Set theory solely exists to facilitate the creation of contrived logic puzzles of no more mathematical merit than a game of sudoku.

Previously >>16899914

Bose was a crank

what math do I need to write cool looking shaders

>>16922249
Define shader

>>16922268
the things that run on your gfx card

What are some cool concepts in the intersection of numerical analysis and linear algebra?

>>16922249
Basic geometry and some linear algebra - the latter is true for most coding though.

>>16921941
fuck miggers

Do you think there exists a polynomial-time integer factorization algorithm? If so, what do you think such an algorithm would look like?

>>16922249
topology

>>16922249
>>16922269
Just learn how to do basic calculations (+, -, x, /) with 3x3 to 4x4 matrices in binary, octa, hexa and decimals.

Is Matt Parker smart enough to do motivic cohomology? Is he smart enough to compute Ext functors? Can he describe what a bialgebra is off the top of his head? Or.... Is he a midwit grifter who can only regurgitate funny math facts and write jeet-tier Python code?

it's crazy how 99.99999% of math was useless until the invention of electricity

>>16922564
It's crazy how you're fucking retarded.

>>16921941
Is there something akin to orientation that has 3 states instead of 2?

your mother is so fat that the axioms of set theory are unable to either prove or disprove that there exists a set b such that it can contain her

>>16922600
Maybe triality?

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>>16922645
The word fat was against the TOS so I had to tell it to say hamburger instead.

so I heard if your elementary algebra sucks you're basically fucked trying to move on to higher math

are there any good algebra problem sets or whatever so I can figure out if I actually know algebra and not just some fucked up version of it from the book I followed?

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What if 1/0 is just another type of imaginary number like sqrt(-1)? Ever think of that, huh?

>>16922700
There are two ways your algebra could suck. The first is you don't even get the point of algebra, you don't understand solving for x, and how x is a representative from an entire class of numbers. Algebra gives you tools for abstraction and speaking in generality. Yes, this is crucial for higher math.

The second is just not memorizing the basic facts (theorems) in algebra like the quadratic formula and the difference of two squares.

If you took algebra in grade school you probably get the point of algebra but you don't remember the important facts. Better yet, you should understand how Algebra give you the tools to *verify* the difference of two squares is as it is

[eqn] a^2 - b^2 = a^2-b^2 +0 [/eqn]
[eqn] = a^2-b^2+ab-ab = a^2+ab-ab-b^2 [/eqn]
[eqn] = a(a+b)-b(a+b) = (a+b) \cdot (a-b) [/eqn]

>>16922709
use case?

>>16922700
>so I heard if your elementary algebra sucks you're basically fucked trying to move on to higher math
Actually, the best way to assess this would be to actually try to move on to higher math. Start reading elementary linear algebra right now:
https://understandinglinearalgebra.org/home.html
Maybe try these interactive challenges, but they include trigonometry and precalculus:
https://www.khanacademy.org/math/get-ready-for-algebra-i/test/x127ac35e11aba30e:course-challenge
https://www.khanacademy.org/math/get-ready-for-algebra-ii/test/x6e4201668896ef07:course-challenge
Try this problem book for the basics:
https://books.openbookpublishers.com/10.11647/obp.0168.pdf
More resources at:
https://textbooks.aimath.org/
https://realnotcomplex.com/
https://github.com/rossant/awesome-math

>>16922700
Ask the AI to test your knowledge

>>16922747
>muh ai
kys

>>16922700
Personally I think solving word problems until you can see how to express them algebraically with minimal effort is the best skill you can develop at that level. Actually performing the baby algebra should be trivial, you can probably fit all the non-obvious rules on a single notecard.

>>16922559
Eh, I used to watch him a lot in high school; many of his older videos are quite nostalgic to me. I don't mind his combinatorics or prime number videos. As for his topology videos, I have no idea - I almost never watch them.
Though the Kellogg's video he released a few days may be the worst video he's ever uploaded. I was about to unsubscribe, but then I remembered he's one of the OG math YouTubers, so I let it slide.

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>x^2 - 3x = -1
>x(x - 3) = -1
>x - 3 = -1/x
>1/x + x = 3

i give up

>>16923010
You should be able to immediately recognize what number plus its reciprocal equals 3.

>>16923010
You're almost there, just multiply both sides by X, then subtract X^2 from both sides, then factor out an X

They cause permanent harm to school children by even teaching them the concept of "degrees" in regards to angles. Any teacher that does so should be imprisoned alongside pedophiles and child murderers.

>>16923010
x^2 -3x + 1 = 0
x^2 -3x + (1+c) = c
x^2 -3x +(1+c) = (x-r)^2 = x^2 -2r*x + r^2

r = 3/2 => (1+c) = 9/4
c = 9/4 -4/4 = 5/4
x^2 -3x +1 + 5/4 = 5/4
x^2 -3x + 9/4 = 5/4
(x-3/2)^2 = 5/4
x-3/2 = +-sqrt(5)/2
x = (3 +-sqrt(5))/2

>>16923010
anon, you're supposed to try to recognize patterns, not just mindlessly apply rules in an infinite loop

>>16923189
Actually, he just needs to mindlessly apply the quadratic equation formula.

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What if you have a square inside of a regular hexagon with unit sides so that three of the vertices of the square are on the perimeter of the hexagon and the fourth one is on the line drawn between two of the vertices of the hexagon like in picrelated.

What is the side length of the square?

>>16922709
Say 1/0 = x, try to prove 2/0 != x. Else x * 0 = 1 and 2. Also does 0 * 1/0 = 1?

>>16923244
let i = 1/0, then 2/0 = 2(1/0) = 2i

OK, I'm studying Group Theory at uni and I love it. I'm a month in and have made a separate sheet with around 40 definitions and theorems that I've come across so far. I'm drowning in them. I can prove only some of them. Do I need to make sure I can prove them all on my own? How do I remember them? I don't want to rely on a sheet of paper laying next to me while I study. I want to be able to do things from memory. Should I memorise them with flashcards? This is my first serious subject. I coasted through Calculus without memorising or doing much work at all, so this is new for me.

if you have to flip the inequality sign doesn't that mean the inequality was wrong from the start?

>>16923544
In what context?
If you're doing something like taking the reciprocal or negative, no

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>>16923544
If negative X is less than 0 then clearly X is greater than 0.

>>16923546
>it is self-evident
fuck off euler I know that's you fucking frog

>>16923560
Euler would never give a proof by "it is trivial".
That's Fermat posting

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>>16923570

>>16923543
Do exercises. Lots of them. You'll remember the ones worth remembered.

>>16922700
There was a book called something like The Huge Book of Algebra Problems that I remember being pretty decent

Math bros, do you have a special place you do math? I'm small brain but also brain hacking and I get most "stuff" done when I've walked to a special location and have some peace and quiet away from a computer. Has to be outside the house. I think I'm gunna start reading some math books for the fun of it. I am a NEET and taking the "learn everything" pill. Math isn't the only thing I'm doing. Please wish me luck. In both finding a quiet well lit place and not being pwned by phd physicists while trying to learn algebra and trig. My 4 books I'll have on hand are elements of set theory by enderton, how to solve, how to prove it, and basic mathmatics. Gunna do some math studying for the hell of it. On top of everything else I do. For what purpose? I don't know. Why try? I don't know. I have yet to master my own life and am like a man stranded at ocean swimming in any direction looking for dry land or a boat to take me to one.

just read math books and stared at the stars. Reading about induction in how to solve it was really interesting. So was learning about heuristics. Endermans set theory is really cool, I just don't know how I could commit something like that to memory. But I do read it from time to time. Reminds me of my old algebra notebook where I wrote down a bunch of rules, which is now missing. Any book specifically about calculating in algebra after you've mastered basic PEMDAS would be great, I still have to practice my basic PEMDAS abilities, specifically long division on paper, practicing long multiplication, etc, calculating exponents, even beginning with rules of exponents. I also like stuff like associative and communicative properties. I would go to my school online curriculum but I would rather just buy a book.

I really enjoyed reading the math books /mg/ recommended. They're dry and boring sometimes but other times they're just as interesting as a novel or videogame. My favorite writing style wise is how to solve it (reads more like a book) and my favorite for actually learning "difficult" concepts is basic mathematics, but it's not perfect. I will go back out later tonight and do more math and report back if anyone is interested.

Another math topic that greatly interests me is mental math, there is a great book on the subject I'll dig up if anyone is interested by an author in my locale. I also have a second hand "the Great Courses" dvd with only disk 2 of mental math to watch. I will report back if I ever.

I'm thankful there is a math general on 4chan, no matter how much I may not like it sometimes. For topics like computer science and game design there is no general at all, and discussion is even worse. Physics discussion is literally non existent. /lit/ is its own beast but similarly grim. The question remains. Can a 4chan user increase his or her IQ to be genius level and beyond through force of will or will fate take them silently into the brainlet night like many. I like to think IQ tests are just a game that can be trained for. Regardless, I have an autistic obsession with learning stupid things for no reason.

Also, I learned the words heuristics and pedagogy. Is there a word for the studies and practices of independent learning? Any schools of thought on the subject? I would like to learn from someone who has mastered learning, if at all possible. Not just any one thing either, but broadly. In so much as the limitations of depth vs breadth and the achieving "mastery" and what that even means.... As well as preserving knowledge, personal biblification techniques, with a auxiliary section on pedagogy and heuristics, or at least references. I would like to understand the forefront knowledge at the most auxiliary and bite sized atomic factoids level, "cosmological" concepts, grand theories, as well as new developments in minute understanding. I am wondering if it's humanly possible to be educated on art, science, technology, world events, etc, be a "well educated person" in todays society, or if it's too much for any one person to know.

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1st year undergrad in math in EU
our uni basically closes off theoretical math for only the best students since day 1 like literally day 1 hour 0 but you _technically_ can sign up for required courses during 2nd and 3rd year it's just that the versions of calculus and linear algebra they make you take are very computation heavy and dont have a lot of proofs.
I'm not the most hardworking person but how would I go about catching up? Besides theoretical specialization we also have data analysis, applied maths and some finance jewery i dont think i'm interested in any of those particularly much

>>16924465
It's all about the holy grail of learning calculus. Which you're supposed to do in high school. Basically PEMDAS -> Algebra -> Trig -> Calculus

Geometry is the fabric being woven. If you're a brainlet you can learn PEMDAS, in like a month. Algebra is where you're supposed to give up. If you can make it past algebra to trig you can actually learn calculus, and be a big boy math literate gentleman. It's not as hard as people make you think. You too can go to grad school. And get the PHD. And be the post doc researcher. Your own lab, out of reach? Some say this is a myth. Many such cases.

>>16924473
not sure if youre replying to the right guy

>>16924465
if you can't do calculus by hand you're not ready. I'd work on this if you're not there yet. 4 years of uni undergrad should just be showing up to class and networking after class. The real study is being proficient at calculus by hand, being able to prove theorems, basic understanding of philosophy etc, if you're already at this level I would pick up chemistry and physics. Or do some wacko courses that may inspire you in the future for your undergrad. Or just make friends at school.

>>16924478
Well I seriously doubt he can do calculus by hand and basic proofs, being a first year math major. It's the bare minimum but there's lots of frauds who haven't even met that bar.

>>16924479
>>16924480
ok just so we're on the same page here, what do you mean by calculus? since i'm in the EU and it might not be 1:1
also i have made ZERO friends and i wont make a single one one because i'm mentally ill

>>16924479
>calculus by hand
that's the only way to do calculus, computers literally can't perform it

>>16924481
Finish the book issac newton wrote on calculus, order a copy of the original text and study it. Supplementary texts may also help, but that's what I'd go with https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica

There are other books by newton, i'm not sure, but calculating motions in the universe and a firm grasp on mythology and astronomy helps. Get a telescope.

>>16924487
calculus by micheal spivock is also very good, /mg/ talked about it historically and everything they recommend is great. I would generally start looking into the bibliography of Issac newton, for calculus. If you want to really learn...

>>16924437
>I just don't know how I could commit something like that to memory
If you've got a lot of time and paper to burn, copy the book down verbatim into a notebook and add your own commentary.

>>16924490
That's probably a good idea

>>16924487
ok but how do i learn how to prove things and learn from reading proofs and find proofs etc

>>16924496
if then therefore, do I believe it? The biggest proof is that there is no proof.

>>16924491
It works great with dry/dense textbooks

>>16924496
But really it's more about making mathematical arguments, and learning logic is probably where I would start.

>>16924500
we had logic during 1st semester and it was the only course i enjoyed. i really, really enjoyed it and did well

>>16924496
If you're really looking into learning proofs, how to solve it/how to prove it are really great. With basic mathmatics and the ocassional math trainer/khan academy/notebook work I'd imagine you could get really far.

>>16924501
schizophrenia pilled

This conversation is getting old but it's been productive. I go now. China numba wan.

>>16924503
i may or may not have schizophrenia symptoms and be schizoid too
i'm interested in systems and determinism and formalizing concepts
i thought when i'd do a math degree we would stop with gay faggotry that are numbers after 2 semesters but apparently thats not how it works here

>>16924505
You should be able to play doom with pen and paper

>>16924487
ive just checked
its in latin
i wanted to learn latin but im not ready yet

>>16924437
>Another math topic that greatly interests me is mental math, there is a great book on the subject I'll dig up if anyone is interested by an author in my locale.
I'm working through "Secrets of Mental Math" by Benjamin and Shermer. It's pretty good. It made me realize there were mental math things I was doing incorrectly my whole life and it's because the methods taught in schools are not good. One of the authors, Benjamin, also has a good book on combinatorics if you're into that.
>>16924465
Don't listen to the trolls telling you to "do calculus by hand" (whatever that even means).

Work through these books:
Calculus by Thomas
How to Prove it by Velleman
Linear Algebra by Friedberg
Abstract Algebra by Judson
Understanding Analysis by Abbott

You can skip the Calculus book if you've already done the material. If you don't like Velleman's proof writing book then try Book of Proof by Hammack. Don't focus on trying to do every question in these proof writing books. Just make sure you understand the material and get to the Linear Algebra, Abstract Algebra and Analysis books as soon as you can. I went full retard and did ALL the questions in Velleman's book and it didn't help me too much. What worked was actually banging my head against actual Linear Algebra, Abstract Algebra and Analysis proofs. You'll probably want to throw in an elementary number theory book too. I've heard good things about Elementary Number Theory by Jones and Jones.

>>16924729
I'll pitch Elementary Methods in Number Theory by Nathanson for a good and accessible number theory book

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What is his life strategy, exactly?

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>2011+15
>not doing your calculus by hand
ISHYGDDT

>>16924729
ill definitely (not) check it out after i come home (i'm a lazy fucking loser)
thanks for sharing though, how to prove it seems to be a common recommendation

I exclusively perform mental calculus.

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>>16924905

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Does this problem have a closed form solution?

Two circles are inside a regular pentagon, the blue one twice the area of the yellow one. A line is drawn across the pentagon being tangent to both circles at the point in which they intersect.

The length of the segment AB is one unit. The problem is to find the length of BC.

How many of you actually have a pure math doctorate or are at least a PhD student?

>>16924729
I hate these lists because they're always some crazy ancient books.
If I don't see Stewart Calculus in these I just ignore. I'm in grad school at a T10 now and I used Stewart for Calc 1-3. For analysis just use Ruin. Its a well worn path and its been done like a million times.

>>16925274
I'm an APPLIED math MASTERS student. How pathetic am I in your eyes from a not-worthy-of-life to degrading-manual-labor-only tier?

>>16925300
>complaints about crazy ancient books
>uses rudin

>>16925204
Idk. Played around with it for a bit. I thought of collapsing the pentagon into a rectangle with an open top, doesn't do much though. You've got more unknowns than equations so it's hard. There's probably a relationship between the diameter of the circle and the side length.

>>16925300
Thomas' Calculus is direct, contemporary competition of Stewart's

>>16925300
Which ones are ancient? Thomas is used as a Stewart alternative in North America. As for the analysis book: I recommended Understanding Analysis to that guy specifically because it's gentle and he said that he has no experience with higher mathematics. Advising him to work through Rudin would be torturous. I used Understanding Analysis as my first analysis book and I found it to be very good. Rudin and the rest of the BORE-baki group suck all the fun and creativity out of mathematics. To me, Rudin is ancient.

Besides, why are ``ancient'' books bad? For undergrad mathematics, you can pick up a book from, say, the 1970s and be just fine. Not much has changed at the undergrad level. Why would it matter? The guy I replied to isn't doing cutting edge research mathematics.

Now let's go through the authors I recommended:
Thomas <- Dead
Velleman <- Alive
Friedberg <- Not sure
Judson <- Alive
Abbott <- Alive

The majority of the authors I recommended are alive and are willing to accept errata for new editions.

Now let's do you:
Stewart <- Dead
Rudin <- Dead (good riddance)

Are there any modern books (say, from the last 200 years or so, and very must have pdfs somewhere online) that teach geometry mainly with straight edge and compass? Or is Euclid still the gold standard for this stuff?
>why
I do math for fun and drawing is fun.

>>16925435
Try:
Jacques Hadamard's Lessons in Plane Geometry (the new edition with Mark Saul's Reader's Companion)
Kiselev's Planimetry
Daniel Callahan's Euclid's Elements Redux

if group theory is real then why aren't there group theory cpus? checkmate atheists

>>16925535
Technically integers are a group, so...

>>16925460
Thank.

>>16925301
I think you'll make great middle-management.

Imperial Analysis

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>the limit of any function as the denominator approaches 0 behaves in consistent and predictable ways
>somehow division by 0 is undefined and unknowable

>>16925795
have you considered f(x)=1/x

>>16925807
basically any time you see me I am internally contemplating 1/x

>>16925814
any progress?

>>16925795
>>16925814
contemplate the following:
1. there are two directions from which the denominator of 1/x can approach 0
2. the behaviors of those two directions are not consistent with each other
3. the inconsistency grows as they approach zero
4. they don't even agree on which infinity they're asymptotically pointing to at 0

>>16925836
Protip: it's entirely consistent when you realize that it simply wraps around the numberline

A superhyperreal number b is said to be positive infinitesimalimal if b is positive but less than every positive hyperreal number

>>16925807
why don't we just say that
lim x-> 0 1/x = {-inf,inf}
it just produces two different numbers at the same time. Let it be a set man.

>>16922700
The book by Aluffi is good.

>>16925848
follow-up questions:
A: What is 1/0 + 1/0?
B: What is 1/0 - 1/0?

>>16925873
if hyperreal numbers sit "between" real numbers, then real numbers aren't continuous
if the hyperreal number system somehow creates gaps between real numbers, then in the context of the hyperreal number system real numbers are discrete

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>>16925882
All numbers are discrete because no number exists until you calculate it.

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>>16925885

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>read proof
>that makes sense
>immediately look away and try to do the proof myself
>uhhhh uhhh uhhhh
>stare blankly at the page for 10 minutes
>give up

Can anyone explain how to use spatial vectors in rigid body mechanics?

>>16925965
there used to be a guy but he died back in '76 so you're on your own

>>16925970
Who?

>>16925970
kek used to happen in biology all the time before gene sequencing

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Why is this wall-hitting man constantly called (on X and in magazines, that is) a "legend" and "the greatest living mathematician alive".
He won some championship, then proves half a dozen notable results in meme subfields, and then after winning some prices in 2010 people just kept calling him a legendary mathematician ... I assume because nobody else is known?
I mean yes, he is a super strong mathematician - but is that really all that exceptional? And is any of his work actually relevant?

>>16926115
everything major has already been discovered, even if the greatest mathematician to ever exist were born today the best he could hope to accomplish over his life would be a few proofs of things already taken for granted

So I just mastered my multiplication tables up to 5. Why are people always saying math is hard? This shit is easy.

>>16926259
just wait until you get to fractions

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I've speedran 6 different calculus books/courses and this is the first time I've seen this explicitly defined. I mean, it makes sense, just weird how most courses never distinguish them (probably because they're "infinitely close" or whatever).

honestly what they really need is just a new number system that allows for blurry numbers with overlap

>>16926239
actually ... good point

(ofc, this doesn't quite explain why people are so needy to call someone a legend. Or "Mozart of Math", as one article of some famous outlet put it)

>>16926264
I'm confident that all calculus texts you looked at - I mean at least Springer texts or whatever - were clear about this.
Basically, just from writing down the definition of f' in terms of limits, you have that \Delta is finite. And no math textbook uses \Delta in place of d if the limit has been taken. At worst, some thermodynamics books may do this.

>>16926323
nta but most calculus courses are pretty bad, they just start using d's instead of deltas because "that's the calculus notation lol" and then they give you a table of rules to memorize
luckily if you just memorize and keep going to higher courses eventually you pick up the skipped details (or you fail out)

>>16926331
>a table of rules to memorize
I like the Keisler book from the wiki, if you actually follow along with it you don't need to memorize shit because everything becomes dead obvious at a glance.

I saw I cool problem on the internet. The dimensions of the cracker are 60 mm by 120 mm. The dimensions of the cheese are 73 mm by 88 mm. It always sucks when some of the cheese is hanging outside of the cracker. So the problem becomes as follows. What is the maximum percentage of the cheese that can be in contact with the cracker when you put the cheese on top of the cracker?

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>>16926361
100%
you bite off the excess

>>16926264
>>16926323
>>16926331
>Teachers of elementary mathematics in the U.S.A. frequently complain that all calculus books are bad. That is a case to point. Calculus books are bad because there is no such subject as calculus; it is not a subject because it is many subjects. What we call calculus nowadays is the union of a dab of logic and set theory, some axiomatic theory of complete ordered fields, analytic geometry and topology, the latter in both the “general” sense (limits and continuous functions) and the algebraic sense (orientation), real-variable theory properly so called (differentiation), the combinatoric symbol manipulation called formal integration, the first steps of low-dimensional measure theory, some differential geometry, the first steps of the classical analysis of the trigonometric, exponential, and logarithmic functions, and, depending on the space available and the personal inclinations of the author, some cook-book differential equations, elementary mechanics, and a small assortment of applied mathematics. Any one of these is hard to write a good book on; the mixture is impossible.
—Paul R. Halmos, How to write mathematics, Enseign. Math. (2) 16 (1970).

>uhhh, you can't just assert that irrational numbers are real
>>"irrational numbers are real because you can divide all numbers into those greater than and those less than it"
>OMG SO TRUE
I feel like most math is just circular logic shitposting

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>>16926409
I was going to shitpost about defining postive numbers as the set of all numbers that can be generated by successive additions of the number 1, but it turns out that's actually the defintion...

>>16926417
how many times do you add 1 to itself to get 0.5

>>16925953
Anki, ginkgo biloba, hill sprints.

>>16926409
The logic isn't circular. Not even natural numbers are "real", they aren't any less "imaginary" than complex numbers. But the use and abuse of these words without quotes can mislead you

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>>16925204

k = radius of black circle
b = radius of blue circle
g = radius of green circle

k/b = Sqrt[1/2]
k/g ≈ 0.362507354396385

I don't get it, this book is called Elementary Calculus but it covers Calc 1-3. I don't see anything called Intermediate or Advanced Calculus?

>>16926520
>I don't see anything called Intermediate or Advanced Calculus?
Advanced calculus can either mean calculus 3 or the same subjects as calc 1 but from the point of view math majors. This latter POV of advanced calculus is also called elementary analysis, and the subsequent advanced POV is not called that, but simply real analysus. It's also possible that advanced calculus can mean a mishmash of complex variables, variational calculus and div grad curl in general curvilinear coordinates for physicists

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>>16924905
>>16924952
math to stop the flow of time?

>>16926418
-0.5 times

>>16926640
At best, I have a song for that:
https://www.youtube.com/watch?v=j-opxZJvIZ0

>>16926640
E = mc2

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>>16921941
Stargate mathematics

god i need to study but i couldnt give less of a fuck

>>16926640
legends say that if you master Advanced Calculus you gain the ability comprehend higher dimensions and freely move through time

I'm rotating a photorealistic microcosm of the universe in my mind as I post this

>>16924905
We have calculator now.

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>>16922275
Fuark, mang
This is hard
>linearAlgebraIntro.jpg

It's not racemixing haram if youre both good at math

>>16927286
Ever heard of [math]\sum{}[/math]?

>>16927318
Oh the thing that makes easy for computers, impossible for shape rotator humans?
No I dont employ the ligma notation cheat code for a formatting of indexers exercise

>>16927318
bro your M fell over you gotta be more careful when typing

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>studying advanced mathematics
>go to solve problem in computer program
>write retarded baby code
how do I actually start applying math to things besides problems in a textbook?

The year is 1969. The air at Berkeley is, like, totally thick with tear gas and bad vibes, and Ted is sitting in his office staring at the boundary of a unit disk because the *System* is a literal prison. He’s scribbling about $F_{\sigma\delta}$ sets because if he can just prove the boundary is manageable, maybe his life won't feel like a divergent sequence.

Then, the door opens. It’s Alexander. He’s wearing these raggedy sandals and carrying the entire concept of *Topos* in his head like a heavy backpack. He looks like he hasn't slept since the 1966 ICM in Moscow, which is honestly a mood.

"Ted," Alexander says, his voice sounding like a sheaf of continuous functions. "I heard you were looking for a limit. But why seek a point when you could inhabit the whole space?"

Ted blushes, which is super embarrassing because he’s supposed to be a hardened critic of industrial society. "It’s just a cluster set, Alex. It’s not a big deal. The Power Process requires me to solve this alone."

Alexander walks over and leans over Ted’s desk. He smells like lavender and pure, unadulterated abstraction. He points to the unrestricted trapezoid on the chalkboard. "See this? It’s not a shape. It’s a **Motive**. It’s the soul of the math trying to find itself in a world of military funding. You’re trying to bound the boundary, but I want to *be* the boundary."

Ted’s heart does a non-measurable jump. "I... I have to go to Montana. I have to build a cabin. The technology... it’s a restricted fibration, Alex! It’s killing the agency of the individual!"

Alexander reaches out, his hand hovering over Ted’s slide rule. "But Ted... what if the 'Power Process' is just a natural transformation? What if your 'autonomy' is just a section of a larger bundle? We could... we could co-limit. Together."

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>>16927505
if you have to ask, your ngmi

>>16927505
I warned you about skipping the word problems bro
I told you dog
I TOLD you about skipping the word problems

>for all of infinity multiplying a number by a number gives a bigger number
>except in the range of -1 to 1
Why is this specific number range magic? And out of all the infinite numbers we just happen to center all our mathematics around that point? Is this proof of God?

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>>16927553
0 is a black hole and 1 is the event horizon.

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>>16927505
Programming is only as powerful as your modeling skills are and how well you can intuit how to abstract the processing of those modeling skills. I can only model my physics word problems like sticky notes. Many CS grads in 2016 were the equivalent of today's ChatGPT habit tracker app devs because they have never modeled something. Physics is the king of models. Nerds git gud with flight sims and programming better programs to program better programming programs for computer mechanics. 3D modeling itself if you make anything hard into easy is powerful, lucrative, rewarding, and capable of solving the hardest engineering problems like tiling vector fields. If you know advanced maths you should know real world demands with a prompt. A cloistered 3d animator can make bank making mining simulations that map onto a mining company's requests. You need to touch grass.stl

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>>16927605
>found the limits of my GPU
>found the limits of my CPU
>turns out it's super sensitive to the nature of the task at hand
>parallel fpga courses are 2 years away from me in ECE school
I feel like I learned something but something too empty to call it something

>>16927557
chat is this true?

>>16927613
I can create a 600GB image in my mind, guess man still prevails over machine.

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>>16925204
>find the length of BC
|BC| = |BC|/|AB| since |AB| = 1.
Thus one can alter |AB| and find |BC|/|AB| instead of |BC|.

>>16926510
definitions:
a = radius of K circle
b = radius of B circle
1 = radius of G circle

system:
b = Sqrt[2]*a
c = Cos[Pi/5]
t = Tan[Pi/5]
(1 – a/c – x)^2 + (0 – y)^2 = a^2
(b – c – x)^2 + ((c – b)*t – y)^2 = b^2
(b – c – (1 – a/c))^2 + ((c – b)*t – 0)^2 = (a + b)^2

approximate solution:
a ≈ 0.362507354396385
b ≈ 0.512662817047358
x ≈ 0.200551144763890
y ≈ 0.0891859430070010

The exact solution is depicted.

>>16925204
I mean, it seems to me that there'd only be one possible way to construct the smallest regular pentagon that contains both circles so intuitively it should be solvable, but I'm lazy

>>16925204
>Does this problem have a closed form solution?
What, yeah it definitely does. Why wouldn't it? You're just doing simple circles and tangent lines. There's gonna be two answers though because there's two ways to draw what you wrote; the picture you showed is one of those ways

>>16926264
I personally have never really mentally reserved the term dy and \Delta y as the true difference and the approximate, respectively, nor would I ever actually advocate for it. If the author defines those two as the true and approximation, then sure, it's reasonable. But \Delta _ just means change in _ for whatever context that isn't referring to some object that is infinitely-small-but-nonzero. I'd recommend that you think of dy in general as the differential, or perhaps something infinitely-small-but-nonzero as most casually do.

I've personally don't recall ever having seen it like that, but maybe it's cause I saw it once and it was never really referred again. Like, when you learn about numerical techniques, I don't think they refer it as dy and dx for the true distances. Seems more like a kiddie intro thing.

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>>16927937
There are two solutions.
2nd solution:
a ≈ 4.43326569164754
b ≈ 6.26958446673129
x ≈ –0.362375379901045
y ≈ –1.64332375459355

>>16928076
nta but chatgpt agrees with the definitions in the image

>>16926418
use divergent series to get whatever number you want bro.
Ramanujan pajeet math is suitable for this shit stuff.

>>16927613
>>found the limits of my GPU
>>found the limits of my CPU
more often than not a better algorithm fixes those issues.

>>16928287
>second edition
outdated trash

>>16928312
>calculus has radically changed in 20 years

Quick, prove there isn't a third set of numbers beyond the rationals and irrationals that create gaps in the set of real numbers.

>>16928336
>what are the infinitesimals

>>16928336
irrational is a catch-all though, any number that isn't rational is irrational

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>>16927937
equation of magenta line: (v – y)/(u – x) = q
q ≈ 3.9396917385398805950300262
u ≈ 0.2005511447638899744885732
v ≈ 0.08918594300700103376141002

say you got isekai'd into a medieval high fantasy world with nothing but a perfect understanding of all modern earth mathematics
what do you do with it

>>16928624
You could be that worlds equivalent of Newton, but that's about it.

>>16928624
>high fantasy
probably get an aneurysm because the magic system is utterly illogical but impossible to deny

>>16928624
Try to be a teacher of an elf princess and fuck her.

Pick an uniformly random [math]n \times n[/math] matrix with entries in [math]\mathrm{GF}(2)[/math]. Given that the entries on the diagonal are all equal to [math]1[/math] what is the probability (as a function of [math]n[/math]) that the determinant of the matrix is also equal to [math]1[/math]?

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>>16928484

|BC|/|AB| ≈ 0.44466044382825726

https://www.wolframalpha.com/input?i=Solve%5B%7Bq%3D3.93969173853988059503%2Cu%3D0.20055114476388997448%2Cv%3D0.089185943007001033761%2Cy%3D%28%28q*Cos%5B%282%CF%80%29%2F5%5D%E2%80%93Sin%5B%282%CF%80%29%2F5%5D%29*Cos%5B%284%CF%80%29%2F5%5D%2Bv%E2%80%93q*u%29%2F%28Cos%5B%282%CF%80%29%2F5%5D%2Bq*Sin%5B%282%CF%80%29%2F5%5D%29%2Cr%3D%28y%E2%80%93Sin%5B%286%CF%80%29%2F5%5D%29%2F%28Sin%5B%284%CF%80%29%2F5%5D%E2%80%93y%29%7D%2Cr%5D

Any of you have experience working with Typst? I'm looking to switch away from LaTeX.

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>math is like super useful
>>what if you got reincarnated in a primitive world but with the sum of all math knowledge
>uhhh, I'd, uhhh... become a teacher?

>>16928786
To be fair, teaching is probably the best thing you could do on a global scale because you're not going to revolutionize an entire nation within your lifetime no matter what your chinese cartoons tell you.

Hell, even being sent back to the 1400s with the total knowledge of all electrical engineering and you even somehow convinced the Queen of Europe to give you her full support and funding, you might make some lightbulbs and write some really good books but you'd be long dead before you ever saw a transistor or Sega Genesis.

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>>16928727

I shouldn't have posted the previous system,
because the following system is clearer.

system:
q = 3.93969173853988059503002
u = 0.20055114476388997448857
v = 0.08918594300700103376141
(s*x + c*y – v)/(c*x – s*y – u) = q
c = Cos[(2*π)/5]
s = Sin[(2*π)/5]
x = Cos[(4*π)/5]
1/r = (Sin[(4*π)/5] – y)/(y – Sin[(6*π)/5])

note:
|AB| = Sin[(4*π)/5] – y
|BC| = y – Sin[(6*π)/5]

>tfw you get to the point where you realize that 99% of mathematicians are retarded
I've made it...

>>16928786
Is there is a good technical handbook to recreate the civilization from scratch?
Like I don't even know how to obtain electricity and how to dig up precious metals.
>"knowledge"
the book is not technical.
full of words, not 1 single diagram or a formula

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>>16929280

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>>16928905
An auspicious placement of the two main circles is depicted.
Now, the tangent between them intersects the vertical side of the pentagon.
Which simplifies calculating the ratio |BC|/|AB|.

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>>16929290
face the wall, anon

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chatgpt literally solved it instantly

>>16929337
Visually and intuitively that answer does not make sense.

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>>16929346

>>16929347
>The blue circle radius = 2r and the yellow circle radius = r
The original image is clearly saying the area of the blue circle is twice that of the yellow circle, not the radius.

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>>16929348

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>>16929351
>wait that's not right
>just subtract 1 from it lol
They're actually using this shit to build operating systems and space ships.

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It's harder than any field of mathematics.

Is the board so dead now that no one is even making new /sqt/s?

>>16929430
No point in making it when everyone posts their stupid questions as individual threads anyway

>>16929398
fact check: true

>>16929430
more like 4chan is dying. Hard to get new visitors with its image and word-of-mouth spread, unlike normie sites

>>16929476
more to do with
>captcha hell just to post from a fresh ip
>flooded with bots despite captchas
>no moderation anymore on most of the boards beyond removing illegal content (boards flooded with shitpost threads, people spamming tranny content, avatarfagging, etc... all ignored)
>/r/eddit raids every round of janny applications and then spends the next several months attempting to change 4chan culture by banning allowed speech they don't like

>>16925204
|BC| ≈ 0.5
a ballpark figure

>>16928727
>|BC|/|AB| ≈ 0.44466044382825726
a fine calculation

>>16929351
>BC ≈ 0.7071
an AI hallucination

>>16929430
I was the only one baking for some time and I was on holiday when the final one died. Everyone just asks AI nowadays, thread was already slowly dying.

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>>16929398

>>16929430
>>16929641
Pretty much. The site is dying by the metric of posts per hour (and many others), and the thread isn't needed anymore. I'd been there many years, but I noticed recognizable posters no longer posting and a few months ago I left as well.
Oh well. Thanks for baking the last few.

>>16929665
DX12 is 98% boilerplate code that you copy/paste from the docs.

>>16929674
> copy/paste from the docs.
Get with the times.
> "Claude, write me a DirectX 12 application."

i struggle to study
has anyone managed to overcome this? i almost want to try to get a tablet so i can study in bed

>>16929749
do you think euler studied in bed with a tablet?

>>16929751
isnt all of math from that time period basically 1-2 semesters of work

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>>16929754
>heh, it took you guys multiple generations to develop calculus, huh?
>well, I superficially memorized some calculus rules in a couple of semesters, guess that means I'm smarter than all of you

>>16929756
The end result is I know more than them. How we got to that point is irrelevant.

>>16929756
strawman

who wants to be my mathamatibuddy and learn math together and maybe kiss

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I just found a weird cousin of the pythagorean theorem

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Can all math be shown as a visual proof?

>>16929945
not rigorous

>>16929954
Proof by "just look at it"

>>16929945
>who is Galois

>>16929979
Someone not for midwit niggers to study.

let C be the set of all correct solutions to the problem
let x be any member of set
the solution to the problem is x

>teachers HATE him

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>>16929979

>On the following Bastille Day (14 July 1831), Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a loaded rifle, and a dagger.....
VIVA LA EL REVOLUTIONEY!

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Gnomons?

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"Ne pleure pas, Alfred ! J'ai besoin de tout mon courage pour mourir à vingt ans !"
(Don't weep, Alfred! I need all my courage to die at twenty!)

https://youtu.be/tvUM1th1z6I?si=VJW606vqQLI144Dk

why come mathematicians never complain about imposter syndrome?

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Well?

>>16930364
I got that it reaches the end in [math]e^{1000} - 1 [/math] seconds.

>>16930364
Instead of
>t [s] = L_0 / v_ant = 10^3 [m] / 10^-2 [m/s],
the classical (not relativistic) time t [s] would be from
>ln(1+t [s]) = 10^3 [m/s] / 10^-2 [m/s]
meaning
>t [s] = e^{10^5} - 1 [s] = e^{100000) - 1 seconds,
which means any positive velocity gets it to the end.

Now what's weird to me is that d/dt ln(1+t [s]) = (1/1+t) [1/s], or generally d/dx ln( f(x) ) = f'(x)/f(x) [1/x] which is the right units, but I wonder how the hell ln( [not-unitless] ) work. And also, \int (1+t)^{-1} dt = ln(1+t [s]) is unitless too which is odd.

There's no issue with e^{f(x)} cause d/dx e^{f(x)} = df/dx e^{f(x)) meaning [f] needs to be unitless, but there's no restriction for ln(x), odd.

For the relativistic case, it seems only an extra term is added to the diff eq, but the solution for t for one frame of reference is gonna be like the func inverse of an integral of another func inverse, so there's no closed form.

>>16930572
eh, that e^x thing also has issues. e^{ ln{ f(x) } } = f(x), so it can have units too. Idk, if there are rules, they're weird

>>16930364
yes.
just from intuition alone it's easy to see: the proportion of total length expansion that is occurring behind the ant is always increasing, whereas the proportion of the total expansion occurring ahead is always decreasing.
the actual numbers aren't relevant, they're just coefficients making the ant take more or less time to reach the end.

Everyone says
>you learned math the wrong way

So how should we have been taught?

>>16930656
Ideally it's the one that helps you learn the best, but people vary in intelligence or learn differently. It's good to distinguish between honors classes, but even then it can never be perfect for everyone.

Also, some people only learn to pass grades, not to understand. *shrug

>>16930656
Euclid's Elements + Complete works of Archimedes + Pythagorean oath.

>>16930572
Jesus christ nigger use the built-in TeX editor.

>>16930572
my eyes

what's the definitive book on calculus?

>>16931119
Calculus as in calculus or as in introduction to analysis?

>>16931136
if I knew wtf you were talking about I wouldn't be asking for a book

>>16931139
Thomas's calculus.
If you wonder why a formula is what it is, finish Thomas's and start reading Spivak's calculus.

>>16931373
Im so dumb, ignore

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Is using AOPS from pre algebra all the way to calculus a better idea than using AOPS at the beginning and then using books like Thomas/Stewart/ Spivak or Apostol once you reach calculus? (I guess I'd be sacrificing a bit of rigor for continuity).

Does anyone have experience with AOPS? what is it like? Im a thirdie and AOPS is probably the best way to reap the benefits of a high quality American education in math if you live outside of America and you are looking for structure and continuity.

>>16931622
Stop asking for the way to learn and just learn. Nobody stop you from learning from other books if the one you're using is shit.

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>>16929430
1/6

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>>16929430
2/6

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>>16929430
3/6

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>>16929430
4/6

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>>16929430
5/6

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6/6

Abel Prize announcement:
https://youtu.be/2dtaywzNsqk

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>>16931660
This year's recipient: Gerd Faltings
Was previously awarded the Fields Medal in '86

>>16931663
>mathematics award
>gives to a physic institution
>he isn't even black or woman (male) or woman (female)
>born in Germany

>>16931650
>>16931651
>>16931652
>>16931653
>>16931654
Useless AI slop. Kys.

>>16929683
this I wrote a small DirectX8 game engine

Hey, first time posting here after lurking for like 16 years, feels weird as hell to click for the first time on a board after all this time on 4chan, kind of exciting even.
Anywa, coming from /agdg/. I realized I actually need math now for my projects. Like most idiots I dropped it in high school once it got hard (EU btw not an amerilard).
I'm looking for resources recommedantions on linear algebra (vectors/matrices), geometry, noise, and splines because I want to do procedural environment generation (terrain, roads, city layouts, buildings) in Houdini / VEX (which is pretty much Python).
If anyone has some cool resources for these, it would be cool.

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recursive matrix equation:
A(n + 1) = [B(n) + C(n) + D(n)]/3
B(n + 1) = [A(n) + C(n) + D(n)]/3
C(n + 1) = [A(n) + B(n) + D(n)]/3
D(n + 1) = [A(n) + B(n) + C(n)]/3

Q = quadrilateral
The (n + 1)th Q is smaller than the (n)th Q.
The Qs must converge to a point.
They converge to [A(0) + B(0) + C(0) + D(0)]/4.
This explains the depicted equation.

I'm reading through a signals & systems book and they keep talking about Euler's formula
obviously it's not hard but when the fuck was I supposed to have learned it? they never mentioned it in any previous course I took

>>16932009
Probably algebra when you learned about complex numbers.

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I'm getting btfo in my proofs course. Got a 70 on the first exam, which I considered a fluke because 20 points was missed from doing a single problem incorrectly (worst part was I realized it ~30 minutes after taking the test and going over it again). Just took a second test today over divisibility, existence/contradiction and induction and pretty sure I failed it. I do actually study decently and know the content, what ends up happening is I blank on a question and forget what method I'm supposed to use or simply misunderstand the instruction like a retard. I can't see myself doing well in this course unless I spend 2-3 hours each day intently grinding over problems to the point where every conceivable way to solve a specific problem is ingrained in my long term memory. But I don't think anyone in the class is actually doing that.
Starting to think this just might be an IQ thing and I'm getting filtered. Fuck my retarded faggot brain.

>>16932213
Try shitposting less and studying more.

>>16932213
You didn't hit have a triple digit IQ day? You know normie highschool teachers take that to get certified right?

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How do you prove that cos(9°) is equal to this?

>>16932485
Both are roots of [math] T_5(x)-\frac{1}{\sqrt{2}} [/math], where [math] T_5 [/math] is the fifth Chebyshev polynomial.

>>16932485
cos(10x) + i sin(10x) = e^(10 i pi) = (cos(x) + i sin(x))^10
Plug in x=9°, expand the RHS, compare the real parts of both sides and solve for cos(9°).

>>1693249
>e^(10 i pi)
Sorry e^(10 i x) of course.

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>>16932485
>How do you prove that cos(9°) is equal to this?
Like this:
https://www.wolframalpha.com/input?i=Sqrt%5B5+-+Sqrt%5B5%5D%5D%2F4+%2B+%281+%2B+Sqrt%5B5%5D%29%2F%284+Sqrt%5B2%5D%29+%3D%3D+Cos%5B%289+Pi%29%2F180%5D
How else?
I just rolled my eyes.

>>16932485
>Screenshot_20260321_125120.jpg
Based on your time zone, you probably posted from one of the following eleven countries.
Finland
Estonia
Latvia
Lithuania
Kaliningrad
Ukraine
Moldova
Romania
Bulgaria
Greece
Cyprus

>>16931843
That's a very specific request that does not map well on to typical math pedagogy. Given how specific the things you're looking to do are, you might want to look for some textbook 'math for graphics/computers' or something. You might also try asking on the technology board.

>>16932213
Just keep grinding it out. It sucks at first, but eventually you start to love it. It will serve you well later on; once you hit grad school, grit is more important than brilliance.

Oh and make sure you're in those office hours every single time. Find a professor you vibe with who can be your mentor. Math has to be understood intuitively before it can be done rigorously, and a good mentor will bootstrap your intuitions.

>>16932551
Dumbass, Kaliningrad is not a country. It's a non-autonomous province of Russia.

>>16931843
https://gamemath.com/

>>16932551
Edited the file name on purpose to mislead you. Jokes on you!

>>16932485
They teach you about the 3rd and 4th roots of unity in school, but you can also find the 5th roots of unity too. z^5 = 1 means z^5-1 = 0 which means (z-1)(z^4 + z^3 + z^2 + z + 1) = 0. Quartics are solvable, so it's doable to get cos(72) and sin(72), which also means you can get cos(90-72=18), cos(9), cos(9/3=3) cause cubics are solvable, cos(1), which means you can get cosine of any integer

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>>16932625
>Dumbass
I know you are, but what am I?
>Kaliningrad is not a country.
Yeah, I know.
It's a Russian exclave.
I didn't concentrate.
Or I failed to check my prose.
I should have written "territories", instead of "countries".

>>16932653
>Jokes on you!
You're ugly.
And no one likes you.

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>>16929934
proof:
https://www.wolframalpha.com/input?i=Solve%5B%7Ba+Csc%5B%281+Pi%29%2F18%5D+%3D%3D+b+Csc%5B%2812+Pi%29%2F18%5D+%3D%3D+c+Csc%5B%285+Pi%29%2F18%5D%2C+a+Csc%5B%282+Pi%29%2F18%5D+%3D%3D+d+Csc%5B%286+Pi%29%2F18%5D%2C+z+%3D%3D+a%5E2+%2B+b%5E2+-+%28c%5E2+%2B+d%5E2%29%7D%2C+z%5D

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Why does this make this repeating pattern of zeros and nines in the first digits?

>>16933283
You can simplify the expression to
[eqn]v = a \left(\sqrt{1 + \frac{1}{a} + \frac{1}{2 a^2}} - 1 \right) [/eqn]

Now if you use the Taylor approximation for the square root [math] \sqrt{1 + x} \approx 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} - \ldots[/math]
you get that
[eqn]v \approx \frac{1}{2} + \frac{a^{-1}}{8} - \frac{a^{-2}}{16} + \ldots[/eqn]
Because [math]a[/math] is so big you can see each of those terms clearly in the solution.

im so fucking anxious about combinatorics exam this shit is fucking blavk magic to me and i specifically chose a harder course im like 1 month behind and if i fuck this one up im out since theres only 2

>>16933477
>he can't count
mcdonald's is always hiring.

What's up with all the jeet-tier troll posts this thread?

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If you put three squares inside a triangle like this and it makes a smaller version of the original triangle, is the ratio between the area of that small triangle and the original triangle the same for any triangle?

Mathlet here, I've returned to ask if this
https://epoch.ai/frontiermath/open-problems/ramsey-hypergraphs

Is 'moderately interesting' as is claimed, or if it's a scam to trick brainlets like me

>>16933923
No. If the area ratio was the same, then the ratio of the squared-lengths are the same and equal to the area ratio.
Choosing any two sides A and B and an angle X from B to A, then the length ratio is R = 1 + 2[A/B + B/A - cos(X)] * csc(X) which clearly isn't constant.
A nicer, more symmetric form of R (although less easy to tell if constant) is [math] R = 1+ \frac{ A^2 + B^2 + C^2 }{ 2*area_\triangle = area_\square } [/math]

I found the center of the triangle given this configuration. Like, you can choose any point within the triangle's center, then trace 3 lines from the center O outward to the three vertices to get similar triangles of varying size. But I don't know if there's any significant meaning to the center in this configuration. But the properties are that

1.) the minimum distance (perpendicular line) of O to any side, say B, is B/(R-1),
2.) the distance from O to vertex v_AB is [math] \frac{ |A||B|*D_{AB} }{ (R-1)*2area_\triangle } [/math] where D_ab is the diagonal of the parellelogram between A and B (so if A and B are vectors at the origin, then D_AB = |A + B| ),

and equivalently
3.) the law of sines ratio [math] \frac{ \sin\theta }{ L } [/math] for all three of the smaller triangles from O to any two vertices is [math] \tfrac{ 4area^2_\triangle = area^2_\square }{ ABC*D_{AB}*D_{BC} } (R-1) [/math]. I don't know if the fact that it's true for all three is something significant or is always true.

If any of yall can find something significant about this (like this isn't the inradius or centroid or wtv, and the center is always inside the triangle), lemme know

>>16934173
Ugh, my shit got deleted

I forgot to add, since the law of sines ratio for any triangle is S = 2[Area Triangle] / ABC = [Area Parallelogram] / ABC, that means the ratio for the smaller triangles found in 3.) is S multiplied by a factor of (A^2 + B^2 + C^2) / D_{AB} D_{BC}

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>>16933923

r = area/AREA

If d = e = f, then r = 1/(13 + 4*√3) ≈ 0.0501801.

If d ≠ e = f, then r = 1/(Tan[ε] + 3/Tan[ε] + 1)^2 where Cos[ε] = d/(e + e).

If (d, e, f) = (10, 5, 7), then r ≈ 0.0247651.

Thus r isn't constant.

Plot[{1/(13 + 4 Sqrt[3]), 1/(1 + Tan[ε] + 3 Cot[ε])^2}, {ε, 0, Pi/2}]

>>16934173
>If any of y[']all can find something significant about this [...], lemme know
hundred dollah bill y'all

>>16934173
I realized it last night that I was incorrect that the law of sines is equal for all 3 triangles. The equation came out as the diagonal of the parallelogram, and I was like "Oh, so there's only 2", but no, there's actually 3: D_AB, D_BC, D_CA. All the equations above should still be correct

If you draw it these correspond to twice the length of the medians; the median isn't the angle bisector but the opposite side bisector, so the intersection is the centroid. Instead of in terms of D_AB, you could write in terms of the distance from the centroid to vertex v_ab.

Lets make new names. The triangle has vertices a,b,c, the center of the triangle in this case x, and the centroid of the triangle y

>>16934177 says the law of sines ratio is
[math] R_{xab} = R_{abc} \frac{ \overline{ab}^2 + \overline{bc}^2 + \overline{ca}^2 }{ 3^3 \overline{ya} \overline{yb} \overline{yc} } 3 \overline{yc} \ \Rightarrow\ \frac{ R_{xab} }{ \overline{yc} } = const [/math]
so
[math] \overline{yc} \sin\theta_{xcb} = \overline{ya} \sin\theta_{xab} [/math]
You could start out with points, a,b,x, and use this relation involving theta_xab and theta_xba to find the last point c. When I was incorrect before, it would've meant that [math] \sin\theta_{xcb} = \sin\theta_{xab} [/math]
But still, look how stupidly useless this relation looks. You're forced to find the centroid too?? It would be such a nontrivial construction process to find c

For any arbitrary center z in the triangle, the following hold:
[math] R_{zab} = R_{abc} \frac{ \overline{ab}\ \overline{bc}\ \overline{ca} }{ \overline{za}\ \overline{zb}\ \overline{zc} } \tfrac{ \overline{zc} }{ \overline{ab} } [/math]
If you want to choose a new center z' then
[math] \overline{za} \ \overline{zb} R_{zab} = \overline{z'a}\ \overline{z'b} R_{z'ab} [/math].
I don't see how this can make a more insightful connection between the centroid and this center x involving the squares of the sides but I feel this would be useful

>>16934900
That last paragraph
>For any arbitrary
is stupidly wrong, please ignore it. I'm done

>>16935003
>I'm done[.]
That's good, because no one even understood anything which you wrote.
We were all scratching our heads; and not because we're monkeys, or have lice.

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Up to how many numbers is this circle possible?

>>16935212
Only becomes possible with n=32 and then the number of solutions greately increases with a bigger n.
https://oeis.org/A071984

>>16929906
>and maybe kiss

>>16934900
Ooh I found out where the centroid comes from. For any triangle ABC, you can stack 4 of them together in a Tri-force-like triangle pyramid so that the three medians of this larger triangle are equal to the three diagonals mentioned before, e.g |A+B|. You can construct a new triangle A'B'C' whose sides are the triforce pyramid's 3 medians, and if you find its centroid (intersect its medians), the 3 inner triangle from the centroid to the vertices have the same side lengths as ABC. These 3 inner triangles have the same area as ABC, as each one is the same as ABC divided in two pieces along a chosen median and glued back together along the side that was cut in half. Since there are 3 inner triangles of A'B'C', that means it has 3 times the area of ABC. I'll call triangle A'B'C' the median transform of ABC.

Separately, find the center of the triangle for this particular problem, call it O. Draw the 3 lines from O that are perpendicular to the sides of ABC, and call their 3 intersections with ABC as a', b' and c'. Then the centroid of a'b'c' is O, and a'b'c' is similar to the median transform of ABC, which is larger by by a factor of R-1, or (A^2+B^2+C^2) / 2*Area_ABC.

Because it's all connected, we can find that
Given the ratio for triangle ABC, R_ABC, the ratio for the median transform is R_A'B'C' = 3(ABC/A'B'C') R_ABC
if you want to do redefine the transform so that it preserves area, it's not that different but you need to multiply by sqrts.
Also, when the centroid O is chosen as the center of a triangle abc, then the ratio for the inner triangle, say R_Oab, is such that [math] R_aOb \frac{ \overline{ab} }{ \overline{Oc} } [/math] is constant, meaning [math] \overline{aO} \overline{ab} \sin\theta_{Oab} = \overline{cO} \overline{cb} \sin\theta_{Ocb} [/math], which means all 3 inner triangles have the same area (of course). There's a relation for R_Oab from R_abc but it doesn't seem too interesting to me.

>>16935497
Oh, and if you do a median transform twice you get the original triangle scaled by 3

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>>16923010

My favorite integral from calculus 2
[eqn] \int \sec^3(x) \ dx [/eqn]
[eqn] = \int \sec^2(x) \sec(x) \ dx [/eqn]
Proceed by integration-by-parts
[eqn] dv = \sec^2(x) \ dx, u = \sec(x) [/eqn]
[eqn] v = \tan(x), du = \sec(x) \tan(x) \ dx [/eqn]
[eqn] \int \sec^2(x) \sec(x) \ dx = [/eqn]
[eqn] \sec(x) \tan(x) - \int \sec(x) \tan^2(x) \ dx [/eqn]
[eqn] = \sec(x) tan(x) - \int \sec(x) (\sec^2(x) - 1) \ dx [/eqn]
[eqn] = \sec(x) \tan(x) - \int \sec^3(x) + \int sec(x) [/eqn]
[eqn] \int \sec^3(x) \ dx = \sec(x) \tan(x) - \int \sec^3(x) + \int \sec(x) [/eqn]
[eqn] 2 \cdot \int \sec^3 x dx = \sec(x) \tan(x) + \log | \sec(x) + \tan (x)| [/eqn]

[eqn] \int sec^3(x) \ dx = \frac{1}{2} (\sec(x) \tan(x) + \log | \sec(x) + \tan(x)| \tag{\bigstar} [/eqn]

God I hate information theory so much, why the fuck do I have to take this shit in bachelors

>why the fuck do I have to take this shit in bachelors
Probably because it's extremely powerful and widely applicable. You'll survive with one less "introduction to category theory for babbies".

>>16935738
Vagueposting is so gay

I heard number theory is mostly self-contained. I want to get good at number theory and have little interest in other branches. I was great at math but never engaged with it after high school. Is there a guide to this I can use by any chance? Alternatively, do you guys have any recommendations for number theory books that will get me from euler theorem noob to grad level?

I need to become a number theorist chads... help me anons...

>>16936142
>>16936148
from my understanding, Burton's Elementary Number Theory is the usual recommendation for baby's first book in the subject

>>16936142
An Introduction to the Theory of Numbers, by G.H. Hardy and E. M. Wright.
It's very beautifully written and has turned people into number theorists since 1938.

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>>16934715
>1/(13 + 4*√3)
= 1/(1 + 2*Sqrt[3])^2
= 1/(1 + 2*Tan[Pi/3])^2

n = 3
P(e) = (Cos[(e*Pi)/n], Sin[(e*Pi)/n])

triangle A:
vertices: P(0), P(2), P(4)

U = 1 + 2*Tan[Pi/n]

triangle B:
vertices: U*P(0), U*P(2), U*P(4)

for each triangle:
linewidth: 3
edgecolor: (0, 0, 0, 1)
facecolor: 'none'

C = Cos[Pi/n]
S = Sin[Pi/n]
Q(o, e) = (C + S)*(Cos[(o*Pi)/n], Sin[(o*Pi)/n]) + (Sqrt[2]*S)*(Cos[(o*Pi)/n + ((e – 3)*Pi)/4], Sin[(o*Pi)/n + ((e – 3)*Pi)/4])

square C:
vertices: Q(1, 0), Q(1, 2), Q(1, 4), Q(1, 6)
facecolor: (1, 0, 0, 1)

square D:
vertices: Q(3, 0), Q(3, 2), Q(3, 4), Q(3, 6)
facecolor: (0, 1, 0, 1)

square E:
vertices: Q(5, 0), Q(5, 2), Q(5, 4), Q(5, 6)
facecolor: (0, 0, 1, 1)

for each square:
linewidth: 1
edgecolor: (1/2, 1/2, 1/2, 1)

>>16935498
>Oh, and if you [...].
Oh yeah?

>>16935765
>Vagueposting
Oh so that's what it's called.
I didn't know, that it had a name.

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>>16936165

If n = 4, then r = 1/(1 + 2*Tan[Pi/4])^2 = 1/9.
The image thereof looks like a 3-by-3 chessboard.

If n = 5, then r = 1/(1 + 2*Tan[Pi/5])^2 ≈ 0.16617848.
An image thereof is depicted.

>>16936167
Doing the median transform twice (getting the same shape 3x the length) also does a 180deg rotation. The smaller version of the median transform that fits inside the triangle is a 90deg rotation (or -90 depending on how you construct the transform). Either way, this means doing the smaller transform twice means you end up with the same shape right side up. It turns out that the median transform preserves the value of R-1 = (a^2 + b^2 + c^2) / (2Area), so the area of the smaller transform is always 3/(R-1) smaller. Doing the transform twice does this area change twice, but length only changes by square roots, meaning that the second smaller transform has side lengths 3/(R-1) smaller.

Since you're drawing a sequence of 2 types of similar triangles that fit inside each other getting smaller and smaller, they should converge at a point. This new center is basically a new problem not necessarily related to the other center or the centroid, but since the triangles are converging at a geometric rate 3/(R-1), it's easy to find where this new center is. I see nothing interesting about the equation for the new center though.

>>16936207
pretty

>>16936207
Funny, the R equation you're getting for the regular polygons are is similar to the equation for the triangle R. Area of a regular n-polygon of side s is A= n*s^2*cot(pi/n) / 4, so 5n^2 / (2A) gets you the 2tan(pi/n) = R-1.

You should try to do it with a an arbitrary convex n-gon not regular and see if you still get [math] \displaystyle \frac{ \sum_{i=1}^n s_i^2 }{ 2*Area } [/math]

>>16936258
>5n^2
meant ns^2 = 5s^2

>>16936253
*3/(R-1)^2

>>16934173
>R=1+A2+B2+C22∗area△
>>16935497
>R-1, or (A^2+B^2+C^2) / 2*Area_ABC
>>16936253
>R-1 = (a^2 + b^2 + c^2) / (2Area)
That's three versions of the same equation.
I hate to disappoint you, but I don't know how you got this equation.

>>16936253
>3/(R-1)
>>16936260
>3/(R-1)^2
And I don't understand the transform having to do with either 3/(R – 1) or 3/(R – 1)^2.

Don't bother elaborating, because this thread is going to be archived soon.

>>16936258
>You should try to do it with a an arbitrary convex n-gon not regular
I wasn't even able to do it for an irregular 3-gon.
Because it was too much for WolframAlpha to handle.
Which does all of the "heavy lifting" for me.

But I was able to do it for an isosceles triangle:
R – 1 = Tan[ε] + 3/Tan[ε]

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>>16936207

If n = 6,
then r
= 1/(1 + 2*Tan[Pi/6])^2
= 1/(1 + 2/Sqrt[3])^2
≈ 0.21539.

What could be next?

>>16936339
Eh, it's alright, it's probably not even true. For a convex n-polygon with n verticies ordered counterclockwise, every 3 adjacent verticies determines a vertex of the new, larger n-gon, so like P(v1, v2, v3) = v2'. The side length of the larger n-gon then depend on two vertices like v2' and v3', which means it depends on 4 adjacent vertices of original triangle. For a polygon with more than 3 sides, these 4 points are independent whereas a triangle only has 3 independent points. This means that the larger polygon for n>3 isn't guaranteed to be similar to the original smaller polygon from where it came, and there's no way I see it for it to have a defined "center" as for a triangle. And if they aren't similar polygons, it'd be odd to assume some factor scaling of R-1 or R.

>>16936371
>4 adjacent vertices of original triangle
*polygon

Is the following formulation of the pigeonhole principle constructively valid?
For any [math]f : A \to B[/math] such that there is a surjection from [math]A[/math] to [math]B[/math] but no injection we can find distinct [math]a, a' \in A[/math] with [math]f(a) = f(a')[/math].

>>16936495
Why care about surjection? If it's 1-1, then there is no two distinct a and a' with equal mapping f(a') = f(a). If it isn't 1-1, then there is. Pidgeonhole is more like if |B| < |A| then f cannot be injective.

>>16936352
If the bigger regular polygon (with N sides) has one unit side length, what is the side of the smaller one created by those N squares?

>>16924424
>>16924424
Some math problems you can think about while walking. Abstract algebra and topology are good for these types of problems.

>>16924444
Digits of truth. Yes. IQ test skills can be trained. Do lots of math and reading and thinking and look up every word you don't know. After a few years of daily effort you'll be improooving.

>>16936841
The ratio is is 1/R where R = 1 + sum of sides^2 / (2* area). The basic same math that applied to the triangles applies to regular polygons cause you can draw smaller triangles that are similar to each other within the polygons and the math comes out the same. It's basically a picture argument you could do if you draw it out. Notice that his area ratio is 1/R^2. You could've started there and gotten 1/R

What math book do you own that satisfies the wizard tome feeling for you?

>>16936980
Number theory in a sense is good for this as well. I spent an entire evening just laying back in my chair with coffee and cig trying to find a name based on my saxophone's serial number. Started with a coprime, one was emrip but both were safe primes with respective germain primes, and these germain primes kept repeating in the euler totient, difference of squares, quadratic residue, and the blum integer decomposition. Then I set up a prime derived alphabet mapped to the Italian alphabet with a prime counting function to assign a leter to each of the germain primes, the quadradic residue and shell defined by the euler totient. I wont tell you her name its my secret, but its pretty cool that there was so much information hidden in dark corners of just a serial number to draw a name from. Had a lot of dead ends that weren't quite right particularly in jacobi symbols but I think I narrowed it down quite right to the things that actually mattered.

Can someone please help me wrap my mind around the following natural inference? I am entirely new to math proofs and learning from scratch.

Not(P) Implies Not(Q)
——————————
Q implies P

I get why P would imply R if P implies Q and Q implies R. But introducing the Not is difficult for me to understand.

>>16936161
>>16936150
Cheers

>>16937244
It might help to think of it in terms of sets. Rather than propositions P and Q, consider "all worlds where P is true" and "all worlds where Q is true".
If Q implies P, then all worlds where Q is true are worlds where P is true. P being true doesn't say anything about Q, though, but the point is that "Q true" is a subset of "P true".
If Not(P), then P is false, so we're not in a "P true" world.
But because we're not in a "P true" world, we're not in any subset of "P true", either, including "Q true". So Q -> P implies Not(P) -> Not(Q).

The reverse direction is easily obtained by just considering their negations instead (i.e. consider A = Not(P) and B = Not(Q), such that Not(A) = P and Not(B) = Q) and applying the same reasoning there

>>16937265
Thank you I think I get it. So Q is a subset of P if Not P implies Not Q, because Not P implying Not Q shows that there is a relationship between P and Q in such a way where you can say P is like a parent of Q. So then if Q is true, that would imply by that relationship that its parent P is true… I think.

>>16937244
Just write out a truth tables for both and see they’re equal. It should be easy to tell that the meaning of both are equivalent to, both Not(P) and Q are impossible

>>16921941
I just now found out while studying for a test I'm taking later on that my calculator can find roots to polynomials AND it can solve systems of equations. I'm so hyped I had no idea that feature was there before. There's a really limited amount of time on the exam so having it be able to do that will save me immense effort and also save me from getting stuck due to minor arithmetic mistakes. I'm pretty excited to discover this but I don't have anywhere else to post so sorry if this is dumber than the usual posts you guys get on here.

I also added some custom programs on there for example one to find modular multiplicative inverses, modular exponentiation, and one that automates doing Euclidean algorithm step by step with each remainder/quotient. I think that's fair because they provide a severely limited amount of time for how many problems are on the exam that are not difficult theoretically but take up a lot of time doing busy work to solve.

Like for example they'll have some recursive function and then ask you to solve it by hand which is just BS. For example BS like: Solve Ackermann(4,0) is an exercise in tedium not a test of knowledge IMO