

… that said, I think it's all right to give mechanical skills training because that suits the most. But they could have told me in the beginning that's what I'm going to receive. Well, at least now I know better and can enjoy learning things I should have learnt half my life ago, and learning them with brains too old for best of mathematics.


>Better to come study in Europe if you're a US wizzy.
I honestly don't know if there are many US undergrads who could hack it in a European PhD program. The level of rigor in an American undergrad program outside of the very top places like Harvard or Chicago is absolutely pathetic compared to Europe.

That being said most of the people applying for grad school in plain old math are probably in it because they're really into it (money-grubbers are going to end up in stats or a financial stream) so maybe they study enough outside of class to make up for their system fucking them over. I can't say.


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If it's any solace to you I've failed 3 courses this week, I kept telling myself I will study for them later, that they're too easy for me anyway. I let that accumulate for about 2 months and will now need to pay back my financial aid, which I will do with another loan. This has happened to me so many times I've lost count, it's a trap I keep falling into.


In case anyone is interested, Ohio State's basic calculus course is starting, again, a new iteration in Coursera …


I've enrolled to that now approximately 17 times without ever hanging around long enough for the course to advance past things I don't already know (ie. the first couple weeks).


>with another loan
do all uni guys voluntarily lend themselves to slavery?



Unless the state provides free universities like in some countries still happen in Europe, that's pretty much the only way to get into one.



So many youtube videos fall into this bullshit. Almost every single math channel I watch is some simplified explaination of a somewhat complex math concept, and they gloss over any of the real calculations. This is fine for getting people who don't know anything about math interested in a subject, but frustrating for anyone who is interested enough to learn more.

A recent example is shown in the video in attached image. He completely glossed over what a^(i*c) means, saying you should watch other videos in the description. Well, I went to the description and those folks used nothing more than stupid visual explanations that didn't really prove shit. After a quick google search, I found a simple answer, which used euler's formula: e^(ic) = cos(c) + i*sin(c)

Then substitute a for e^[ln(a)] to get: a^(b+ic) = a^b{cos[c*ln(a)] + i*sin[c*ln(a)]}

You can easily see if you set b = 0, you get a value that will lie on an complex valued unit circle. But no, no one didn't bother to post this simple 5 minute explanation and just went for some stupid convoluted visual bullshit that confused me more than anything else.


Maths can't be explained in layman's term and vulgarised like science can be.
You can get the general idea of some theories in physics or even more in biology but what lies behing are mathematical concepts.
And these take time and effort to understand, every detail counts too. It's basically pen and paper, reading books and solving loads of exercices until you get the feel and intuition of the concepts you just learned.

As feynman said on his lecture of mathematics and physics, trying to make sense of physics or maths by explaining it with words (or pretty pictures) is like trying to explain to someone what music is. Impossible to get across.


anywiz gone from math to EE? how did you do? do the two inform each other? thanks




I didn't exactly do that, but something close. I double majored in computer science and mathematics. I first started with computer engineering (literally EE with CS… looking back I don't see why it wasn't a graduate program that accepted EE or CS graduates), but after realizing I did not give a fuck about electric circuits I switched over to CS in my sophomore year. Chose to double major in math because I literally just had to take 4 extra math classes to get a math major.

Anyway, the extra math classes sure as shit didn't help with any of my CE classes. The math classes I was required to take in the CE curriculum were all I needed to understand stuff. If you taken any higher level classes such as topology, real analysis or number theory, I'd reckon that shit will be 100% irrelevant to EE.


what turned you off about circuits?

i'm just like, "i want to create something of value and EE will get me there, probably".

been dicking around with dc theory in electrician school, and it's fun. i think a lot of fields can fulfil the urge to problem solve.



All the stuff I was learning about circuits I figured I could just find out on my own using books and items that I could buy at radio shack. I figured it to just be a lot more boring than figuring out the optimal solution for logic gates and figuring out math tricks to create more efficient circuits. I suppose that is the core of computer engineering, but I liked the computational theory more than creating circuits.

Like >>35526 said, pure mathematics is the framework of just about every theory in other field. I guess some people realize they like the mathematics more than what the mathematics is being used for. I was one of those people.


>I liked the computational theory more than creating circuits.
yes i think we are the same however just to clarify, by "creating circuits" do you mean "physically creating circuit boards".

i always wonder if pure math people struggle with feelings of immanence: as engineer you're not quite that, but it does seem there's a theory/practical experience divide and some of us are better at one than the other.

do you still work in CS and do you find your ability to zoom back and see the big picture useful. is there some part of you that likes to see the practical results of your work.



Yea, I hate doing the physical stuff, but I knew that the manual bread board shit was only for college. That was quite a few years ago so many college students have access to 3d printed PCBs now.

Creating the circuits on paper was just a huge meh affair for me. Didn't really hate it, but didn't like it either.

As for my job, yea it is is CS related. Data scientist so statistics and machine learning algorithms are what I use all the time. The end result isn't as satisfying as an engineer, but I like the day to day work a lot better.


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I'm reading Spivak's Calculus, and it's been four months since I started limits chapter, and still don't finish the execises; I've solved only half of them. Should I start next chapter and try those exercises later?



You can't learn math without doing the exercises. However, completing all the exercises in the textbook is almost never necessary, but without a teacher to tell you which ones to solve, it is best to just do as many as you can. If you can't figure out how to do some of the harder exercises, it should be safe to skip. Things that you really need to learn will probably show up later in the textbook, forcing you to learn it eventually anyway.


Yes, no need to burn yourself out.


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Stalling too long with the same stuff will burn you out as noted in >>35688.

Either move along or at least take another book or find an online course about the same subject for exercises for the same subject.

I often find that using just one book is not cutting it for me. If I get stuck with more than one exercise, I try to find another view on the same subject from another book. Once I get back to the original exercise I got stuck with, I often find myself able to figure them out.

But you really should be doing exercises and a lot of them. Without doing lots of shit with just pen and paper is irreplaceable when it comes to math.

If you're afraid that your neighbours find your half-finished and incorrect solutions while going through your trash, just shred all papers in very small pieces. Or better yet, burn them like I do.


> Here are a series of MIT OCW courses that will help you learn calculus:

I can understand most of what he's talking, but I can't solve even pre-calculus problems from the net or books. What am I doing wrong?


>The end result isn't as satisfying as an engineer, but I like the day to day work a lot better.
sorry i lost track of this thread.
if you're still here, would you mind expanding on that a bit? i think what you're saying is that you enjoy having a concrete finished product?

electrical work is satisfying in this respect; also spending time with people who have good business and time management skills has been a growth experience.

i'm about to start taking EE classes at my local cc and it's a very hands-on course; considering that i enjoy spending time on my own solving math problems, am i about to make myself very miserable?


Then you don't understand.



Math is a very analytical discipline so if you haven't learned how to identify a problem and what is needed to solve it, you will not be able to accomplish anything. I don't think that can be taught… you just need to practice. At any rate, you should post stuff here that you are having trouble with. I am sure someone will help you out.



>If you're afraid that your neighbours find your half-finished and incorrect solutions while going through your trash, just shred all papers in very small pieces. Or better yet, burn them like I do.

Stuff like this is why we need a math thread. Glad to see others are as paranoid as me when it comes to finding your shitty work.

I use a whiteboard so I don't waste paper. Also if I arrive to an elegant solution I just leave it up on the whiteboard to bask in it for a few weeks.


What's your motivation for learning math, wizards? I've always liked math but I don't have much of a reason to outside of recreational purposes. I started teaching myself calculus last year and I got all the way up to applications of derivatives (alright, so not that much… but it's probably at least 8 weeks of material in school) but I stopped because I got bored or something, I can't recall. I think I even posted in one of the math threads here. I dream about building a curriculum for myself but I don't have a specific purpose other than "I have fun learning" but it's not enough to carry on once the novelty of it dissipates.

I know some of you are students but surely there are people who aren't in academia (NEETs I imagine, since wageslaves (presumably) don't have the time to learn math outside of school) who want to learn math for some reason.



I don't think I would care about math if I didn't have to learn so much of it in college. I don't see how anyone can give a damn about math without at least gaining a solid grasp on calculus.


for me it is physics and as was said, some fields/topics in mathematics use calculus.
but i do not learn page after page. when i do not understand something, i open the book and try figure out what is going on.


I am essentially mentally retarded


I just think it's beautiful. And it's fun.
True, you don't get very far unless you enjoy it.
I personally don't like calculus. It's just computation. It's really useful to know how to compute that stuff, but if you don't put it to use or if you don't provide context to it, it's pretty much empty.
I like Real Analysis for that reason. It is the actual semantic content behind calculus, not just the computation, and it leads to the interesting results for which calculus is just a tool: I'm talking about projective geometry and topology.
Maybe you should go explore something else for a while, either Analysis or go right into Projective Geometry (it's really nice). It should provide a nice refresher and put your calc skills to use.


i think the majority of this people on this sub are intp personality type and this thinking style corresponds well with interest in problem solving.

i'd also say a lot of people here tolerate solitude really well and math is an entertaining way to fill the void as normies would do by superficial interactions with others.

i'd advise you all to try to make a career out of it if you can. the world is much kinder to weirdos with a talent.


>on this sub
You spend too much time on reddit.


Wizchan must have been linked to or mentioned somewhere. Sudden uptick in greentexting and posts like these.


Fuck, I want to be smart. I feel so inadequate and inferior with my low IQ and even worse self-discipline to study.



I really don't think math is about being smart. Sure, smart people are able to understand things quicker and do better in school because of this, but I think anyone can understand math given enough time. Math just requires people to understand basic shit and then build off that. If you fail to understand something at the lower level, you will not be able to progress. Math is the embodiment of building on top of existing logic.

Anyway, I am just learning about transcendental numbers. Are transcendental numbers important, or just one of those neat things in math that don't serve any real purpose yet? I would think numbers that have no algebraic root are fairly important, but I can't really think of a situation where needed to knowing precise transcendental numbers is important. Or is it just so you don't waste your time trying to solve an impossible problem such as squaring a circle?


I don't even believe there is more intelligent people and less intelligent ones. I think we all have the same potentian and we just need to have the proper foundations. The same goes with talent.

I don't think there's much to do with trascendental numbers, besides showing that R is uncountably infinite. And perhaps some interesting ways to calculate them like all the methods that exist for calculating pi.
Besides that, nope, I barely even remember they had a name.
I mean, there are a couple such numbers that are very important, such as pi and e, and you can prove that they are trascendental, but besides that? I haven't seen much attention paid to them.


Slow paced intro book I've read on graph theory: http://a.co/74odbzj

Faster paced grad text on graph theory I liked more because I prefer concise fast-moving texts:

The first one is cheap as heck as are all Dover math books. Dover is my go to because of the cheapness.

The other can be pirated.


File: 1493585423870.pdf (1.9 MB, Ulrich Knauer-Algebraic Gr….pdf)

Second chapter of Ulrich's Algebraic graph theory studies graphs as matrices, wich is an approach suitable for CS and all programming-related applications.


So 3blue1brown is uploading a series of videos on calculus. His linear algebra videos really helped me gain the proper intuition behind linear algebra concepts. I am a visual learner so seeing those animations made things really clear to me.

After watching the attached video I was thinking about the radius of convergence and I couldn't remember how the hell do it. I mean, I should know since I took calculus and real analysis in college but the knowledge just escapes me. This caused me to head over to khan academy, and watching a video for 10 seconds jogged my memory, but I decided to just do every single subject they have on that website since I *should* know all of it. If I forgot how to calculate the radius of convergence what else did I forget? Doing all of these exercises is making remember shit that I haven't done for years. It is a bit frustrating to have to do basic shit (basic number line counting and such) when you obviously need those skills to solve higher level problems, though. I figure I can power through all this stuff in a week or two.


>After watching the attached video I was thinking about the radius of convergence and I couldn't remember how the hell do it

You do the ratio-test method (http://tutorial.math.lamar.edu/Classes/CalcII/RatioTest.aspx)


Bumping this thread cause I don't want it to die. I really have no where else to talk about math other than this place here. No idea what to add to the conversation, though. Been a lazy piece of shit for a while so I haven't been working on anything.


what's the best way to prep for little rudin? I don't have much experience with proof writing and it's a bit abstruse to me to be asked to prove things since I am not too familiar with proving


File: 1496898248526.pdf (2.14 MB, Ethan D. Bloch (auth) -Pro….pdf)

>what's the best way to prep for little rudin?
> I don't have much experience with proof writing
First of all, you must read a one or two books on proof writing, like Ethan D. Bloch's Proofs and fundamentals: a first course in abstract mathematics or A transition to advanced mathematics, by Smith, Eggen, and Andre (read first five chapter of each one). There are problems or concepts in the former that are explained in the last, so they both complement each other.

After that, you must study calculus in one variable and linear algebra, then multivariable calculus and a little introduction to topology; only after studying all this, you should be ready to read little Rudin.

By the way: since set theory's concepts are used in all branches of mathematics, one must study a whole book on that topic; doing this can help you understand more easily topics like the least upper bound property appearing in calculus.


It's a form of art. It's not necessary a purpose. I personally just enjoy the beauty inside math.


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After a month of hiatus, I skip the rest of the exercises and read next chapter; it was a supprise how easily I could solve most of them (just skip three); even read Three hard theorems chapter and have solved half of the exercises.

Because I like maths. No more reason needed… at least it was my main and only reason when I was studying my math degree. Now, after some years of lack of study, I have forgotten what once I learned. I must get a job, and I'd like one on computing (in the math sense) but here nobody needs someone like me, who studies it seriously; instead, statistics is the only branch of mathematics companies need, but I'm not good on that and due to it I've been studying calculus in order to get strong basis to fully understand measure theory, statistics and all about financial mathematics.



Math is an arcane art to most people, and the bulk of your knowledge is only useful to other mathematicians. It is like being a travelling wizard and everyone just wants to take advantage of your fire spells to start fires and shit because that is the only thing they know that you are good for.


I'm interested in delving into mathematical logic, is it going to worth the time?



Math is logic, but logic isn't math. You can move onto philosophy once you understand logic. What discipline you choose depends on your personal interests.


It depends on what you consider to be worth. One must study mathematical logic in order to study other branches of mathematics, like computation, set theory and hyperreals (at least when researching its axioms soundness). Other people study it just because, for no particular reason but enjoyment.

If you want to apply it to real problems, then you should study more mathematics because by itself has no applications outside the realm of pure mathematics. Topics like semi-Thue systems and recursive functions, wich are part of mathematical logic, have been applied successfully in computation, particularly on parsing theory, wich leads to de development of the front-end of compilers.


mh, what? logic isn't philosophy either. you can define logic as the foundation of mathematics (category theory, set theory etc.) or some sort of meta-mathematics (model theory etc.)


File: 1502353855215.gif (3.73 KB, 827x432, 827:432, pearsons.gif) ImgOps iqdb

Which is best, pearson or spearman correlation?

I am using pearson and it seems better than spearman because it suits better for 2 linear variables that more or less positive, but aren't both made to calculate the correlation between in 2 variables? Spearman is more used if the correlation goes negative but why the fucks it matters if both are supposed to work with correlations?

Pearson's correlation is my favorite but I am writting a book that will include correlation and should I use both?


Did anyone get taught about the hyperbolic trig functions in highschool? I remember them just being tossed around in college like they were something simple that I was supposed to know. Actually, now that I recall imaginary numbers weren't even introduced to me in highschool. Did I just go to a shit school?


Philosophy is nothing but drawing conclusions through logical arguments.


So I just graduated from college. I guess I am qualified as a mathematician now but I feel as if I am a terrible one. I remember theorems and shit, but I always have to look them up to execute them.


No you aren't qualified as a mathematician. A mathematician … does maths for a living.
Schoking I know.
Once you graduate college you need to do graduate school and get a phd then a post-doc and then get a job as a mathematician. Finally then you can call yourself a mathematician.
Right now you have a maths degree, that's it.


Anybody can call themselves a mathematician, anonymage. Some stupid arbitrary standard about having a Ph.D to be called a mathematician is bullshit.

If an engineer get an engineering degree, and works at x company, is he not an engineer? Anybody who uses pure mathematics as a job, any job, is a mathematician.


You are correct, but having a physics degree I would feel weird calling myself a physicist. I feel like only an active scientist has the "right" to call himself physicist/mathematician/chemist.


Can a math wizard please help me? I'm trying to solve a problem where I have the number of people that need pills and I have the average number of pills that those people eat for a week. Is it possible to find the total amount of pills I will need in a week with just these 2 information? I'm trying to figure out a way to find the total amount of pills that will be needed but I'm too dumb, haven't done math in over a decade. Also, where is a good and free place to start learning math?


Multiply the number of people by the average number of pills needed?


>…have the average number of pills that those people eat for a week. Is it possible to find the total amount of pills I will need in a week…

Did you explain that right? It sounds like you have your answer


avg = (total number of pills distributed)/(total number of people)

avg*(total number of people) = (total number of pills distributed)



Meant to quote >>39523


In the problem I have the average amount of pills the people need each week, and the number of people that need pills. I need to total amount of pills that should be ordered for the next week. Is there any math way to solve this?


it's litteraly the definition.
If an engineer works as an engineer he's … an engineer. Yes.

If one uses maths at a job he's often doing "engineer" type work. If it's maths for the sake of maths and he does it in industry yeah that's a mathematician.

The guys working on secret encryption stuff at the nsa would be an example.



I gave the solution here >>39529 so did >>39524

Multiply the average amount of pills people need each week by the number of people who need pills.



There are things such as unemployed physicists and mathematicians. If you consider that your primary skill, then I think you can call yourself one. I guess you aren't officially one unless you are currently getting paid for it, or got paid for it in the past.


File: 1506242963387.png (150.17 KB, 547x287, 547:287, 1506035247708.png) ImgOps iqdb


Though, according to Bertrand Russell in the Principles of Mathematics, Math and Logic are identical.


File: 1507988486095.gif (9.56 KB, 1087x438, 1087:438, Double_Integrals.gif) ImgOps iqdb

I wanted to bump this thread, since I enjoy it so much, and so I decided to update for any interested wizard. I'm currently working my way through double integrals, so I can find the area under 3D surfaces of the type z=f(x,y).

I'm finding double integrals to be pretty easy, especially because the concept isn't particularly groundbreaking at this point (unlike when I first saw anti-differentiation). Are there any other anonymages working on similar concepts, of Calculus in 3D? I have heard that triple integrals are very difficult to get down, and the last big concept of multi-variable calculus.

Has anyone here taken graduate level Calculus, or a proofs course? I guess that's the next step, after Differential Equations.


I never did double or triple integrals before my electromagnetic physics course.
Am I the only one not getting the triple integrals meme ?

There's nothing hard about it. Calculatin the areas and volumes in EM was actually fun and fairly intuitive.


Do they teach you a method on how to do it? I'm just trying to think how it could be done, and what it actually calculates. I guess I could peek ahead, but I'm trying to get good at sketching the volume under the surface that is getting calculated.

Unfortunately, I know absolutely nothing about physics, nor do I want to. I have looked into mechanics, but it seems to stray too far into word problems and concepts. I prefer plain mathematics, where all I do is understand the theory behind x concept, and maybe the proof, before then applying it.



If you truly understand integrals and what they are calculating, then you shouldn't have any problem with triple integrals. Triple integrals are no different than double, which really aren't no different than single. You just add an extra dimension each time.


File: 1508105769726.png (51.49 KB, 529x486, 529:486, 2017-10-16-000937_529x486_….png) ImgOps iqdb


Take a look at a vector/complex analysis book. Triple integrals, and in general integrals of n real variables, are defined the same way as double integrals, so they have the same properties. The methods for calculating them are similar.



Thanks for replying to me, and I apologize for being slow in my replies. I read the chapter on triple integrals, and did some problems; not too bad at all. If you ignore the algebra, these concepts seem very basic to understand and implement. You have an inner integral, within an inner, encompassed by an outer, right?

I also watched some Khan Academy to get it down, and I practice along with him.


File: 1508513814101.jpg (17.68 KB, 400x399, 400:399, 1508213652039.jpg) ImgOps iqdb

> I prefer plain mathematics, where all I do is understand the theory behind x concept, and maybe the proof, before then applying it
>Khan Academy
>triple integrals are […] the last big concept of multi-variable calculus
Wizie, if you really want to understand mathematical concepts, particularly those about calculus, you should stop watching that kind of videos and throw your books away. Start reading Spivak's calculus book; if you understand that book you should be able to easily understand multivariable calculus… actually, Spivak's book on multivariable calculus (Calculus on manifolds) is too thin compared with engineering books on the same topic.

If you decide to read them, I'll be reading this thread if you post any doubt.


any books for beginners to logic?


I did a bachelor degree in mathematics and now i just hate it and am so sick of it


File: 1508911205303.jpg (35.4 KB, 331x499, 331:499, logic.jpg) ImgOps iqdb

Enderton's A Mathematical Introduction to Logic is the classical recommendation; I have read a few pages, but I prefer Richard E. Hodel's An Introduction to Mathematical Logic.



School is pretty effective at sucking the fun out of everything.


are you helping high school kids with their schoolwork or could you be able to build a career on maths?


Not him, but the only mathematics people want now are statistics. All you do is help scientists out: biologists, chemists, etc. If he enjoys pure mathematics, he can't do shit. It's really a shame.


Can anyone recommend a field of mathematics that I can study recreationally that also has interesting applications in daily life? I'd like to study something challenging, but not feel like I'm stuck dealing with endless abstractions all the time.

I was thinking probability theory might be a good starting point. Any other ideas?


I'm probably not the best guy to say this since I'm a highschool dropout who's never gotten past basic algebra, but interesting applications in real life is pretty broad when it comes to math. I mean for instance I think I recall (please someone correct me if this is wrong) reading something about how something like topographically related manifolds/laticies(?) mathematics is surprisingly useful for understanding many real world things, even the idea of creativity or idea synthases. To me I find that to be something which I do apply in how I think about a lot of things in the real world, but that might not be something you find as real world interesting.

Though I realize you probably meant you want something you can use in your day to day life, it's still a bit hard without knowing what you are interested in. A NEET who likes to dream up imaginary universes with physics and such is probably going to find more use with some kinds of applied maths, than others, and still other people will have other preferred fields.

Sorry if that sounds like I'm being condescending or stupid or anything, I'm not intending to be, I just think it might be helpful if you might clarify a bit more what you might find interesting or useful in your own life.


It depends on what you consider "interesting" applications. You should start studying basic mathematics (listed below) and then choose the one you like the most.

>Number theory

>Euclidean geometry

>(Naive) Set theory

>Analytic geometry
>Linear algebra

>Calculus of one variable

>Calculus of several variables
>Differential equations
>Partial differential equations

>Mathematical analysis

>Measure theory

These all are elementary mathematics. Set theory is used in every brach of mathematics (except category theory) so you must read at least one book about it.

>I'd like to study something challenging, but not feel like I'm stuck dealing with endless abstractions all the time

That's impossible. Current applications make use of advanced mathematical fields, like statistics, for wich you need calculus (of one and several variables and both differential and integral), mathematical analysis, measure theory and, of course, probability. A daily application could be financial mathematics, but in this case you'd only need high school-tier books.

But.. thinking about it, digital design, computer architecture and everthing about computers are really simple applications; you only need to know number theory, geometries (euclidean and analytic) and logic. Linear programming is also very simple, as it is elementary operation research.


I guess I should have talked a bit more about my background. I did an undergrad degree in engineering so I already covered most of the topics you listed except the last 3. Of course, this would be "engineering math", not "math math". I've forgotten most of it, but I wouldn't feel out of my element if I had to pick them up again.

I like the suggestion of statistics. I'll look into that.


statistics is not maths imo.

I had to study Measure theory before studying probability and it was the most abstract thing ever. I'm sure there's plenty of real world applications, but you won't escape them if you're going to study maths
And as you pointed out, engineering maths is not maths at all.


File: 1509223279262-0.pdf (2.14 MB, proofs_and_fundamentals.pdf)

File: 1509223279262-1.pdf (3.42 MB, [Douglas_Smith,_Maurice_Eg….pdf)

File: 1509223279262-2.pdf (3.59 MB, statistical_methods.pdf)

Even considering your background, I think you should read at least one book on logic, like Proofs and Fundamentals, by Ethan D. Bloch, which will teach you how to make proofs and consecuently, understand a mathematics book. Reading the first three chapters is mandatory, and reading the fourth and fifth one is desirable. Also you should read the first five chapters of A transition to Advanced Mathematics by Douglas Smith, Maurice Eggen and Richard St. Andre, because this complements the former, and clarifies some doubts that could arise in Bloch's book. Once you have read these books, you should be ready to read any book on mathematics (provided you know the background).

Now, statistics has many applications so there are many books on this topic whose approach depends on the student's needs: some are stupidly easy, and some are ridiculously difficult. Since you are an engineer, I think a good book could be Freund and wilson's Statistical Methods because it explains each topic like a recipe but also deduces the tools it offers you; it has nothing but the most basic results, with no tons of theory, just the exact portion.

>the most abstract thing ever
There are even more abstract branches of mathematics.


File: 1509225257489-0.pdf (1.04 MB, Probability essentials.pdf)

File: 1509225257489-1.pdf (4.17 MB, DeGroot M., Schervish M.-P….pdf)

File: 1509225257489-2.pdf (2.21 MB, probability.pdf)

Adding more books:
>Probability Essentials
Explains only the most basic theory and leaves the deduction of some important results to the reader. It has a few exercises, but can be too abstract.

>Probability and Statistics

I loathe this kind of books: it has hundreds and hundreds of paragraphs explaining a single concept and then thousands and thousands and thousands of numerical exercises that are special cases of only one general formula, making the study too boring.


This seems to have an approach among the former two: concepts explained in one or two paragraphs, examples, a few numerical exercises and abstract exercises.

You should take a look at these books and read the one you like the most, then you could read a book on statistics with more theory.


>the most abstract thing ever
Hyperbole my wizzie. Topology and group theory topped it but still, I couldn't make any practical sense of the little probability I heard from that course.

I was just pointing out that studying maths for applications might be an end goal, but the actual study of the beast will be all quite abstract.
t.failed third year maths and have to decide tomorrow if I try again or abandon mathematics for a few years or maybe even forever (at least learning it in uni).
Sorry for the blogpost at the end.




Wew I remember creating this thread. Didn't think it was that old.



Have you improved in math since then?


Any helpful stuff for someone who forgot everything from highschool?


Paul's Online Math Notes.


File: 1522590028725.pdf (9.99 MB, The Maths E-Book of Notes ….pdf)

This book is one of my absolute favourites that I still run through for refreshers. I've never found the retention of tricks and routines particularly easy to hold onto, ever since the earliest stuff in school.



I got a job, which ironically enough caused me to quit studying math and improving.


im having trouble with using rotation to calculate whether something is within field of view

basically i'm using the degree i'm rotated and the degrees of objects always rotated to face me. it is easy enough for a 180 degree field of view to just subtract 90 from my degree of rotation to determine if it's beyond my left peripheral, and adding 90 for the right side, but i don't know what to do when this gap jumps goes above/below 0 degrees because it goes from 350 to 10, or 10 to 350 in just a 20 degree rotation. using simple math this gap is 340 degrees, but i know it's only 20 degrees. but i don't know how to account for this jump

for example i am facing 0 degrees north, for something to be in my peripheral (if the objects are always facing me) it will be rotated between 270-90 degrees. i don't know how to deal with it the degrees around 0 and the restarting of its counting from 359 to 0 to 1

i have been trying for 2 days now to solve this, but i don't know how. it's like i need a new system that doesn't jump from 359 to 0 to 1 or vice versa. i dont think i described the problem well enough for anyone to understand. i don't get how to make the program understand 350' is only 20 degrees from 10' basically



Try using quaternions instead.


can't even do algebra, i don't think that's on the table for me unless i get seriously spoonfed the info

i think maybe i can read about modular arithmetic, it seems relevant from what my tiny brain grasped off the wikipedia page


So you want your program to jump from 10 to 350, through 0 instead of 20..100..300? Program as in code? Can you post the method and some test cases?

t. Js monkey passing by



To rotate a point around a quaternion you just need to use the formuls qpq^-1, where q represents the quaternion that is your rotation and p is your initial point to rotate. p^-1 is the inverse, or opposite rotation. -θ along the axis of ration. All of this his is explained in the wikipedia article here:


It may be a bit hard to read if you do not understand it, but the info is all there.

Quaternions can be thought of as a representation of rotations in 3d space. For example a rotation 30 degrees around the x axis in quaternion form is cos(30)/2 + (i + 0j + 0k) sin(30/2). Of course you can do a rotation among any arbitrary axis, which is represented by i,j, and k.

To code this you will also need to know that i^2 = j^2 = k^2 = ijk = -1. You can use algebra to derive the identities, jsut be sure that you do not assume that ij = ji, it doesn't. Here is an example of what to do to find out what ik equals.

ijk = -1
ik = -1/j
j*(ik) = j(-1/j)
jik = -j/j (can reduce to -j/j = -1, but that gives us nothing useful!)
ik = j/(j^2)
ik = -j

Or you can do what everyone else does and just use these rules:

ik = -j
kj = -i
ji = -k
ki = j
jk = i
ij = k


it's already modular 360 after i normalized it to positive-only values, so it does that, goes from 350 to 0 to 10, and when rotating the other way it goes 10, 0, 350. but i don't know how to use basic arithmetic for something already using modular arithmetic. with basic arithmetic the difference is 340, in mod360 it is only 20 if what i grasped from reading that wikipedia page on modular arithmetic is correct

for a 180 degree field of view basically i take the degree of rotation of the player, this can be between 0 and 359, and subtract 90 to obtain the boundary for rendering the left side of the screen, and add 90 to do the same for the right side of the screen. if an object facing the player has a degree of rotation between those values it renders

>Program as in code?

>t. Js monkey passing by
well i'm using a visual programming thing but i don't actually have code for this at this point, just the relevant values i decribed. if you can word something with javascript i can probably understand it and use the same javascript math expressions

thanks for the help, i will try and decypher it. i don't know if this being 2d matters or makes it simpler also


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Let's see… I'm not sure whether or not I understood your problem, but I think it is something like this:

Let A be your viewpoint, and let p and q be the lines demarcating your field of view (red shaded region) and Θ the angle between them. It seem there are a set of objects in the whole field, some of them facing you, like object T, and you want to know whether or not it is in your field of view afer rotating, say, φ degrees (positive or negative) whit respect to A, right?

I think images explain themselves.



I didn't know you were doing things in 2D. Quaternions are for 3D space, regular complex numbers will do in 2D, but I dont think your problem even requires that. Seems like your problem is that you are assuming that your left side is always going to be less than the right side, which you found out is not true.

Just add a check to see if right side limit is less than the left side. If this scenario is true, check for values on the interval [left side limit, 359] and [0,right side limit]


I'm probably not the best guy to say this since I'm a highschool dropout who's never gotten past basic algebra

Same here. Someone help me not be retarded.


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Is there a method, book or theory which can help me setting prices? I mean, how to derive a price for a product given some data (like surplus, expired and defective items, quantity of items I purchased at a given price and quantity of (the same) items I purchased at a new price, etc.). I'm looking for some mathematical foundation to do this, but most of what I've found are explanations like "take into account market, promotion, objectives and other subjective stuff with no mathematical foundations and then use this shitty strategy based on what you think could be a good choice". I want to read something with real maths, based on statistics, calculus, mathematical finance and serious observations.

Assets pricing seems to be the most resembling stuff, but I'm not sure whether I should spend my time studying it as my only purpose is manage a small grocery store.


this may be?
not a math person, never used it, probably 100% wrong
Seems to be useful in the world of wall-street finance, soviet planned economies and shelf-space optimization for department stores though.


Thanks, wizzie, but no, it does not applies here.


what a shame, good luck wizzie.


I don't think there is a specific equation to calculate that, so it would probably be best to use some form of machine learning. If you have (or can make) some exemplary data showing 'good' examples of pricing and some bad examples of improper pricing, it can turn the data into a model, and gradually improve itself as you give it feedback. Although mathematical, there is ready-made code out there that you can essentially throw data into


Reading these threads just makes me sad honesty because I used to be a math(s) student at university but I was too dumb for it and too depressed so I just dropped out and became a useless fucking NEET. It's a shame because I still recognise math(s) as really beautiful and I'm sort of jealous of all you guys in this thread who actually have the motivation to make your way through these textbooks and actually reach an understanding of this shit, or even gain a qualification from it.

I remember this. Just felt like endless examples.


Could you tell me more about linear programming and the vehicle route problem?


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Well… integer linear programming is not a topic I'm versed in, but since I'm currently a neet, I could explain it to you if you allow me a few weeks (yes, I'm too lazy, and besides that, it's been a couple of years since the last time I studied it).


File: 1527424925173.png (51.19 KB, 1072x557, 1072:557, linearprog1.PNG) ImgOps iqdb

We had to do something really similar recently.
It was about a traveling merchant or something like that where you have 5 locations you have to visit and in the end have to return to the first one.
The problem starts with having locations and "distance" or values it takes to reach another from each point in a matrix, kinda. I'm not the brightest I'll just write down what we did.

I just made this from what I remembered, it might be flawed, or not at all what you are looking for.
The final part of the tree graph I tried to take pictures and write it down by hand, hope its readable.


You basically start from location 1. and branch into all available locations. Choose the lowest distance one (if two identical, have to check both). The repeat for the next one.
The one on the picture is already starting from 1->2 because as you can see in the table that is the lowest distance one. (Distance was 0 while the others were 2, 2, 7)


By the way I just randomly read your post there asking for this, I'm not the one you asked it from. This is the first time I contributed to this thread.
Sorry if it was irrelevant or wrong.


I hope this is the current maths thread. I was wondering what other wizzies think about number theory?


Wow, I made this thread ages ago. This was back when I was in college. I graduated and got a job since creating this thread.

Surprised it isn't dead yet.


I remember writing this post. 3blue1brown actually recetly uploaded a video on quaternions. Love his videos because he has a nice way of explaining things. There is no rigor, but his videos provide you with the intuition needed to completely understand the concept, which is something almost all textbooks lack.



Number theory is the king of autismo studies, I think. It is only useful on accident.


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I found out about it when I first heard about RSA and am now focusing on it since I hope to become a cryptanalyst.


This is an interesting talk about the sphere.


Second part.


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I want to study math. I know a little and I did fairly well on the GED. Which courses do you suggest I take on Khan Academy, and in what order?


i am mathematically retarded.


File: 1545705624630-0.pdf (2.14 MB, proofs_and_fundamentals.pdf)

File: 1545705624630-1.pdf (1005.65 KB, Morris D.V., Morris J - Pr….pdf)

>Khan Academy
Well… I'd never suggest someone to take courses on that page, specially if that person wants to seriously study maths. A better option is studying books on how to do proofs, like
>Proofs and Fundamentals, by Ethan D. Bloch
>A Transition to Advanced Mathematics, by Smith, Eggen, and Andre

They both teach you the basics on logic, how to argue, and the basic proof methods. Then you'd be ready to study something more challenging, like all the books writen by Ellina Grigorieva or those by Titu Andreescu.


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Does anyone have a Mathematics self-study resource guide similar to pic-related?


I disagree with you about that. I used Khan Academy several years ago to help prepare me for going back to university, and it helped me immensely. I learned almost everything I needed for first year computer programming, a second year statistics course, and some first year math courses, plus first year physics. All without paying any money. I know there are better sources out there but you're likely to have to pay for them.


Is all that a part of some combo program, or did you choose every course yourself, knowing what to learn beforehand?


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I didn't technically know everything I was going to learn before hand, I just knew the names of the courses and looked up recent syllabi hoping it would be the same. So I did take a bit of a gamble, but it was Intro to Programming on Khan Academy and that's why my courses said they were as well. But I didn't mean to say I learned everything I need to know from Khan Academy. For programming I did, but the others it was more of catching up from high school.


It doesn't matter anyways though because I ended up dropping out halfway through my second year because I couldn't keep up with the material and couldn't understand it well enough and quickly enough. At that point there was nothing Khan Academy or any other website could do, trust me I googled endlessly.


Huh. Took intro online courses on math and physics when finishing high school too, which rehashed all the school program and started the first year university topics. And a programming course. The website was a local startup and they were offering these for free at the time. Attended the university for three semesters too, also got lost and dropped out. I think the mage who wrote against the course advice meant that this medium ties you to itself, and after that year and a half you can't find your own way, expecting more info to appear in the same format. From the university programming classes, I understood that just like online, the staff lectures and exercises leave you with a very incomplete, barely comprehensible version of what they're supposed to teach, while saying fuck em and reading textbooks and open source made these classes the only where I succeeded, along with English. Unfortunately in the math I wasn't that interested, didn't do this and screwed up.


What kind of career path can you expect with a Mathematics degree (from a reputable UK University)? Is it even worth getting a degree?


If you stop at a bachelor degree there isn't much. Some finance banking shit. Add some physics and there's meteorology. High school teaching if you add some teaching certificate (not sure how UK does it). A lot of people learn programming but most mathematics is not really applicable


AI/machine learning is very hot right now but you need computer science knowledge with it, and it is mostly stats


if you understand math at a high level its just kinda comfy to see what is happening. thats how it works for me anyway


Any other wizzies working through Spivak's calculus at the moment? Would be nice to discuss things every now and then to bolster motivation if I ever get lazy.
Not that I intend on relying on others, just figured it would add to the fun. Trying to do all the problems unless it is explicitly mentioned in the text that I should probably wait for some.
I'm taking a regular non-proof based single variable calculus course at community college as well as basic stats, but obviously that's not too difficult or time consuming to prevent me from doing other things.


As long as you apply it then you can make a good living doing pretty much anything. Learn how to program well and learn a particular domain that you are interested in applying mathematics to or that relies on mathematics.



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How about we try solving some Putnam problems and post our solutions?


Has anyone went through the book "How to Prove It: A Structured Approach"? I'm teaching myself how to solve proof's and I could barely get through the introduction. I don't know if I'm dumb or if this is expected when getting into proofs. Has anyone had a similar experience?


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Is Ted Kaczynski right about math not being a worthwhile activity?


kaczynski has never been right about anything in his whole wretched life.


That sounds very mundane and… immature for a "genius".


This reminds me of a wikipedia article I read on mathematic aesthetics https://en.wikipedia.org/wiki/Mathematical_beauty

Apparently most mathematicians like math for the minimalist, raw beauty of formula's and numbers.


I'm this poster. I decided to drop the book since it was going to slow and I'm going to jump right into Rosenburg's Discrete Mathematics and It's Applications. Skimming through the pages and comparing them to How to prove it, they are teaching the same things, just more in depth, which is how I prefer it. I might go back to how to prove it later for more practice but It's unlikely because of time constraints and because I'm mainly studying them to get a better grasp on Introduction to Algorithm's. If I can adequately understand Algorithm's then I will focus more on that than any unnecessarily higher level of math needed for the goal of improving my programming.


I'm working through spivak very very slowly (usually I do one big burst for an entire week and then forget about it for a month), so far I'm on chapter 3 after 3 months. The problems are very difficult, the average amount of time it takes for me to finish a single problem is ~30-45m, some taking an entire day to figure out. Like many math books, I found that somehow applying the solution to the last problem often helps a lot, and the biggest issue is usually trying to understand what the question is asking you rather than trying to prove something (when you don't even know what to prove).



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Let's discuss Max Tegmark. Did anyone know him before they read my post? The most important thing I would like to talk about is one postulate that goes like this: "all mathematically consistent structures exist in reality." This statement follows from the principle that the universe is entirely composed, "woven" from mathematics. Now, I have a question. Does this mean that somewhere there must be a universe in which an exact copy of me was lucky enough to live in a virtual reality with full immersion, in which the world perfectly adapts to the needs of each user? In the film Matrix, the virtual reality was the same for all people, but I'm talking about a private and individual, one that was created only for my copy. Because why not? Imagine what will be possible in a thousand years. I'm sure that computers and AI will be so advanced that everyone will be able to leave this world for the sake of diving into a simulation in which they will get everything they want. I even think that people from birth will immediately connect to the simulation, because this shitty real world will not be needed by anyone.
Generally speaking, does this mean that there are all sorts of universes that can only be imagined? I do not quite understand what do you mean "consistent". Contrary to what? After all, in another universe, there may be other mathematics that will not be comparable to our own, then what is it about? This is a really interesting point. I first started talking about virtual reality and my copy, because I am very dissatisfied with my life and would like to know if my second self (or third, or fourth, or fifth) exists in another world that gets everything it wants.
I came here because I don't understand mathematics at all, and I need the opinion of someone who has a deep understanding of the subject and can tell me how right Max Tegmark is.


How many here tried to study mathematics at uni and failed out? Count me in, I wonder who else went through the same experience.
It was just too intellectually hard for me personally.


STEM in university, at least at the undergrad level, is all about doing problem sets over and over again until the process of solving them becomes automatic and ingrained. The assignments and exams tend to be recycled from year to year with minor variations, so if you can confidently solve all previous problem sets, you will very likely do well on the course.



I think a major issue when it comes to "pop math" is the assertion that the universe is based from mathematics, rather, it's predominantly the other way around. We've taken reality as an inspiration for conjectures, theorems and sub fields. Euclidean geometry arose from observations of the real world, where you can consider geometric objects as being a set of points within a vector space having 3 orthonormal bases. From there came Euclid's postulates, and we discovered that the parallel postulate doesn't actually hold in spherical/hyperbolic geometries (like Earth) somewhat recently (it took Gauss, arguably the smartest mathematician in history, to figure it out). The need for calculus arose mainly because physicists wanted to explore the notion of instantaneous rates of change. Likewise Hilbert spaces (generalizations for euclidean spaces) were developed so that QM could work. Formulae like e^pi*i, for example, are more a consequence of notation rather than some fundamental property. Like 3blue1brown explains in the video, a substantial amount of mathematics is a consequence of reality, and there isn't much of a point to theorems that aren't useful (of course many "useless" ideas become incredibly useful in the future, like graph theory).


Perhaps but the amount and difficulty of these problem sets/past exams were too hard for me to overcome(topology and number theory kicked my ass).
Why argue about this though?


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Damn, am I dumb. Even the first chapter of "Basic Mathematics" is too complicated for me. I'll have to start relearning arithmetic because I don't remember how negative numbers work nor how to multiply effectively, seeing as I just use a calculator for everything these days. According to Khan academy that material is pre high school level. Looks like I have a long way to go.


I accidentally came across the following statement in one of the encyclopedias, but the proof was not there: adepts of abstract algebra can prove that 2+2=1. Can someone explain to me how this happens?


I don’t know about 2+2=1, but I’ve seen some for 1=2 and other such stuff. Usually things like these come down to disguising an error mathematical logic. For example disguising a division by zero by using variables, or applying both a 2nd power and square root to a negative value and assuming it comes out unchanged.



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I'm trying some competition papers online, the first question is pic related. The answer says that f(x)=1 for all x in Q>0 (aka f[x] = x^0), but surely that's not right? Can't f(x) also be √x? That leaves x√y = x√y, unless I'm missing something.


It can't be √x because the values of f(x) should be rational.


Oh, that's disappointing. God, I'm such a retard.


A Proof That The Square Root of Two Is Irrational


I took an undergrad graph theory course last semester and absolutely loved it! I think I want to do research projects in the topic to consider whether I'm fit for it at grad school. We followed Introduction to graph theory by Douglas West in class, it's an amazing book. You can also check out Reinhard Diestel's book, it's also nice.


Oh this was an IMO problem: https://www.imo-official.org/problems/IMO2018SL.pdf

Pretty neat solution and relatively easy to follow. I would have never been able to come up with that myself though, and considering it's an IMO problem I doubt you should get frustrated if you overlook something.


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i hope to read them one day.


Challenge accepted.
Seriously though, thumbed through Bourbaki and seems pretty decent. I might read it.
I can't imagine who in his right mind would read principia though. It looks like it was written for a computer (it kinda was). Perhaps one could reframe the whole work to be input to Coq or something like that. Of course imstead of sets it would require Type theory, which would be an improvement as a matter of course.
Makes me wonder about the nature of such work. Is it to be read by humans? Or perhaps only incidentally the way a computer program is?


Hey, Vsauce. Michael here.

Skeletons are scary and spooky, but you know what else is? Niggers.

According to the U.S Justice Department, in 2006, 32,443 succubi of Caucasian origin were raped by men of African origin.

That same year, the number of African American succubi raped by Caucasian men… was… zero.

In fact, 90% of all interracial crimes in the U.S. are committed by blacks… against whites…

So what if all blacks were to… suddenly… disappear from the U.S.?

Murder would go down 49.7 percent, welfare recipients would go down 40 percent, SAT scores would go up about 100 points, the average IQ would go up 7 points, and AIDS victims would go down a staggering… 67… percent.

Significant changes for race that only makes up 13% of the population.

In biology, races are distinct genetically divergent populations ‘’within the same species’’, with relatively small morphological and genetic… differences.

Populations can be described as ecological races if they arise from adaptation to different local habitats or geographic races when they are geographically isolated.

However, if sufficiently different, two or more races can be identified as… sub… species.

So how long do two races have to be isolated from one another before they're considered separate species?

Earlier this year, Archaeologists found artefacts in a cave on Western Australia’s Barrow Island dating back more than 50,000 years, making it Australia’s earliest known site of human occupation.

In contrast, the domestication of the dog began just 15,000… years… ago.

This distance, some several thousand miles between the mainlands of Australia and Asia also caused another kind of distance.

Genetic… distance.

Genetic distance is a measure of the genetic divergence between species or between populations within a species.

Dogs and wolves have been found to have a shorter genetic distance between each other than Negroids and non-Negroids. The distance between Eurasians and Africans being even larger than Homo Sapiens and Homo… Erectus.

Well, it kinda makes sense. I mean, can two races with diverging skin tones, eye colors, behavioral patterns, intellectual and athletic capabilities even be considered the same subspecies? Is it even a question? Is it even a question you're… allowed to… ask?

In October 2007, geneticist James Watson, best known as one of the co-discoverers of the structure of DNA, and the familiar double-helix model we all had to learn in High School, was lambasted by the scientific community for a response he gave in an interview regarding the divergence of intellect between geographically isolated populations.

"There is no firm reason to anticipate that the intellectual capacities of peoples geographically separated in their evolution should prove to have evolved identically," James wrote. "Our wanting to reserve equal powers of reason as some universal heritage of humanity will not be enough to make it so".

The response resulted with a suspension of Watson's administrative responsibilities and was forced to… cancel… his book tour. And tragically, in 2014, Mr Watson auctioned his Nobel Prize medal he won in 1962, stating that "no-one really wants to admit I exist".

Harsh. But people tend to react this way to ideas that go against their own personal world view and… cognitive… biases.

Confirmation bias. It's the tendency to search for, interpret, focus on and remember information in a way that confirms one's… own… preconceptions. This effect, stronger for emotionally charged issues, warps your interpretation of data in a way that keeps you from… being… wrong. There is something fundamental in our minds that makes us hesitant to question ideas that we've… already come to a conclusion to.

We all do it. But who could blame us? That's just part of being a member of the human… species.

And as always, around blacks, never relax.


I enjoy this pasta.


Why do whites never have ghettos?


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Is this image trying to say that S is not a subspace of R³ because these reasons?:
>S is a set in which element 1 is any arbitrary real number, element 2 is any arbitrary real number squared, and element 3 is zero.
> multiplied by c is not necessarily the square of an arbitrary real number. In order to be in the set S, the second element must be a square of an arbitrary real number. Because the second element, when multiplied by an arbitrary real number may or may not be square after the operation, it may fall outside of S and so under rule one it is not a subspace of R³.
> plus another arbitrary real number squared is not necessarily another square. Because of that, it may fall outside of the set S and thus must not be a subspace of R³ under rule 2.
The image is a little ambiguous and my math skills aren't very good, so I want to be sure I've got it correctly.
When I say arbitrary real number or a I do not mean that there is a consistent value between them. a in the first element could or could not be different from the a in the second element, and so on. Just being clear.


>>S is a set in which element 1 is any arbitrary real number, element 2 is any arbitrary real number squared, and element 3 is zero.
I meant to say that S is the set of vectors in which element 1 is any arbitrary real number, element 2 is any arbitrary real number squared, and element 3 is zero.


Correct. And if the a in the first and second places are not to be the same, use different variables. Use a and b instead.


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Studying math by programming


Ehm… that's not stuyind math, that's studying programming, or, at most, numerical analysis, for wich you don't need to understand the maths behind the algorithms.


I have a very poor understanding of 3x(n>1) vectors and matrices in any other context than as an array of numbers that may or may not represent points and those points may or may not be positions. For 3x3 vectors I can intuitively imagine constructing something like a right prism or triangular pyramid, but more complex ideas are difficult to imagine in any other way than "yeah it makes sense."
So, what are some interesting concepts that can be represented by 2 or more 3-dimensional vectors?



Yeah it's the resource I used. It's good, I think I'm just too much of a brainlet for linear algebra. The calculations and formulae themselves are trivial but I can't visualize anything but NxM matrices where N<4 or transformations that aren't from ℝⁿ to ℝⁿ. It's making going any further a real slog when I have to imagine everything as abstract numbers, especially n to m-space transformations when they're used in nearly every problem.


It is kinda hard to help if you do not provide specific examples. Matrices are basically nothing more than arrays of vectors. What those vectors represent varies depending on the use.

I would venture to say that most people do not understand higher dimensional transformations, though. You know know what works and apply it. Only thing that is easy for me to grasp about dimensions > 3 is that they contain more space per unit cube.


As an example imagine a 3x3 matrix multiplied by a 3x2 matrix. The product is a 3x2 matrix. What is a way to physically interpret the loss of a vector and preservation of dimensionality? 3blue1brown explains that this can be thought of as a 2-dimensional plane mapped in 3 dimensions, but that doesn't explain the process of the transformation. What is a physical process by which a concept that can be thought of as a matrix with 3 3-dimensional vectors be transformed into a matrix with 2 3-dimensional vectors? It can't be projection, because projection would lose a dimension while preserving vectors.
Another example would be multiplying an MxN matrix by an Mx(N+z) matrix. How can a transformation like this be interpreted as a physical phenomenon? I understand they can be interpreted however is necessary for the application, but as I said I'm not okay with thinking of matrices and vectors as arrays of numbers and transformations as arcane functions to use on those arrays to get more useful numbers. I'm not really asking "what are matrices/vectors", but instead "what is at least one way that these transformations can be explained as a physical phenomenon." Knowing some real physical phenomena that can be represented as a linear transformation that increases or decreases the dimensions or vectors of something will make intuitively understanding linear transformations a lot easier.


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I think it is kinda hard to understand because matrix notation is backwards. You can think of matrix multiplication A*B = C as a transformation of the unit vectors in B's space. A tells you how to transform the unit vectors in B's space (standard notation is to every unit vector transformation be represented by a column. col1 = X_t, col2 = Y_t, col3 = Z_t), and the result C is where the vectors are transformed once you "stretch" the unit vectors by A. Example shown in my screenshot. That is me stretching the X unit vector by 2 so all of the X values are multiplied by 2.

If A is a 3x3 matrix and B is a 3x2 then B are the two points you want to transform using transformation matrix A. If A is 3x2 and B is 2x2 then A is describing which plane the 2d points in B should go. 2nd attached pic is me putting the points on the XZ plane. 3rd is putting the points on the XY plane.


Yeah, I can understand the mechanics of linear transformations when used in rotations, translations, projections, or other abstract numeric operations, but seeing them as physical processes is the part I am having difficulty with. I suppose by your answer it's really only going to get more abstract from here, isn't it?



In that case the matrices aren't really anything special, they are just a way of rewriting the equations in your system, if the system happens to be linear. You can probably think of them as a stretching and squishing of vectors in the state space, but what that actually means is entirely dependent on what you are trying to model. A lot of the time the columns do not actually mean anything other than that is how it must be for your equation to be represented in matrix form, though. I'm sure someone who is crazy about number theory could probably ramble on about it is related to pascal's matrix or some other obscure shit, though.


Yes. It's called recreational mathematics.


I really am delighted by aspects of information theory and often basic number theory. It fills me with a sense of wonder learning about some of this stuff. Like I am learning actual fucking witch craft; Being able to pass along secret knowledge. Being able to touch the boundaries of order right up until the bounds of noise. There's so much useful stuff hiding under the surface and its all so elegant.

The sad part is practically none of it is taught in schools. Parts of it might be found in undergrad degrees. But mostly maths in undergrad degrees are about turning you into a good little robot wagie who can churn out mundane results like a calculator. In contrast: research math is all about abstract bullshit that will never see the light of day and often has contradictory 'results' anyway. So while you might cover some cool stuff there. It's definitely not the focus. The advantage you have doing this on the side is being able to learn the actual cool shit.

My favorite result is 1 + 1


File: 1617801015185.pdf (79.21 KB, lebesgueMeasure.pdf)

I wrote up some notes on Lebesgue nonmeasurability. I think I tried to make a separate thread about it some time ago. I'm still stuck thinking about why it's not a bigger deal than it is. In my mind it should be up there with the other 'popular' math paradoxes–or at least up there in popularity with stuff like Banach-Tarski or the Klein bottle.


Thanks, that was useful. I'd been wondering for some time what Lebesgue integration was, and never bothered looking into it.


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Practicing writing proofs. Problem is from Mathematical Proofs: A Transition to Advanced Mathematics, Fourth edition, by Chartrand, Polimeni and Zhang.


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Made it simpler.


This one is hardly readable, but it all boils down to a matter of style. I understand your phrasing in term of P(x) and Q(x) is a didactic device from te book, but you ought to state it more directly using prose.
The second step in your proof obscures the links between the data. Instead say x^2 lies in the interval (0,1) and -2x lies in (-2,0).
Lastly, on step 3, a double negation is always harder to read than a positive statement.
Actually, you can't use k for both numbers because you're constraining your proof for consecutive numbers (eg 4 and 5), so the right hand only
applies to such pairs of numbers.


Still, he's figured out how to use LaTeX and align things well, and he's actually got the idea of a proof down in that the proof is technically correct. For someone starting at this, he's starting out pretty good.

You're right about the second point on >>57945 , this proof is assuming consecutive integers.


(I just remember when I was first starting this and got shat on to high heaven and I pretty much just quit the field entirely, so I wanted to add some positive points)


Oh I do not mean to discourage anyone. But i myself found myself wanting someone to critique my proofs lest I miss anything important.


This isn't a proof as it does nothing to demonstrate that all even integers are equal to 2k where k is some integer, and the same for odd integers and 2k-1


This is 500 pages of chemical kinetics theory, examples, and data. I want to do my capstone on something similar, so I will dig through this to understand the basic theory, and retrace the author's steps in terms of math/computer stuff


It's a bit like me. I was also trying to study maths for a while. I didn't put enough effort and I had to quit.


The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy


>it took Gauss
wasnt that Lobachevsky?


Well, it's assumed that Gauss already knew about it since when Bolyai developed it and his father, Gauss's friend, sent a letter to Gauss to explain about his son's discovery and Gauss reply was not kind:
"To praise it would amount to praising myself. For the entire content of the work…coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years."
And that he already knew about it. Gauss was an arrogant person, btw, he didn't even wanted any of his sons to do mathematics because he knew they would never be better than him. Anyway, Lobachevsky is another person who developed non-euclidian geometry(around the same time as Bolyai) and it's believed that Gauss after learning Russian(at an old age), decided to read Lobachevsky's works which he became interested(it's said that he learned Russian to be able to read this very work, but I have read a russian paper that disagree with that and has valid arguments using Gauss's letters and journal).
Now, the interesting part is that by the time they accepted non-Euclidian Geometry, they realized that they have been using non-Euclidian Geometry for a long time in astronomy: Spherical Geometry. Which is why for hyperbolic Geometry we know who developed it: Bolyai and Lobachevsky; for Spherical Geometry, you are not going to find "the man who started it" since it has been developed since Ancient Greek, they just never realized it was non-Euclidean Geometry.


ig it's complicated like everything in life
btw are you this guy >>51927?


Nope, this >>60183 was my first post in this thread.


>The need for calculus arose mainly because physicists wanted to explore the notion of instantaneous rates of change
Not really. Calculus was already studied way before this, the concept of the area under a graph and the tangent to a point in a graph were both studied for many years. It just so happens that it could also be studied for instantaneous rates of change because it turned out that "finding the tangent to a point" and "the average velocity at a time t as t goes to zero" are the same thing.
>Likewise Hilbert spaces (generalizations for euclidean spaces) were developed so that QM could work.
Hilbert Spaces existed way before QM, it makes no sense that "it was developed so that it would work for a field was not even born yet".

It's not that the universe is based from mathematics, it's that using mathematics is the best way to calculate our observations of the world. Many fields of mathematics were(and are) developed without any relation to the real world, until someone almost 2 centuries later found out he could use groups in QM too, for example.

>there isn't much of a point to theorems that aren't useful

This is flawed since you never know when something is useful. Maybe we shouldn't have studied non-Euclidean Geometry since Euclidean Geometry worked quite well and there was no use for non-Euclidean for a long time.


I'm not an historian of mathematics, but I think your statement:
>Calculus was already studied way before this
only applies to integral calculus. As you pointed out, integral calculus (in a primitive form known as "the method of exhaustion") was already been studied by Eudoxus and Archimedes in ancient Greece, way before Newton and Leibniz. But before Newton and Leibniz, the problem of tangents was approached through very different and ad-hoc methods that have very little to do with our present-day notion of derivative. The idea of a difference quotient really is based on the definition of average velocity (the difference in the dependent variable represents distance, and it is divided by the difference in the independent variable that represents time) and as one takes the limit one obtains instantaneous velocity. It's really hard to deny that calculus was influenced by the study of the physical world. It is no coincidence that Newton worked both on calculus and physics.


I've just been realizing how much of group theory is applied in solving the Rubik's cube. So much so, that I really wonder why subgroups, generators, commutators, centers, and centralizers are not discussed in terms of the Rubik's cube.


Also, while I'm posting in this thread, I feel I should mention…I used to spend a large amount of time studying mathematics, and got burned out when I realize my return for the time I put in was so small. Now I feel this odd sense of jealousy (?) when I see others studying it. I've been trying to get over this, but don't know how.


Group theory goes well with many puzzles, anon. Many puzzles allows you to make a movement and another that gives you back the original position(so you have an inverse to every move), composition of moves are allowed and give the same result no matter the other, the identity is usually no move at all and so on. What is sad is the lack of books and professors teaching group theory in terms of puzzles. Mathematics is, in one way, solving puzzles and so it would go so well.


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> composition […] give the same result no matter the other [sic]
If only that were true in the general case.
> The Rubik's Cube group is non-abelian as composition of cube moves is not commutative; doing two sequences of cube moves in a different order can result in a different configuration.


Yeah, it's jealousy. How do I get over this jealousy?


I spent years studying it too and got no return. However, I just see people as foolish for wasting their time on it since there's a 99% chance they will also get no return. Your jealousy is odd to me


Yes, I was wrong. I was thinking about abelian group for some reason. My mistake.


Here's a small challenge. The solution has already been posted on another site, but if you've seen it please don't spoil it for others who might want to have a crack at solving it themselves.

Start with a 3x3 tile. At each level take the result of the previous level and place four copies of it in a 2x2 matrix, with a gap of one third of a tile between them. Then add one new tile in the center, which will overlap each of the four blocks for one third of a tile. Here's the result after two iterations:
┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐
│     │ │     │ │     │ │     │
│   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │
└───┤     ├───┘ └───┤     ├───┘
┌───┤     ├───┐ ┌───┤     ├───┐
│   └─┬─┬─┘   │ │   └─┬─┬─┘   │
│     │ │   ┌─┴─┴─┐   │ │     │
└─────┘ └───┤     ├───┘ └─────┘
┌─────┐ ┌───┤     ├───┐ ┌─────┐
│     │ │   └─┬─┬─┘   │ │     │
│   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │
└───┤     ├───┘ └───┤     ├───┘
┌───┤     ├───┐ ┌───┤     ├───┐
│   └─┬─┬─┘   │ │   └─┬─┬─┘   │
│     │ │     │ │     │ │     │
└─────┘ └─────┘ └─────┘ └─────┘

Repeat this process indefinitely. At each level consider the ratio of the area covered by tiles to the area of the minimal square enclosing that level. This enclosing square is simply the one determined by the outermost tiles.

The question is: does the series of ratios converge, and if so what is the limit?


Let's see if the third level fits into a wizchan post.
┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐
│     │ │     │ │     │ │     │ │     │ │     │ │     │ │     │
│   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │
└───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘
┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐
│   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │
│     │ │   ┌─┴─┴─┐   │ │     │ │     │ │   ┌─┴─┴─┐   │ │     │
└─────┘ └───┤     ├───┘ └─────┘ └─────┘ └───┤     ├───┘ └─────┘
┌─────┐ ┌───┤     ├───┐ ┌─────┐ ┌─────┐ ┌───┤     ├───┐ ┌─────┐
│     │ │   └─┬─┬─┘   │ │     │ │     │ │   └─┬─┬─┘   │ │     │
│   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │
└───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘
┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐
│   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │
│     │ │     │ │     │ │   ┌─┴─┴─┐   │ │     │ │     │ │     │
└─────┘ └─────┘ └─────┘ └───┤     ├───┘ └─────┘ └─────┘ └─────┘
┌─────┐ ┌─────┐ ┌─────┐ ┌───┤     ├───┐ ┌─────┐ ┌─────┐ ┌─────┐
│     │ │     │ │     │ │   └─┬─┬─┘   │ │     │ │     │ │     │
│   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │
└───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘
┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐
│   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │
│     │ │   ┌─┴─┴─┐   │ │     │ │     │ │   ┌─┴─┴─┐   │ │     │
└─────┘ └───┤     ├───┘ └─────┘ └─────┘ └───┤     ├───┘ └─────┘
┌─────┐ ┌───┤     ├───┐ ┌─────┐ ┌─────┐ ┌───┤     ├───┐ ┌─────┐
│     │ │   └─┬─┬─┘   │ │     │ │     │ │   └─┬─┬─┘   │ │     │
│   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │
└───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘
┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐
│   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │
│     │ │     │ │     │ │     │ │     │ │     │ │     │ │     │
└─────┘ └─────┘ └─────┘ └─────┘ └─────┘ └─────┘ └─────┘ └─────┘


File: 1657512976955.png (3.41 KB, 386x257, 386:257, ClipboardImage.png) ImgOps iqdb

is it 2/3


File: 1657533576404.png (113.13 KB, 1477x1499, 1477:1499, level4.png) ImgOps iqdb

Correct. Do you have a proof sketch? Or did you eyeball the first few ratios?


File: 1657568103824.png (Spoiler Image, 4.06 KB, 348x359, 348:359, ClipboardImage.png) ImgOps iqdb

just a guess, i drew them in paint and then selected the black pixels and looked at the areas


File: 1657568537700.png (Spoiler Image, 39.67 KB, 454x738, 227:369, ClipboardImage.png) ImgOps iqdb

this is maybe a more acceptable method though


File: 1657570186144.png (398.7 KB, 2958x2971, 2958:2971, level5.png) ImgOps iqdb

>>62378 >>62379
That's pretty neat! Looking at the ratios and seeing them stabilize is a valid method for obtaining the value of the limit. (R's completeness gives us convergence of Cauchy series.)


File: 1657836872148.png (52.87 KB, 1035x3000, 69:200, ClipboardImage.png) ImgOps iqdb

i dont know the proper terms for this stuff but these shapes i will call racetracks

any configuration is allowed except a 2x2 filled in space, this keeps them looking like 'tracks' or paths. is there a rule that determines for any given racetrack whether or not you can visit all the tiles while avoiding tiles you already visited? and if not, how many tiles are left over that it can't visit using the most efficient path

im thinking it has something to do with the number of 'holes' or islands in the track, and the number of tiles with 1 neighbor, 2 neighbors, 3 neighbors, 4 nieghbors. like some kind of ratio between them? i also think by describing the shapes as # of holes, and # of neighbors, you can build different looking racetracks but as long as they have those in common they are functionally identical. maybe some neighbor amounts are more important then others, im not sure


oh the colors of the stuff just visually reprsent tiles with 1 neighbor (yellow), 2 neighbors (green), etc. and the numbers below the shapes are for the neigbor tiles, holes, and bad tiles that cant be reached. i thought maybe by coloring them the solution would jump out, but it hasn't



good lord i dont understand anything in those articles, time to scrap this idea


File: 1657913330189.jpeg (62.17 KB, 594x516, 99:86, images.jpeg) ImgOps iqdb

Souds like the problem of the birdges of Königsberg [1], also related to pic which was all over the internet a few years ago.

[1] https://en.m.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg


I solved this with wizards 8 years ago. It's possible, you just have to think "inside the box" and in terms of mirrors


File: 1657913883850.png (194.78 KB, 594x516, 99:86, ClipboardImage.png) ImgOps iqdb

i remember than on old 4chan long ago, i recall the best meme answer was one gigantic line


File: 1657918029566.png (16 KB, 433x155, 433:155, IMG_20220715_154411.png) ImgOps iqdb

I've been trying to hack this since it was posted, but I have to admit I'm a gigantic retard and I haven't got anywhere near. The most far I've gotten is a big complex expression (pic related). I thought O was getting near but that big square at the denominator is beyond my skill. Back to the drawing board I guess? If I keep going I'll either have to juggle polynomial coefficients and take the limit to infinity, ie a fucking mess.


File: 1657948833736-0.jpg (Spoiler Image, 1.15 MB, 1944x2592, 3:4, IMG_20220715_234246.jpg) ImgOps iqdb

File: 1657948833736-1.jpg (Spoiler Image, 1.04 MB, 2476x1858, 1238:929, IMG_20220716_000437.jpg) ImgOps iqdb

File: 1657948833736-2.png (Spoiler Image, 133.68 KB, 1222x759, 1222:759, IMG_20220716_001856.png) ImgOps iqdb

Nevermind, I solved it. Turns out I was on the right path, it just took some number juggling. I also used the data from >>62379 to verify my formulas, and a bit of lisp to do the tedious work. Here's my proof, I rushed the last step, I just verified it with lisp.


File: 1657971859477.png (1.36 MB, 5919x5915, 5919:5915, level6.png) ImgOps iqdb

>>62409 >>62410
I am unable to read your handwriting / low contrast combo. The formula in >>62409 is very nearly correct as a starting point. What we seek is the limit at infinity. The minor problem is that your meanings of n in the numerator and denominator are out of step. When n=1 the numerator yields 41, but the denominator yields 9. The correct denominator for a numerator of 41 is 49. I assume this has been corrected in the handwritten notes and they proceed from a correct starting point, but I am unable to read those.


File: 1657978696517-0.png (Spoiler Image, 39.92 KB, 1269x439, 1269:439, equation1.1.png) ImgOps iqdb

File: 1657978696517-1.png (Spoiler Image, 40.53 KB, 1304x409, 1304:409, equation1.2.png) ImgOps iqdb

Yeah, sorry about that, here is the updated equation and the pass to the limit substracting 1 from the top and adding 1 to the bottom.
After that, I just noted it would be easy to write it in base-2, yielding fractions of the form 101000/1000000 = 0.1010 ~= 1/3. as seen in the lisp output in >>62410.


that's so cool i wish i understood math like that. i feel some sort of collective pride regardless being on the same website as people who can do stuff like this


> equation1.1.png
This numerator is a beautiful reorganization of the numerator from >>62409 but the denominator is still off. When n=1 you get 41/36 instead of 41/49.

> equation1.2.png

The ratio on the right, wrapped in a limit, is correct. This is despite the denominator of equation1.1.png being off, because you rightly kept the dominant terms.

> After that, I just noted it would be easy to write it in base-2, yielding fractions of the form 101000/1000000 = 0.1010 ~= 1/3. as seen in the lisp output in >>62410.

This base-2 view is a very nice idea. Looking at the ratios and seeing a consistent pattern is a valid method for obtaining the value of the limit.

However, since the final part of the proof was done numerically, we have to consider the call for an analytic-only proof sketch to still be open.

╭─────╮ ╭─────╮ ╭─────╮ ╭─────╮
│     │ │     │ │     │ │     │
│   ╭─┴─┴─╮   │ │   ╭─┴─┴─╮   │
╰───┤     ├───╯ ╰───┤     ├───╯
╭───┤     ├───╮ ╭───┤     ├───╮
│   ╰─┬─┬─╯   │ │   ╰─┬─┬─╯   │
│     │ │   ╭─┴─┴─╮   │ │     │
╰─────╯ ╰───┤     ├───╯ ╰─────╯
╭─────╮ ╭───┤     ├───╮ ╭─────╮
│     │ │   ╰─┬─┬─╯   │ │     │
│   ╭─┴─┴─╮   │ │   ╭─┴─┴─╮   │
╰───┤     ├───╯ ╰───┤     ├───╯
╭───┤     ├───╮ ╭───┤     ├───╮
│   ╰─┬─┬─╯   │ │   ╰─┬─┬─╯   │
│     │ │     │ │     │ │     │
╰─────╯ ╰─────╯ ╰─────╯ ╰─────╯

(rounded corners)


File: 1659504814863.png (532.08 KB, 602x699, 602:699, Squid_Girl_Ikamusume_holds….png) ImgOps iqdb

I started doing pre-calculus, as well as physics. It's somewhat relaxing to do maths, I find it more interesting then the physics im doing. I wonder if I could just learn calculus in a couple of days, if I tried really hard. I kind of just want to learn to maths and get to the upper levels out of curiosity


I've never understood what that's supposed to be. I recently read a calculus book and it doesn't seem to have much prerequisites beyond basic competence in symbolic manipulation (aka middle-school algebra) and passing familiarity with trig maybe.
Anyway you could very easily learn the essentials of calculus in just a few days, but it's a bit of a deep subject and you'll find you need to learn more in depth as you go.


pre-calc is basically just a remedial/refresher course.


It's basically highschool stuff lol. Im just a brainlet so I forgot. I pirated a bunch of textbooks to help me


File: 1661338994754.png (60.23 KB, 1358x468, 679:234, wizchan.png) ImgOps iqdb

It's because putting things to the power of i like this create an image of a rotation.

Now that you have this knowledge try and solve this? (No googling this is a known result but its fun!!)


Hey anon! You have gone very far in your journey congrats!!

you are at a cross roads however to go further into calculus you have to take two steps back before you can take a step forward.

I really recommend James Munkres topology. (http://mathcenter.spb.ru/nikaan/2019/topology/4.pdf)

It's an advanced book however Go through the intro!! its 70 pages long and if you have gotten this far into calculus I reckon you can figure it out (use outside sources)


File: 1661427721898.png (Spoiler Image, 109.59 KB, 1280x996, 320:249, spoiler.png) ImgOps iqdb

> i^i

(Also, avoid the trolls who claim that f(x)=e^x is not a function.)


Isnt this statement true for all x in the reals?


It is.

(x-1)^2 + 1 != 0


(PEE * 2) * (POO * 2) = PEEPEEPOOPOO


I was amused by all the Novel AI/Diffusion AI stuff coming out recently, and started wondering about how automated proof solvers have developed since the development of machine learning and neural nets.
It's still really primitive, and I feel like the group isn't using the full complexity of machine learning that a lot of other machine learning projects have created, but:
Proverbot9001: 2417 / 11729 proofs.
TacTok+ASTactic: 1377 / 11729 proofs.
ASTactic: 1142 / 11729 proofs.
They're getting a lot better a lot faster. Do any other anons think we may soon be reaching the point where you can just click a goddam button and a proverbot spits out a proof for you? Imagine if the Riemann Hypothesis got proved by a bot. What kind of shit would that storm up?


The four color theorem was first proved by a computer and it took a long time until mathematicians accepted it.


I think these are categorically different. The computer-aided proof of the 4 color theorem was proof by exhaustion as I understand it ("This isn't a proof, this is a phone book!"). These proofs being spit out by these neural nets are not necessarily that, as they're using actual proof strategies. Yes, they're way too detailed for publication since they pedantically show _every_ step, but it's possible to take the output from proverbot and rewrite it as a typical proof (i.e., use it as a REALLY good proof assistant). Granted it's tedious and a LOT of work, but it transforms work that requires Terry Tao/Ramanujan level genius into "keep your head to the grindstone"/"anyone can do this as long as they keep at it" work.

I also wonder if there's magic that would be able to use as a training set:
inputs - The low-level Coq proof output from proverbot.
outputs - The corresponding human written proofs.
And via some GPT-ish magic, generate a standard human proof. Call this "publish-or-perish" bot.

Take the inverse of publish-or-perish bot to feed proverbot a question, proverbot makes a proof, and then publish-or-perish bot outputs a human-readable proof.

If I were to go more into dreamland, something I wonder about whether proverbot would have this ability as it stands now is…
- Put the statement "There exists an algorithm with a (success rate>proverbot9001) and (runtime<proverbot9001)" into proverbot9001.
- Hopefully, the proof generated by proverbot9001 is constructive, in which case, take the algorithm generated by the output.
- Loop on the previous two items.
- Use this process to generate a theoretically optimal proverbot.


When you do self-referencing statements like that you'll run into godel's first incompleteness (and halting problem, busy beaver, or komolgorov complexity by proxy which are all equivalent)


File: 1673208215471.png (516.3 KB, 586x512, 293:256, ClipboardImage.png) ImgOps iqdb

Well that didn't take long


File: 1673208861383.jpg (83.36 KB, 586x512, 293:256, wizard_intelligence.jpg) ImgOps iqdb

read instructions!!!!!!!!!!!!!!


I need to stop multitasking..


This is impossible:
Represent each room as a vertex, and also the outside with a vertex. Let each door be an edge between the vertices as depicted. The rooms in the top left, top right, bottom middle, and outside, then, are all odd degree vertices (5, 5, 5, and 9), thus there is no Eulerian path.


File: 1673510907137.png (501.92 KB, 594x516, 99:86, ClipboardImage.png) ImgOps iqdb

just go through the doors more than 2 times and ur good


Funny how it says "hard but possible!" even though it's a basic result in graph theory that it isn't.
And people still keep trying.
I do wonder why they keep trying.


Good graph theory textbook? It wasnt obvious to me


File: 1673624329688.png (36.69 KB, 644x401, 644:401, Oekaki.png) ImgOps iqdb




it's completely possible. the problem is sufficiently vague that you can interpret the rules and contraints in countless ways. if you don't understand what i mean, i could make a list of various assumptions and solutions using them

you might argue this is cheating, but you are doing the same interpretations and making equally baseless assumptions trying to turn it into a math problem

you could argue the real problem to solve isn't the original image, but the largely agreed upon mathematical intepretation, but no one has posted that.


this is correct. The solution to the puzzle is as follows:

The original full image labels the puzzle as an autism test. The line-through-doors Autism test doesn't explicitly state that you can't cross lines, but we just assume so because that would be more of a challenge. Those with autism would take the instructions literally, without nuance, and not think of any potential rules that aren't written so they would solve it by crossing lines. So those who "beat" it are the real autists, while those who struggle by not crossing are normal


My professor recommended "Introduction to Graph Theory" by Robin Wilson, it's free from the University of Edinburgh online. The proof I laid out depends on the chapter "Paths and cycles" where Eulerian paths are discussed.
I like graph theory and wanted to give the proof I do not care about whether it's solvable within the bounds of the vague questioning.
Can you show the solution with crossed lines then?


File: 1673651073358.gif (Spoiler Image, 1.9 MB, 1920x1669, 1920:1669, crossedgif.gif) ImgOps iqdb

>Can you show the solution with crossed lines then?

Here. Try it for yourself first. the trick is to start somewhere going outward, then loop all the ways back around the whole puzzle.


I stand corrected. Also nice job with the gif.
I like that caricaturization of myself, I appreciate the effort you put into it, too.


funny, the video in the OP is just about the puzzle everybody is talking about


i was looking up jomon dogu earlier today and it is frustrating trying to see any patterns in how the dogu developed over time, across different regions of japan

is it crazy to use a map of japan as the ground, and then stack dogu above their region of japan, chronolgically? this way you can see where and when themes and patterns of the dogu emerge

i tried arranging stuff in 2d but there is no direct connection to the region of the map, you need the extra vertical axis to stack them i think

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