/sci/ - Science & Math

Are Braid Groups a fun subject?

>>10557993
They seem interesting from what I've learned about them at conferences. I want to learn more about them.

>>10557993
Yup, I'm kinda interested in braids due to their relation with knot theory

>>10557993
>Are Braid Groups a fun subject?
Why don't you try it and find out?

Threadly reminder to work with physicists.

>>10558260
what is harder graduate degree pure math or physics?

>>10558254
Will do.
>>10558280
Depends on the university.

What is the topology equivalent of Abbott's Analysis?

>>10557306
Is it possible to become an amateur mathematician today? By amateur I mean someone who does not hold a graduate degree in mathematics/physics/engineering...

>>10558967
Well what's stopping you? Just study and practice in your free time

What good resources are there to learning calculus 2 other than the basic textbooks? Any books/youtube lectures? And yes, checked the wiki

>>10558967
Yes.
A degree in engineering wouldn't help desu.
>>10558990
>>>/sqt/

>>10559055
what the fuck would they even research?

Daily Putnam Problem >>10559430

How do I show the identity
[math]
exp[\hat{A}+\hat{B}] = exp[\hat{A}] + \int_0^1 dz exp[z(\hat{A}+\hat{B})] \hat{B} exp[-z\hat{A}] exp[\hat{A}]
[/math]
?

>>10559523
Write down some powerseries probably

Hey. I am very unhappy with my life currently. I used to be happy I did not choose to study math like I once considered because I thought it would have been too boring and tedious. But recently I wonder ... maybe it would have been better after all. I am very intelligent but I think I am wasting it all. I also dislike engineering work and am so unhappy.

Are there any good books you would recommend that teach me proofs? Like a book with 100 problems of increasing difficulty that I can try to prove one by one and see example proofs? The only thing I do so far is occasionally solve some putnam problems.

This problem has been haunting me for a little while. Took an advanced calculus course, it was an extra credit and apparently I'm the only one who got it right because I used taylor expansions to compute it instead of just using derivatives/L'H.
Not homework help, but I'm curious to see if there's a way of solving it without using Taylor expansion:
[eqn]\lim_{x \to 0} \left(\frac{2-2cos(x)}{x^2}\right)^{1/x^2} = \frac{1}{ \sqrt[12]{e}}[/eqn]

>>10559598
How to Prove It by Velleman, it's on libgen
Haven't used it myself but looks like fun beginner stuff

>>10559611
The main trick is to rewrite it as

[math]\exp \bigg( \lim_{x\to 0} \frac{\log \big( \frac{2 - 2\cos(x)}{x^2} \big) }{x^2} \bigg)[/math]

From there it's just a matter of applying LH enough times.

>>10558914
Topology without tears

>>10559613
>beginner stuff
I hope it wont be too easy. But maybe I need it that way. I do math my own way since I lack the academic training. That works great for problems with a specific answer like some putnam, where I know I did it right because the end result is correct. However, when it's just "show x" its a lot harder to prove that I answered it right.

>>10558280
actually, it's CS

>>10557306
>the set of reals is an infinite set of equivalence classes of Cauchy sequences
Why has there never been a good, clear, non ambiguous definition of the set of real numbers? Is it because it doesn't exist?

>>10560873
what is not clear about that?

is yukariposter here?

>>10560900
See: >>10558185
The filename is right and the speech quirks are right.
>>10560873
Axiomatic definition is clean.
>>10560328
There are people out there who actually struggle with point-set topology?

>>10560891
Way too complicated definition for such an intuitive concept

>>10560918
Cauchy sequences are very intuitive though. You're probably mistaking your intuition for Q or something with intuition for R.

>>10560918
I don't think you realize how complicated real numbers actually are.
They have a simple characteristic property (archimedean totally ordered field which is either complete or satisfies the LUB property, both are equivalent), which essentially means "Basically Q, but every monotone sequence has a limit".
To prove that such an object actually exists, you need some sort of construction based on Q that satisfies this, and if you really think about the Cauchy sequence or Dedekind cut constructions, you will see that it would be very hard to make it simpler.
The Dedekind cut construction is essentially forcing the least upper bound property to hold, and the Cauchy sequence one is forcing completeness.
tl;dr: Q is an essentially algebraic object, and you want to do analysis on it (have compactness/completeness properties). It requires forcing completeness to occur, which is what the Cauchy sequence definition does.

>>10559523
guys? is it too easy or too hard? oh and obvious but those are just the first two terms in an expansion, forgot to mention that

>just found out I can't become an actuary as a backup plan since my country requires a fucking four years actuarial sciences degree for you to take a fucking test to become an actuary
>>10561068
It's long and I'm confused about the dz being in the beginning of the integral and the hats everywhere.

Daily Putnam Problem >>10562381

>>10560918
>take the metric completion, the most obvious analytic construction imaginable, of Q, the most obvious field imaginable
>somehow this is unintuitive
if you literally think of R as fucking infinite decimal expansions you're just looking at the Cauchy sequence definition (taking the first n decimal places gives the nth term). What else do you think real numbers are? The fuck else do you want an infinite decimal expansion to mean if not the limit of the finite parts?

Someone explain me what I am doing wrong with this exercise.
Wolfram said it's wrong and I want to understand, without a truth table, why I am incapable of fruitous results.

>>10562573
Why should it be false?

>>10562581
By the truth table and the solutions I inferred that this statements or assertion is NOT a tautology.
I completely missed to write that.
And I am not sure how to show that via logic laws.

>>10562587
It is a tautology

>>10562592
Then this book I use to learn discrete mathematics is full of errors

>>10562665
Post the bit you think is wrong

>tfw graduating (undergrad) and leaving all my cool department people behind
feels surprisingly bad desu

I am reading three mathematical books and one book on logic.

Von Neumann’s Book On Game Theory
Fibonnaci’s Liber Abaci
And Ptolemy’s Almagest

Has anyone attempted the Almagest? His cosmological model is almost more complex than the reality of the astrophysics models we currently use. It’s quite impressive

faggish old people weed shit die old ppl

>>10562937
I too dislike old professors that should've retired long ago.

>>10562760
ive been thinking about this too, ill have taken every math class here by next year, so ill need to find somewhere else by then
im sure its for the best dude

What's the difference between a topos and a category?

>>10558283
Standard tier university. Math or physics. What is the hardest

>>10558260
Physicists keep abusing category theory and its pissing me off.

>>10563517
a topos is a category with "axioms"

>>10563555
Checked. Could you please expand on the matter?

>>10563517
>>10563628
The category of sets has a lot of nice properties from the point of view of logic and foundations (e.g. products, power sets, characteristic functions). Topoi are meant to generalize these properties in a way very much analogous to how commutative rings generalize the properties of the integers.

>>10563628
In the plainest sense, a topos is a completely self contained structure that behaves equivalently to set equipped with a logic. Except, where sets and logics are seen as different from each-other, the topos essentially generates these things.
All the useful theorems from category theory and the algebra's that stem from it also apply, thus we can explore properties of the sets and logics, that we couldn't before from set theory and FOL.
Also since it is a category with objects, we can basically do whatever we want to it and synthetically generate weird logics by adding richer features to the objects. It's a wonderful playground for logical semantics.
"by default" an elementary topos has the properties of higher order intuitionistic logic / no excluded middle.
Syntactic topoi even allow for the translation of one branch of math into another as long as their categories are equivalent up to isomorphism. Want to use some weird arithmetic geometry for a PDE? Just make sure they're in the same syntactic topos

That's one point of view on them, but you can take others and use them for all kinds of math. They are basically a foundational formal semantics.

>>10561068
Are the a-hats and b-hats matrices?

>>10563799
>>10563805
Thanks a lot!

>>10557306
is this proof bullshit?
http://www.rxiv.org/pdf/1405.0023v3.pdf

>>10563928
if it was legit it would have been verified and published. It's also reasonable to assume that if elementary proofs of famous conjectures existed they would have been discovered ages ago.

>>10564037
that is my hunch as well, but i am interested in some certainty... i am not confident in my own proof-reading capabilities. would be also interesting to see exactly how it is wrong and stuff cause i cant find any feedback published on the article

>>10562573
literally just think about it. "if it's not a wednesday, then it being a wednesday means it's raining"; this statement will always be true.

I will NEVER understand the proof of the completeness theorem because I'm a fucking MORON idiot LOW IQ simpelton

>>10564160
This but unironically for all proofs I try to read.

I perfectly understand integration by substitution dealing with definite integrals, it's all rigorous and clean. I don't understand how it works for antiderivatives, though. How is it so that [math]\int f(x)dx = \int f(g(x))\cdot g'(x) dx + C[/math]?

Miss me with that
[math]g(x) = u[/math]
[math]g'(x)dx = du[/math]
bullshit.

>>10564492
An indefinite integral is just a definite one with the lower bound at zero and the upper bound at the variable, plus the value of the antiderivative at zero (the constant of integration)
Also, if you think about it as an antiderivative, it's just the chain rule applied to the derivative of f's primitive, which is f.

>>10564554
This

>>10564160
what one are you trying to read, dont they have modern "translated" ones

Let [math]\hat{f}[/math] denote the Fourier transform of [math]f[/math]. How would I go about proving that [math]\widehat{fg}=\hat{f}*\hat{g}[/math] (under suitable assumptions) without using the Fourier Inversion Theorem?

Brainlet here
Is it possible to enjoy learning calculus? Other subjects like linear algebra are fun as fuck but for some reason I just want to shoot myself whenever I read anything related to limits or derivatives. Am I missing something that makes it interesting, or is it just something to be powered through to get to the fun stuff?

Consider some real inner product space [math]\mathcal{V}[/math]. Given a set [math]S \subseteq \mathcal{V}[/math], its dual set is defined as [math]\{ x \in \mathcal{V}^* : \langle s, x \rangle \geq 0 \; \forall s \in S \}[/math], while the set [math]\{ x \in \mathcal{V}^* : \langle s, x \rangle \leq 1 \; \forall s \in S \}[/math] is often called the polar set.

Is there a name for the set [math]\{ x \in \mathcal{V}^* : \langle s, x \rangle = 1 \; \forall s \in S \}[/math]?

>>10557306
ok anons i remember reading somewhere that all of math basically comes down to linear algebra. is this true? why?

>>10566001
I don't think it's true.
A lot of applied maths is linear algebra (ex: machine learning). Linear stuff is somehow easy to understand so it is often useful to approximate stuff with linear functions (see derivatives, integrals, rep-theory, etc). So in some sense most of the low level maths everyone sees heavily depends on linear algebra.
There's lots of maths that isn't linear algebra though.

>>10564372
What? Show us the simplest proof you can think of that you don't understand.

>>10565802
It's a matter of taste. I enjoyed studying calculus, but I think LA was just ok.

>>10566020
i remember asking a grad student or something cant tell if he is memeing but he affirmed it too

Daily Putnam Problem >>10566116

>>10565978
That set is nonempty only if S is a set of lower dimension than V, since [math]<s_1, x> = <s_2, x> = 1 \implies <s_1 - s_2, x> = 0, \forall s_1, s_2 \in S [/math]

If you can represent C as 2x2 real matrices why is real representation theory distinct from normal representation theory?

>>10557306
Yo math dudes, complex numbers popped up around the renaissance approximately, as well as a bunch of other new maths.

What are the mathmaticians working on now a days? When will they find the next complex number, or then next 0, or the next derivative/whatever? What's the state of the art in math?

>>10566339
is this what people unironically think math is?

>>10566339
Infinitesimals were only rigorously defined in 1960s

Don't fall for the p-adic number meme.

>>10566339
This has to be the most normie post on mg since its inception

>>10566001
I mean a lot of higher level math is based on abstract vector spaces

>>10566001
Maybe "linear" math as those are the only problems we can solve for the most part.

>>10566343
Math is pretty unique among the stem meme fields imo in that normies have absolutely no clue what mathematicians do.

>>10566339
Real Numbers in the Neighborhood of Infinity.

>>10566339
Sup senpai. So recently this Galois dude has proposed some theory about how fifth degree polynomials can't be solved, which is absolutely insane.
It's some cuh-ra-zy shit about permuting roots, check it out.
>>10566343
Yup.

Show me something that confuses you.

>>10566343
Usually people ask me if I solve equations/do computations all day

Failed mathematician here, cant even publish anything of mild signifigance and have lost all intrest in research, is there a way to recover or should i go ahead and just end it all

>>10566555
there's no hope
kill yourself

>>10566001
It's hyperbole, but not based on nothing. Linear algebra is well understood and it's said that pretty much everything you want to be true is true in linear algebra. A common technique in many more advanced topics is to reduce problems to linear algebra.

>>10566594
Kk

>>10563928
Yes. In the "proof" that z^n = a^2 + b^2 admits no solutios, several incorrect assumptions are used.

>Author is prof. at Zimbabwe's national university

Lol

why is probability theory so tedious and boring and difficult
I don't want to fail my exam but it's just soooo boooring I can't even open my notes and look at it and I have to learn how to use the calculator and statistical tables to deal with data sets fuck that's so annoying
boooooooring

>>10566630
you realize there is a world outside of proving useless esoteric results on fictional objects, right?

I’ve tried doing several Putnam/IMO problems but it never leads anywhere even though I understand the subject it’s on.

Are there books not in the meme list that better prepare you for actual problems?

>>10566679
>useless

>>10566681
Anon... I have some unfortunate news for you...

>>10566683
the intersection between math that's useful and math that's interesting is very small
and some random theoretical physics bullshit doesn't count as "useful"

>>10566681
>Putnam and IMO
>actual problems
You better be memeing me. Anyhow, Tao's.
>>10566658
>tfw no cushy job teaching students at Zimbabwe's national university how to grill burgers with money and compute homology groups while publishing joke proofs of the Riemann conjecture with shitty Tooker references
>>10563534
My bad, I was banned for making the topos thread.
After carefully comparing my local uni's math and physics curriculums, I've concluded that physics is a bit harder.

>>10566679
There is nothing for me now. I think I'll just cash out rather then waste resources trying to be somthing I was never meant for

>>10566669
>probability theory
>"calculator and tables"
excuse me?

>>10566686
What would that be.

>>10566703
I meant “actually difficult problems” as they do tend to make you think. Thanks for the recommendation.

>>10566339
Maths has moved a lot since then. The majority of mathematicians don't have anything to do with numbers. It's hard to describe the forefront because you need the equivalent of an undergrad degree to understand it.

>apply to two math phd programs
>both don't return a decision before april 15th
fuck you, it's not like I wanted to work as a perpetual slave for four years for a piece of paper or anything

>>10567820
>want to do a master in the netherlands
>tell assesor that we should finish before march 1st so I can apply
>tells me that we will finish in before that
>almost may and we haven't finished even though everything is wrote already but she doesn't send me corrections because she keeps taking breaks
AAAAAAAHHHHHHHHHHHHHHHH

>>10567820
Any university that hasn't sent decisions at this point just has a lazy shit admissions committee that didn't bother to inform the rejects.
Decisions were made months ago, and competitive applicants for the school heard months ago. A few more maybe squeaked in off the waitlist in March. Nobody is going to get an acceptance to any school that's not total shit this late into April.

Hello /mg/. /prog/ mathlet here.
I want to be able to solve more Project Euler-style problems.
I've hit a kind of wall around problem 70, where I can no longer just compute the answer in the most obvious way.
Basically every problem is computationally infeasible if I try to compute them as they are described.
What area / branch of maths would help the most in reducing the required computations for these kinds of problems?
I've heard number theory is a good place to start, but what else?

>>10568581
It depends problem to problem. There's not really a single branch of maths that will help you the most. You're honestly probably better off learning more computer science.
How are you trying to compute the answer to question 70?

>>10568581
it’s a hodge podge of topics that commonly go under the umbrella term of ‘discrete’ mathematics

for your purposes, i think you’d enjoy something like https://cses.fi/book/book.pdf, it’s more of a straight line to your goals and interest. you can always dig deeper into discrete math if desired afterward

is there a point in learning applied or computationally based calculus if you’re already comfortable with delta epsilon style proofs?

>>10568591
To be more clear, I'm not stuck on exactly problem 70. I have completed the first 50 problems, and then about half of the problems between 50 and 70.
In any case here's what I think a very naive solution would be:

Iterate through all numbers up to 10 ^ 7.
Calculate φ for each n by trial division or something.
Check if they are permutations of each other by casting to strings and checking how many occurrences of every digit.
If yes, store n.

Iterate through all n where n = φ(n).
Find the minimum.

Post the answer (assuming it even finishes calculating, which I doubt), open up other people's answers, see some really cool trick that solves it in 5 lines or even by hand, cry.

>>10568596
My comp. sci. is relatively solid. And while in a lot of cases you can use comp. sci. techniques to speed up a solution, it's usually within the same order of magnitude at the cost of a lot of lines of code and overall added complexity to the solution. It feels very penny wise and pound foolish to try to optimize down some of these problems.

>>10568624
Naive computation will never finish.
I think the trick to 70 is using the multiplication formula for φ. You only need to compute it for prime powers and just multiply it all up. I'm having trouble putting this into code though.

>>10568600
Yes

>>10568624
you could always just poke around the relevant topics in section III for a few minutes, even if your comp sci is solid. it’s more project euler type mathy in that portion

>>10568638
what is it? i’m on mobile so please dont be so vague as to bait me into solving yet another captcha

>>10568641
If n is a product of prime powers q_1, q_2,...,q_k then phi(n) = phi(q_1)phi(q_2)...phi(q_k)

>>10568639
Oh huh, it specifically mentions the phi function. I'll take a look through the book and make a note of anything else that's unfamiliar.

>>10568635
Even that requires prime factorization of all 10^7 numbers it seems, based on this: >>10568645
Seems like it might be too slow still, but given that it's asking for the result with the lowest ratio, it seems like it's calling for an exhaustive search of all n where n = φ(n). No idea if it can be narrowed down further.

>>10568652
>Even that requires prime factorization of all 10^7 numbers it seems
This isn't implausible if you build it up instead of calculating naively.
>it seems like it's calling for an exhaustive search of all n where n = φ(n)
There is no n>1 with this property.
I think you're on the right track with pruning using minimality though.

>>10568660
>There is no n>1 with this property.
Sorry, where φ(n) = permutation(n).

>>10568652
I can compute it for n<1,000,000 in 4 minutes
Trying 10^7 now, is there supposed to be a time limit?

take the sans-serif regular pill

>>10568645
I'm too much of a brainlet to understand this notation.
p|n are the distinct prime factors right?
So if we have the prime factors (1, 2, 3) this would expand to
(1 - 1/1) * (1 - 1/2) * (1 - 1/3)?
I don't see how that gives φ(87109)=79180 as is the example on Project Euler.

>>10568744
1 is not considered prime. The prime factors of 87109 are 11 and 7919, so you get 87109*(1 - 1/11)*(1 - 1/7919) = 79180

>>10568641
Numerical analysis and and computer algebra are crucial to actually computing things in the real world, which is something that is useful no matter what type of math you want to do.
For example, computing with elements of a group, computing eigenvalues/eigenspaces, computing intersections of ideals etc. or on the analytic side, solving differential equations numerically and doing modeling, these are all tasks that mathematicians have to do.
Knowing the tools and how they actually are implemented can help you better use them in your own work, as well as giving you a better understanding of the abstract objects involved.
For example, dimension in algebraic geometry is defined in a very abstract way and all the proofs relating to properties of the dimension are so abstract that it is, on the face of it, a highly nontrivial problem to determine the dimension of a set defined by a given bunch of equations.
It turns out that computer algebra softwares do that very well, and knowing how they do it can help you do it yourself when you have simple cases to treat.
Similarly, really understanding how to numerically solve differential equations in a sound way (being familiar with numerical schemes, stability, consistency) will give you analytical flair that you can use even if you work on more abstract problems.
Basically, the more hands-on experience you have with objects, the more comfortable you will be when you have to use them. Computational math is a way to get that kind of hands on experience.

Daily Putnam Problem >>10569073

>>10569082
/mg/ is a serious research mathematics thread, we don't care about autistic contest mathematics here

>>10568624
> In any case here's what I think a very naive solution would be:
That would definitely be a naive solution.

The first thing is to note that you aren't trying to enumerate all n for which φ(n) is a permutation of n, only to find the solution such that n/φ(n) is a minimum. How does that help you? Well, it means that you want φ(n) to be as large as possible for a given n, which means that you want n to have as few factors as possible (φ(n) <= n-1 and φ(mn)=φ(m)φ(n) <= (m-1)(n-1)=mn-1-m-n <= mn-1).

If you could find a prime number which satisfies the requirement, then n/φ(n) would be n/(n-1) which is barely greater than one. But there can't be any prime solutions: prime numbers never end in a zero, and subtracting 1 from a number which doesn't end in a zero changes the last digit while leaving the other digits unchanged, so will never produce a permutation of the original number. So the next best thing would have n as the product of two primes (the example given is 11*7919).

So now you only need to iterate over pairs of primes whose product is less than 10^7. So find all a s.t. a is prime and less than sqrt(10^7), then find all b s.t. b is prime, b>a and b<10^7/a. For each a,b, check whether (a*b)/((a-1)*(b-1)) is better than the best solution found so far (using x/y>z => x>y*z as multiplication is faster than division), and if so check whether a*b is a permutation of (a-1)*(b-1) (this is the expensive part, so it's the last thing you check).

That will be just about fast enough (for me, a Python version finishes in 59 seconds), but some additional analysis lets you constrain the search space further. In particular, neither prime will be small; so it's faster to count down than up, and you can stop when you know that all remaining solutions will be worse than the best solution found so far; I got it down to ~7.4 seconds.

>>10569096
>φ(mn)=φ(m)φ(n)
This isn't always true anon.
>So now you only need to iterate over pairs of primes whose product is less than 10^7.
Why is this true?
Why can't there be a better solution which is a product of three or more primes?

>>10569088
>/mg/
>serious research mathematics thread
o i am laffin

>>10558280
>pure math or physics harder?
maths is harder, no doubt about it, as an academic career. Its abstractness makes it more difficult to keep your motivation about it, whereas in physics everything you do has real world implications that you can't seperate your study from.

In physics, you WILL achieve progress, in a meaningful way. "Pure" physics in that sense does not exist the same way "pure maths" does. You can study prime numbers till the end of your life, and not contribute a single thing to society, nor know anything new about how the world works, but there is no topic in physics whose study won't end up helping humans live better and more efficiently, and whose study won't personally improve your thinking and knowledge about the world.

In terms of "which is a more difficult subject to study?" I'd say physics. Maths has a lot more tools at its disposal, and it isn't nearly as interdisciplinary+multidisciplinary as physics. You need math for physics, but you don't need physics for math.

TL;DR: Which one requires bigger brains? Physics. Which one requires more dedication and effort? Maths.

>>10569139
>being this much of an undergrad

>>10569088
/mg/ is a serious thread about the applications of motivic cohomology to 1ccing Touhou 16 on lunatic, we don't care about research here.

how can I apply math to touhou? I want to get the PCB and MoF world records

>>10569181
hsifs is easy as fuck lole
>>10569209
talent

hmm i wonder what the point of the gamma distribution is hmm

>>10569181
>>10569209
>>10569222
/mg/ is raizing territory, fags.

Why isn't 0 a field? It's abelian and I can invert every nonzero element :^)

>>10569679
a field is defined to have an additive identity 0 and a multiplicative identity 1 != 0

>>10569687
sounds like a cope desu

>>10568862
I get that it's useful, but won't I learn how to compute things by doing proofs and more theory based stuff? Anyway, thanks for the good answer. I know what to do from here.

>>10557306
What's the most sophisticated maths topic that you could talk about with a 5-10 year old, with the most ease?

>>10569701
Elementary category theory

>>10569679
It's a convention. The zero ring is so different from every other field that it would require restating many theorems as "let F be a field with at least two elements." It's much easier to just rule it out altogether, even if that makes the definition of a field seem a little ad hoc. It's very much analogous to why 1 isn't considered to be a prime.

>>10568492
I don't mind that I got rejected.
I just have a problem with the fact that both of them are too lazy to tell me to fuck off.

>>10569701
Oh, easily algebraic topology. Homotopy probably most directly, but probably also plenty of homology.
>>10569826
Elementory category theory is not very sophisticated and extremely difficult to motivate to someone who hasnt got a broad mathematical background.

>>10569117
> >φ(mn)=φ(m)φ(n)
> This isn't always true anon.
It's true if m and n are coprime, e.g. if both m and n are distinct primes or powers thereof.
> Why can't there be a better solution which is a product of three or more primes?
It's not that there /can't/ be a better solution, but as a first approximation more factors means φ(n)/n is smaller. For numbers having roughly the same magnitude, the highest values of the totient will be for primes, followed by semiprimes, and so on. Each factor p^k reduces the totient by a factor of (1-1/p). More factors = more reductions, while a higher k means a lower p (for the same magnitude), which means higher 1/p and lower 1-1/p, i.e. a greater reduction.

It's not absolutely clear that the best result will come from a semiprime. But this is a recreational mathematics problem, not a PhD thesis. So having easily excluded the possibility of a prime solution, the sensible approach is to find the best solution for a semiprime and submit the result. And guess what? It's the correct answer. PE doesn't require rigour, it requires "good enough". You get another attempt if it's wrong.

Having obtained the best semiprime solution (2339*3557=8319823 => φ(8319823)=2338*3556=8313928 => n/φ(n)~=1.0007), I suspect that it would be fairly straightforward to prove that no value less than 10^7 with three or more factors can have a lower n/φ(n) ratio, irrespective of the permutation requirement (back of the envelope calculation: cbrt(10^7)=~215.44, (216/215)^3=~1.014, (216/215)^2=~1.009).

Is there a math branch that study system of multivariate polynomials of degree higher than 1?

>>10570827
Algebraic Geometry

>>10570849
Wait, algebraic geometry is not about geometry? That’s misleading.

>>10570872
It was originally about complex manifolds and algebraic curves, but then Weil and Grothendieck happened.

>>10570898
Thanks anyway anon

How do you deal with the fact that some people will always be smarter than you?
I’m smart enough to do what I need to do, but I always feel inferior comparing myself to true geniuses.

>>10570898
So, what's it aboutnow

>>10571096
>So, what's it aboutnow
categorical autism

>>10571114
What's the funnest stuff then mate

>>10571090
>I’m smart enough to do what I need to do

That's what matters anon. I'm not smart enough to do what I need to do and its killing me.

>>10571090
>How do you deal with the fact that some people will always be smarter than you?
By not being an insufferable egoist

>>10571090
>How do you deal with the fact that some people will always be smarter than you?
Nobody feels ashamed of packing an 8-inch wiener because there happens to exist a 1 in a million mandingo with a 12 incher.

>>10571242
...well, no I do...
in fact thats an incredibly lucid description of me

>>10571114
Excuse me?

>>10562665
Yeah this all checks out, Not.p implies (p implies q) is a tautology of classic logic.

>>10570872
It's about the geometry of objects defined by systems of polynomials equations

Daily Putnam Problem >>10571410

>non-brainlet, but bad with girls
>ask dad what to do
>don't worry anon, if you get a good job, become financially independent and buy a house, you'll be a catch
>10 years later
>cushy job at investment firm
>got all my shit sorted out
>still can't get girls

>tfw stability doesn't imply attractivity

>>10562573
>law of excluded middle
yikes

>>10571935
Get /fit/. No amount of money will make a decent girl date a lardass or a skeleton.
Also, how symmetrical is your face?

>>10571090
imagine caring that there are like 0.01% of people in the world who are smarter than you
like come on now that's just pathetic

>>10571945
>law of included middle
oh no...

>>10572247
Why is that pathetic

>>10572297
If you think denying the law of the excluded middle implies the oposite, you are using excluded middle brainlet.

>>10572247
.01 * 7,700,000,000 = 77,000,000 people. That's a lot of people.

>>10572361
>law of the possibly included middle
Oh no.

>caring about females at all when a fucking chatbot and a sexdoll and a surrogate mother will be a much better alternative to life.

Anyway, I am stuck at this point.
Anybody knows what I should do?

>>10572389
Do what this guy did >>10566750

>>10572389
No. Shaming is not working on me.
Go be jealous somewhere else V find yourself another victim

project euler protip: memoization

https://volafile org/r/w86qfmug

>dream of being a research mathematician
>realize there's people in my fucking group that mentally overpower me 100:1
>realize there's probably people who could overpower them

all it took was one number theory intro class where the professor has us compete in solving a couple of problems at the end every time to realize just how much pure intelligence and mental capacity actually vary between individuals

Any good books on linear algebra or discrete mathematics? I just started learning matrices and I want to go more in depth.

>>10572674
"Linear Algebra Done Wrong"
It's free

>>10572645
It doesn't fucking work like that you retard.

>>10572716
Brainlet

>>10572645
Just study more

>>10572716
Yes it does. If you disagree you're either in hard cope mode, or you're one of the those people yourself.

>>10572707
why would you want to learn linear algebra incorrectly?

What kind of part time job can I get with a BS in math? I want to get a Master's but I don't have money

>>10573053
you can't :)

How does someone get an intuitive feel for abstract algebra. Once polynomial rings and modules start showing up I kind of get lost in what's going on.

>>10557306
I can take either algebraic topology or differential topology next semester. I'm under the impression that differential topology would have more applications in physics. Is this correct?

>>10573078
Take a day off and visit the subject later.

>>10573095
Yup, can't do relativity properly without Riemannian, and Riemannian require differential topology.
>>10573078
How the fuck do polynomial rings confuse you?

>>10572361
Im not the same anon, and I cant really figure out what you're trying to say given thus chain of comments, but if youre implying that the law of excluded middle is a necessary principle of reasoning, then I'd suggest you look into independence proofs and intuitionistic logics.

>>10572707
linear algebra done wrong fucking sucks

>>10572196
>/fit/

>>10572196
What if I can't build muscle?

>>10572389
true you fucking retard
if p implies q, and q implies r, then obviously p implies r by transitivity of implication, whatever gay ass name you give that in your fancy """formal logic""" classes

>>10573078
there is none, that's why it's called abstract. if it had an intuitive physical representation it would cease to be abstract.
Also just be smart l0l

>>10573129
He's saying not p implies p is equivalent to the law of excluded middle, so saying that if excluded middle doesn't hold then there necessarily exists a possibility other than a statement or its negation relies on the law of excluded middle, a contradiction.

>>10573249
You can.

>>10573479
prove it

>>10573488
No problem.
*unzips dick*
*pisses on your face*

>>10573498
this is possibly the greatest math proof I have ever seen

Any new theorems on the category of axioms?

>>10572674
the best theoretical LA books are Axler's Linear Algebra Done Right (he also has lectures on YT) and Valenza's Linear Algebra: An Introduction to Abstract Mathematics. It sounds like you want Applied Linear Algebra, which I don't have any recommendations for other than David C Lay's, just because there's lectures for it: https://www.youtube.com/playlist?list=PLNr8B4XHL5kGDHOrU4IeI6QNuZHur4F86

For Discrete Maths, see Thomas Van Drunen's or László Lovász books (Discrete Mathematics: Elementary and Beyond is particularly readable and brief for an introduction). I didn't like Epp's or Rosen's. MIT also has the free "Mathematics for Computer Science" pdf, it's over 1k pages but I've heard it's good.

>>10573078
Work through examples. Take whatever theorem applies to K[X1,...,Xn] and run the statement and proof on Q[X,Y]. You should have already learned to work with simple polynomial rings like Z[x] in high school, they just called it algebra

is anyone else /model theory/?

>>10573863
>Axler's Linear Algebra Done Right
cringe

>>10573329
In fact "p implies p" is not equivalent to "not p or p". The former is a weaker condition and is implied by the latter, but the implication doesnt go the other way around in some non-classical logics since "p implies p" is true in intuitionistic logics while "not p or p" is not.

>>10573891
It maybe a meme but definitely not cringe. Literally Harvard use that as textbook

>>10573891
>Here's your linear algebra textbook bro

>>10571090
take some psychedelics and free yourself of the ego

>>10573982
I never said it was, I was talking about the statement "'not p' implies 'p'"

Is there a mathematica for poorfags? I just want to check some solutions.

>>10573883
I just learned that the spectrum of a Boolean ring is the space of models of the associated Boolean algebra.

Anyone want to help a brainlet work through a project euler problem?
https://projecteuler.net/problem=91

The search space is only 50*50*50*50, so iterating all combinations is possible.
Take all points where 0 < x, y <= 50.
Pairwise combine these points with themselves to get all combinations (P, Q)
You now have duplicates of every point because OPQ = OQP, so prune all combinations where P.x >= Q.x or P.y >= Q.y.
This also gets rid of invalid triangles, such as where P = Q.
We now have all valid triangles, so count the ones that are right triangles.

Is my thinking correct?
Also what's the best way to check for right triangles? Using a^2 + b^2 = c^2 is a bit iffy because you have to convert to floats, and float representation is inexact, and square roots are just approximations.

>>10557306
Currently in Calc 2, I'm a Physics major freshman. For this class we have to do a project on the proof of Bessel's Equation. Most people are half assing it but I'm finding it interesting. Only problem is the project isn't any math, just an essay on who Bessel was. Unironically, is there any point to a math teacher assigning an essay that's just a biography?

>>10574781
>Unironically, is there any point to a math teacher assigning an essay that's just a biography?
No. Because you don't get better at mathematics because you wrote a history paper on someone who did math.

>>10574750
I feel like that’s too much work.

Look at the examples it gives for search scope of 0-2, what do you notice about the relationships of all the x,y pairs? Those are the only sorts of relationships that can form right triangles. Try to abstract them as much as possible.

>>10574750
For checking for right triangles you could just check the two segments connecting to the origin and then you know at least on of the other segments that are attached has to be perpendicular

Check slope of OP, PG, and OG, one of the segments is going to have a slope perp to one of the other ones. (Check your valid triangle list)

>>10573985
Harvard is a meme.

>>10575005
and yet each and every math grad from harvard is better than you and your buddies at your shitty low-tier college will ever be

>>10574750
> Take all points where 0 < x, y <= 50.
Not quite. You also want the points where either x or y are zero (but not 0,0 itself).

> You now have duplicates of every point because OPQ = OQP, so prune all combinations where P.x >= Q.x or P.y >= Q.y.
No, that's discarding valid triangles. Just make a list of all 51*51-1=2600 points, then iterate over all unique pairs (i.e. 0<=i<2600, i+1<=j<2600). If you want to operate on the coordinates, you need a total ordering e.g. discard any points where (P.x>Q.x OR (P.x==Q.x AND P.y>Q.y)).

> Also what's the best way to check for right triangles? Using a^2 + b^2 = c^2 is a bit iffy because you have to convert to floats
No you don't. Bear in mind that (x2-x1)^2+(y2-y1)^2 gives you the square of the length of the edge between (x1,y1) and (x2,y2), which is what you need for Pythagoras. You don't need the actual edge length for anything.

Another option is to use the dot product of edge vectors, dx1*dx2+dy1*dy2=0 if the vectors are perpendicular. Either way, each triangle needs 3 such checks (any of the three angles might be a right angle).

>>10575019
Eeeeeeeeeeh
No

>>10575019
cope harder faggot, nowadays it's just a matter of who has the best networking skills

>>10575027
> implying that harvard doesn't offer you the most prolific social ecosystem where to network your way up to academia

>>10575058
Well yeah.
https://www.theamericanconservative.com/articles/the-myth-of-american-meritocracy/

brainlet here, anyone else who feels or have felt anxiety trying to study math? i've always had this feeling of fucking up and wasting my time because im too stupid to do it, in highschool my education regarding math was pretty shitty now that i look back (i.e teachers resigning and not having math class for some weeks or months in some extreme cases).

>>10557306
>Extract from letter 14, 14/06/83 from Alexander Grothendieck to Ronnie Brown

>Your idea of writing a ``frantically speculative" article on groupoids seems to me a very good one. It is the kind of thing which has traditionally been lacking in mathematics since the very beginnings, I feel, which is one big drawback in comparison to all other sciences, as far as I know. Of course, no creative mathematician can afford not to ``speculate", namely to do more or less daring guesswork as an indispensable source of inspiration. The trouble is that, in obedience to a stern tradition, almost nothing of this appears in writing, and preciously little even in oral communication. The point is that the disrepute of ``speculation" or ``dream" is such, that even as a strictly private (not to say secret!) activity, it has a tendency to vegetate - much like the desire and drive of love and sex, in too repressive an environment.

Thoughts?

Fuck Riemann and fuck Darboux.

>>10575219
>Fuck Riemann and fuck Darboux.
Do you really need to swear?

Hello math titans, another brainlet here. I study philosophy and the history of ideas, so I know a lot (relatively speaking) about the historical development of math, but I suck balls at actual math, and my ability to follow the history falls apart in the 19th century, especially the 20th, because it obviously gets too complicated.

I have tried learning math before, and I got up to basic college calculus or so. But I was miserable, because I was learning it from standard textbooks that assume you want to get the bare minimum technical information, learn it by rote and repetition, and move on to more complex practical applications as quickly as possible. I think that killed it for me.

So I'm going to make learning math into a hobby this year, take it slower this time, and try to follow the history of its development, starting with the Greeks and moving through the middle ages and early modern period before I get to the moderns. Obviously I'll have to balance plain old textbooks with other materials, but this is roughly how I want to do it. I'd like to read about the philosophy and epistemology behind the developments as I go along, too, so nothing ever stands out as "this is just how they did things, after this guy invented it." I want to know how and why he invented it.

I know this is a longshot, but did anyone else learn this way? Or have any recommendations for someone who wants to? I'm willing to take it really slow and methodical, and read a lot.

>>10575236
>Or have any recommendations for someone who wants to?

>>10575241
I'm going to end up as a copypasta when I try to follow this image literally and end up coming back to /sci/ a year later schizophrenic and still shitty at math aren't I

>>10575241
>quantum cohomology
fuck off rapcak

>>10575278
>rapcak
a what

They see me rolling!
They hate it! Hahahaha, sorry guys I couldn't help it!

Sorry I'm not good at math. Just here because I thought I might brighten your day!

>>10558260
is that supposed to be hilbert in the math room

>>10575295
see>>10575298

>>10575328
Sorry. I was just trying to make this a brighter place. If you want I should leave? Is better than be told insults.

>>10575027
How dose one network in math these days anyways? Seems like shoving a bunch of math autists in a small cramped room is the quickest way to high pitched shrieking

>>10574923
I agree that there's a pattern to it, and it feels like it should be possible to generate triangles based on those patterns. I also feel like it should be possible to generate every possible "shape" that an integer right triangle can take on, and then plot it on the 50x50 board to see if it fits, but that also doesn't feel like the easiest solution.

Maybe the two together, find all pairs of angles that make up a right triangle, since (0, 0) -> (1, 50) is the smallest angle you can make on this board there should be only say a couple of thousand combinations of angles that add to 90. But this would be a lot of work too, and would involve imprecise angle calculations because of floating points again.

Or how about this: Do an exhaustive search with 1 of the points, i.e. place P on every possible coordinate, and then generate every right triangle based on OP. Which I'm not sure how to do either, hmm.

>>10575021
>Not quite.
>No, that's discarding valid triangles.
>No you don't.
Very good points.
>dot product
Oh shit that's right.
I guess searching through the possibilities is easier after all, you just have to not be a brainlet.

>>10575608
but... i am a brainlet...

>>10575608
>dot product
Wait a second, everything is perpendicular to (0, 0) right? So you would have to use some other method to check whether OP or PQ is a right angle.

>>10575639
Same.

>>10575674
Is this correct?
P dot Q accounts for either P or Q being right angles, so the only thing remaining is whether O is a right angle, which is easy enough to check because that means that either either OP or OQ has to run along one of the axes?

>Be bad at maths
>Take physics 101 in shitty community college and fail because I suck at math
>Change major to mathematics because I decided that I want this one fucking weakness forever abolished to do that I'm going to put myself in situations where either I pass or I get kicked out of college
>6 years later walk out with my master's, finally math is no longer my worst subject.
>Now I can go and do almost anything I desire
>Family asks what I desire to do next
>I want my PHD in math
Have I become the mask? I have nothing left to offer the discipline but I want more? Why?

>>10575021
>Just make a list of all 51*51-1=2600 points
This isn't correct. This gives a list like ((0, 0), (0, 1)), ((0, 1), (0, 2)), ...
But what about ((0, 0), (0, 2))? (Somewhat bad example because P = (0, 0) isn't valid but you see the issue anyway.)

Total unique pairs should be something like
51 * 51 positions for the first point, multiplied by
51 * 51 for the other point, minus
(51 * 51) * 2 for where either point is (0, 0), minus
(51 * 51) * 2 for when the points are equal.

>>10575236
>I know this is a longshot, but did anyone else learn this way?
No way.
>Or have any recommendations for someone who wants to?
>I want to know how and why he invented it
>follow the history of its development, starting with the Greeks
>moving through the middle ages and early modern period before I get to the moderns
You will never get to the moderns this way. Tradition is good and wrong, but a newbie investigating this stuff will not make out anything good. As long as you stick to (under)graduate mathematics, there is no good reason to go off trail. Many books explain the original idea behind a concept in the modern language, and your job as a student is to give an interpretation to what you read.
Tracing back history and the reason why someone introduced a concept is way harder than undestanding the idea itself, so it is not that useful.
So if I had time, I would rather read some chapters from Clelland's book on Cartan formalism than the book from Borceux on the 35 lovely centuries on (differential) geometry.
>>10575241
Second.

>>10575236
Read Weil's book on number theory.

>>10575805
>I have nothing left to offer the discipline but I want more? Why?
Social pressure. Greed for prestige

>>10575674
The dot product of the edge vectors, e.g. OP·PQ=0 iff OPQ is a right angle. OP=P-O=P, OQ=Q-O=Q, PQ=Q-P. IOW, P.x*(Q.x-P.x)+P.y*(Q.y-P.y)=0 if OPQ is a right angle.

>>10575706
> P dot Q accounts for either P or Q being right angles,
P·Q = (P-O).(Q-O), so P·Q=0 iff POQ is a right angle. But you can handle that case specially; POQ is a right angle iff P.x=0 and Q.y=0 (or vice versa).

>>10575821
> This isn't correct. This gives a list like ((0, 0), (0, 1)), ((0, 1), (0, 2)), ...
It gives a list of points, not pairs of points. Once you have that list, you enumerate distinct pairs with e.g.
for (i=0; i<pts.size(); i++) {
....P=pts[i];
....for (j=i+1; j<=pts.size(); j++) {
........Q=pts[j];
This guarantees that you don't have P=Q and also that you don't enumerate each pair twice. This is simpler than having four loops (P.x, P.y, Q.x, Q.y).

Bear in mind that even with this naive approach, it only takes a few milliseconds to check every possible triangle. If the grid was much larger, you'd enumerate ratios and calculate how many triangles have an edge with that ratio. E.g. for P.x/P.y=2/3, that gives you P in {(2,3), (4,6), (6,9), ...} with the size of the set obtained by dividing the size of the grid by the larger number and rounding down. You then know that (Q.x-P.x)/(Q.y-P.y)=-3/2 (product of slopes is -1 for perpendicular lines), so you can easily calculate the number of valid points Q for each P, rather than enumerating every pair of points and testing.

Could you explain to me why math students have an inferiority complex towards cs students?
We don't even think about you, so why are you all so obsessed with us.

>>10575926
you guys are actually useful

>>10575934
I'm well aware, but how does that make the typical math fag angry?
Its not our fault math majors chose their major, they should've known what to expect.

>>10575926
You’re projecting.

>>10575964
where's the linear operator?

>>10575926
>We don't even think about you, so why are you all so obsessed with us.
No one:

Comp fags: Why do you guys hate us so much, huh? huh? im so smart
arent i so cool, i get to copy paste other peoples work, im smarter than math

you should get a personality
also >>10575968
>cs fags have only taken linear alg, imagine my shock

>>10575896
Thanks a lot, I finally managed to get it right. I swear I'm not usually this retarded, I just haven't touched vectors (or maths) in years.
Here are the fruits of your labor:
>>>/g/70633003

>>10576033
Wait, I thought Comp Sci fags and Math fags don't work together, as >>10575926 said!

>>10576039
You leave me out of this, I graduated many years ago (although admittedly from comp. sci.) and have no need to project anything onto maths students.

>>10575964
>>10576019
You guys make constant threads about cs because you're obsessed. Pathetically so.
Why is that?
You never see us make threads about you because we simply don't care about you. We don't suffer from the same inferiority complex you people do.
So why do you feel so inferior?
Why us?

Your laughably snide remark about linear algebra is telling of your inferiority complex. You need to lie to yourself to justify your childish obsession with people who care nothing for you.

>>10576039
I should've just said people on /sci/, but you get my point.

You people seethe with an inferiority complex.

>>10576161
>I should've just said people on /sci/
Yes, you should, but you didn't. Instead, you came to our general to spit these vile words at us, like the dishonest prick you are.

>>10576223
Awww I'm sorry for hurting your feelings anon, no need to cry.
I wouldn't have to come in to your safe space if your ilk wouldn't obsess over us so much. So please stop with the obsession. Thanks.

>>10576161
>I should've just said people on /sci/, but you get my point.
Mathematiians use "we", not "I".

>>10576138
Literally the only time I hear about mathfags “feeling inferior” is when some CSfag makes up shit like this.

People who study math are too busy actually learning to think on a higher level to give a shit.

>>10576308
>Literally the only time I hear about mathfags “feeling inferior” is when some CSfag makes up shit like this.
Why then does your ilk make thread after thread hating on cs?
Its because your inferiority complex, just accept it.

Supposing I have a level set of some function that implicitly defines one of the variables as a function of the others, i.e. [math]G(x,y,z(x,y)) = K[/math]. Would it be correct for me say that [math]\frac{\partial G}{\partial x} = \frac{\partial G}{\partial z}\cdot \frac{\partial z}{\partial x}[/math]? "Cancelling" makes sense, but I'm having issues justifying it.

>>10576494
Lmao no, you should at least remember the multivariable chain rule.

Daily Putnam Problem >>10576533

>>10575206
He's right that there should be more room for talking about stuf that is not entirely formalized yet.
I don't think we should all be writing papers in that way, because we don't want to become like the physicists.

Am I missing something here or is this just a really annoying way of saying >0?

>>10577038
it's stronger than just being greater than 0. it means you have a uniform bound away from 0. Consider that 1/n is always greater than 0 but in the limit its 0 vs 1/n + 1 which is uniformly greater than 0 and in the limit it stays greater than 0.

>>10577038
it's arbitrary bro

How do I learn how math "works"?
Everything I google math related just comes down to "use khan academy", but it doesn't seem to have any part specifically explaining how or why 1 + 1 = 2.

>>10577206
>but it doesn't seem to have any part specifically explaining how or why 1 + 1 = 2.
If you need an entire lesson explaining why 1 + 1 = 2 you have a severe learning disability

>>10573479
I couldn't build any muscles in spite of years of lifting

>>10577211
>why does 1+1 = 2?
>npc explodes

>>10577206
How math 'works' is the main question of the field of mathematical foundations or methods to formalise mathematics. However, these fields use rather involved mathematics on their own, so you would first have to learn mathematics (get very familiar with elementary proving techniques, at the least) before you can appreciate how it can be made to 'work'.

>>10575241
based and endorsed

>>10576494
>the animeposter can't do calculus
What a surprise.

>>10575241
>Galois theory
>High school
What did he mean by this?

>>10577477

> tfw I unironically undertsood the proof of the Abel–Ruffini theorem in highschool

>>10577206

if you really want to know, google "peano natural numbers"

Say I were playing DnD and I had to half a damage role that would normally involve multiple dice (eg, 4 d4). Is there a statistical difference between rolling the normal number then halving the result and just rolling half the number of dice? I think the means should be the same but will the variance differ?

>>10576515
>tfw was literally never taught this and made myself look like a dumbass in front of my complex analysis professor

take the expected value of each dice (in this case 2.5), multiply it by 4 to represent each dice (giving us 10), then divide by 2 to represent half that roll for damage (10/2=5)

>>10577206
Well for 1 + 1 = 2 there are really two things going on.
One is that the natural numbers are very, well, natural in that they make up an immediate fact about our physical reality. 1 + 1 = 2 in this sense is just an obvious truth, if I have one apple and then add another apple I have two apples. Even a toddler understands this.

However for the purpose of mathematics we don't want to rely on obvious physical intuition because that might not be logically sound. Instead we construct a precise definition for what we mean by "1", "2", "+", etc and logically prove the statement. The typical way this is done is define 0 to be some primitive object, then define the "successor function" S(n) that takes a natural number and outputs the next one. Then 1 is defined as S(0), 2 as S(S(0)), 3 as S(S(S(0))), etc. Then we can define addition as a function plus(a, b) that operates on this representation.

You can read more here:
https://en.wikipedia.org/wiki/Peano_axioms

>>10577779
>>10577853
Oh cool, I actually know most of this accidentally by learning CS, despite not knowing algebra or calculus at all.

It's also very hilarious that my brain is turning what I'm reading into "code" in my mind and then I'm parsing through it and figuring out backwards.

Add(x, S(y)) = S(Add(x, y))

Cool that I can create "add" by doing this.

>>10577978
You will enjoy functional programming, like Haskell. It's all about doing what you're doing there.

What’s a decent book to start pure mathematics?

Everything I have learned from school and college thus far is applied and proofs are never even thought of.
I know up to and including elementary Laplace transforms

>>10577810
There might be a small difference if you round the number after halving it (whether you round up or down will add some bias toward higher or lower rolls respectively). Otherwise both should define roughly the same Gaussian distribution.
>>10577206
Sounds like you might be interested in set theory and mathematical logic, which are, among other things, the backbone of Peano's construction of the natural numbers.
>>10577853
An interesting book about the relationship between (human) intuition and mathematics is Lakoff and Núñez's "Where Mathematics Comes From".

>>10575236
hhi anon, this post is funny because i just started doing this right now

right now i am going through a more pop-math book like this: http://jwilson.coe.uga.edu/EMT725/References/Dunham.pdf

afterwards im probably going to go through this book that teaches much more stuff:

https://www.springer.com/gp/book/9781441960528

i have a shit ton of downloaded books for this journey too but i feel like this will take me literally years to complete so i dont really recommend

[math]\color{RED}{\heartsuit}[/math] Daily Putnam Problem >>10578703

>>10578727
[math]\mathrm{\LaTeX} \text{ not working? :(}[/math]

>>10578733
>[math] \color{RED}{\heartsuit} [/math]
No, it just bugs sometimes when you don't put spaces between everything.

>>10566001
I've heard that everything in math is either linear algebra or combinatorics. In the first case it's trivial and in the second it's hopeless

>>10573498
high iq post

https://projecteuler.net/problem=85

Why does (1 .. x)(1 .. y) give the correct number of rectangles in the grid?

>>10578890
To be clear I mean the product of (1 + 2 + ... x) and (1 + 2 + ... y)

>>10578903
Nevermind I found an explanation.
https://www.mathblog.dk/project-euler-85-rectangles-rectangular-grid/

why is mathematics so full of weebs? yuck

>>10578971
>why is mathematics so full of weebs? yuck
t. reddit

Thoughts on the CTMU?

>>10578971
>weebs
Do you kiss your mother with that mouth?

>>10578323
Yeah, I already use FP with Elixir, again accidentally because I like the concurrency aspect of it all.
I didn't ever really plan to learn all this stuff, I just naturally am interested in why it works, it's very simple and easy to understand.

But when I see some massive graphs and weird squiggly lines I am so confused, but I am not slowly getting it and what all the little symbols mean.

>tfw brainlet

Daily Putnam Problem >>10579894

>>10573316
thanks m8.
very appreciated

>picks up a copy of LADR
This book was written for retards.
>picks up a copy of Lee's Smooth Manifolds
This book was written for children.
>picks up a copy of Fuchs-Fomenko
This was written for the cultured and tasteful.
>picks up a copy of Gelfand-Manin's Homological Algebra
This wasn't written for anyone.

What's actually the problem with having a generating set for a module that's not linearly independent (not a basis)?

>>10580723
Consider Z as a module over Z with base {2, 3}.
Essentially having non-unique coordinates is extremely pathological.

>>10560873
>definition
That's not 'just' a definition of R. That's a construction. And there are many many many of them.

You can just say that R is a nonempty, totally ordered field that satisfies the 'completeness' property. That is, the least upper bound property/monotone convergence property/Bolzano-Weierstrass property/Cauchy convergence property/nested intervals property/"intermediate value theorem" property/etc etc etc. Pick what suits you the best to get the intuition.

To show that your assumption is non-vacuous, i.e., that in fact there is a nonempty totally ordered field with the completeness property, you have to construct the real numbers, so that obviously is going to be weirder that just defining it.

The most natural way to define R is as sequence of of numbers between 0 and 9 (decimal expansions), but to me Cauchy convergence is the best. BW is definitely the weirdest one.

>>10580723
Well you lose a huge chunk of the theory of matrices: because there are no well-defined coordinates, you do not have a one-to-one correspondence between matrices and linear maps, which makes everything a LOT more complicated.

How do you prove that in a noncommutative ring an ideal P being prime is equivalent to any one-sided ideals A and B having the property that [math]AB \subset P[/math] implies [math]A \subset P[/math] or [B \subset P[/math]?

>>10581308
I dunno, did you try just proving it?

>>10581323
yes but I'm a brainlet so I couldn't

>>10581327
Yeah but did you try like, writing out the statement AB is contained in P in extensive form or whatever?

>>10581308
Well what's your definition of prime in noncommutative rings? Because what you've said is usually the definition