/sci/ - Science & Math

>>15081116
Is Abbott's Understanding Analysis a good introductory RA book? I'm currently on chapter one and I find the way the book proves stuff in handwavy/nonformal way is really frustrating. Please tell me it does get better in later chapters, I'm considering switching to Cummings right now.

that album is such a meme

>>15081271
there’s nothing handwavey about that proof. no matter what you do it’s going to be a little ugly because 2/3 = 4/6 and you’re supposed to avoid counting the same set twice, and 1/0 doesn’t exist.. etc

How much math can you realistically learn and master within a 2 month period?

I'm a grade 11 STEM student (18 yo) and I want to learn all of calculus up to basic undergraduate analysis. Is it possible to do it within 1 summer break? My current level is on infinite series, btw.

>>15082231
Don't think to much, just do math, calculate a lot, try to be creative. Routine is increadibly important in learning new stuff in mathematics, independently of the level you're at. "Mastering" something is never really possible.

>>15082270
>Mastering" something is never really possible.
How comfortable should I be with the concepts presented in a chapter to move on to the next? My usual rule of thumb is to perform exercises this way: do the first item, if you got it correctly, skip 1 item, then if you got that correct as well, skip 2 items and so on. I always do the applied word problems at the end of the set.

>>15082231
Unless you're a genius math savant (you're most likely not, seeing as you're asking a question about learning math on 4chan of all places), you might be able to cover little more than the bases of real analysis and calculus.
Far too many people have started learning calculus on their own with the intention of getting into real analysis and have gotten stuck on limits and/or differentiation.
I'd recommend you to pay very close attention to every subject introduced in your book/course, go through as many exercises as possible (with particular emphasis on hard-looking ones), and to try to use various resources if necessary for the sake of getting a good grasp of many concepts and techniques you might have to use. It might not hurt to learn set theory and the basics of symbolic logic, but it might be better to refrain from looking into other branches of math until you've gotten a good, solid understanding of Calculus.
Most university students take real analysis after 3 semesters of Calculus, so try not to rush through it.

>>15081271
>le author tries to make his proofs long and have thorough explantations
>le narcissistic undergrad calls it hand wavy thinking it makes him looks smart

What evolutionary pressures are responsible for the genes that make good mathematicians existing in the first place?

Has anyone of you read Baby Rudin from cover to cover

>>15082279
>applied word problems
I have no idea what that is, but if you're (very) comfortable you can of course skip stuff. Just don't skip too much.

>>15082231
>pic
Too close to home…

>>15082231
You can do most of univariate Real Analysis within 3 months, including construction of numbers and all that, but you'll have to rush it. If you already know some basic Calculus, I would recommend you jump straight to Real Analysis with Abbott.

I think it is very weird that:
[eqn] \int_m^{m+ \delta} (m + \delta) f(x) dx \geq \int_m^{m+ \delta} x f(x) dx [/eqn] if [math] f (x) \geq 0 [/math] regardless of sign of [math] \delta [/math]

>>15082047
>>15082292
I'm a literal brainlet who just finished Velleman. I just think Velleman did the proof better using the fact [math]\mathbb{N}\times\mathbb{N}\sim\mathbb{N}[/math]. Anyway I just finished the exercises. It's better than I expected, he seems to leave some of the important theorems as a guided exercise, which I like. I think I'll give chapter 2 a try.

>>15081271
Apostol Analysis is better

>>15082668
>le symbolic proofs are hecking better!!!
Abbott's proofs are pedagogic. They are meant to show how to think about the whole thing. Symbolic proofs are devoid of any soul.

>>15081271
>switching to Cummings
kek he's much worse

What exactly is the difference between a function and an operation?

>>15083648
Notation
They're synonyms but unless you're supremely autistic you don't write things like addition with function notation

>Can't do math for long, back starts hurting.

>>15083648
An n-ary operation is a special type of function whose domain is a subset of [math] S^n[/math], and codomain is [math] S[/math], for some set [math] S[/math].

>>15083801
do sports more often

>>15083648
>What exactly is the difference between a function and an operation?
No difference.
>>15083801
Code?

>>15084120
That would mean doing math less, which is exactly my point.
>>15084485
Why would I code?

>>15081116
Why the fuck are you allowed to take away values from a (supposedly) equivalent equation? It has no applicability in real life. If you're looking at an equation, it should either be true or false.

>>15084485
>Code?
Stop being a coomer and do maths instead.

>>15085149
What do you mean?

Just realized today that calc 3 isnt advanced math. FUCK bros, i started learning math last year being excited about higher math courses that I thought were in the threshold of higher math. I was so excited for calc II because everyone was bitching about how muh hard it was and now that I have reached it, is just baby math. (still exciting at times though)


At what point can I safely call myself a math person? when i reach abstract algebra?

>>15082281
>Far too many people have started learning calculus on their own with the intention of getting into real analysis and have gotten stuck on limits and/or differentiation.
Neither of those concepts, nor any of the other concepts in an introductory analysis course, are very hard to grasp. I felt that certain concepts were hard to apply in problems once an actual calculation arose (uniform continuity for example). But I don't think limits or differentiation really fall into that category. The epsilon-N definition of a limit is pretty straightforward, and applying it to a derivative is almost always the same. Do algebra until you're no longer dividing by zero. Apply the easy limit laws that you know.

Filthy undergrad here, should I go into Functional Analysis (more specifically Operator Algebras) or Evolutionary Computation?
I'm clearly better at EC, and it's probably a more profitable field, but fun anal is so much more interesting, but I'm not exactly that good at it.

>>15086119
>and it's probably a more profitable field,
Where is this information?

>>15085779
don't gate keep yourself. if you got past calc1 and understand it, I would say youre already a math person since most people can't get past college algebra. I read college algebra has a 50% drop rate.
take calculus 4, or differential equations as some call it. that's where it really. opened up my eyes to what you can do with math.
but if you got through calc3, you're a math person in my eyes. but yes, in the grand scheme of things, calculus is a tool. not, advanced mathematics.

>>15086119
always go with what's interesting to you. do you know any CS? if you get a math degree and throw on your resume that you know CS, you'll get jobs easily. a math major is seen as a problem solver, which employers really dig.

Is there such a thing as an inverse geometric series? So let's say there's a dude walking every day, he reaches 27 units of distance total by the end of the 4th day, each day he walked 20% more tha nthe previous day. I've been trying to figure out a way to write it so it works backwards from 27, I'm not smart enough though.

>>15082231
You don't really need calculus to do analysis, it provides more grounding/justification for your proofs. I would try them concurrently as your not gonna run into the reasoning for multivariate calc until you get to the end of your undergrad analysis book.

But your gonna need a good sense of how to do a proof and what sets are to do analysis. Both of which should be covered in the first chapter, if this isn't enough for you the extra resources at the end of the chapter should provide some books on these subjects.

In lecture 28 of Arthur Mattuck's MIT differential equations course, professor Mattuck talks about the fact that the general solution to a 2D system of ODEs is a linear combination of linearly-independent vectors. In that context, says something along the lines of "It's easy to show (by superposition and linearity) that any linear combination of these vectors is a solution. What's hard is to show that EVERY solution is a linear combination of these vectors"

Is it really so hard? Can you not just use the existence and uniqueness theorem to show a bijection between the (2D) space of initial conditions and the (2D) vector space of solutions which you've found?

>>15082376
nah
i pretty much skipped real analysis and went straight to topology, functional analysis etc and filled in the gaps as i went along
all's well

what's your go-to comfy branch of maths that you study when you wanna chill out
for me it's combinatory logic

have to decide on modules for the incipient term soon. undecided whether to do Game Theory or Quantum Information.
thoughts?

>>15087004
I got laid once by semi-plagiarizing the game theory scene from Beautiful Mind (told a girl she was too ugly for my chad wingman so she was stuck with me)

>>15086995
I usually go back reading something I should know but don't such as pic related.
Chapter 2 is where the fun starts

>>15087010
right, so game theory will get me laid. is there any way to get laid via quantum information?

Is there a big checklist somewhere of all of the tools one should be competent with before calculus? Completing the square, trig formulas, complex conjugates, logarithms etc
I don't know if what I learned back in high school has any gaps or not.

>>15087089
I don't think you'll need complex numbers at all for Calculus I-III. You might need conjugation, polynomial division, and various other rationalization techniques, however.
Also, you'll definitely need substitution, trigonometric identities, logarithmic and exponential functions, partial fraction decomposition, and conics. I doubt you'll need it much, but it might be good to know how to use Pascal's triangle to expand a binomial.

You'll most likely learn a few more algebraic tricks along the way while learning calculus.

Ho hoo hooo I bet /mg/ couldn't even solve this!

Just kidding, I bet you could. It's a nice problem (In my own opinion which may be wrong). The book says it's from euclidean ramsey theory. Anyhow, here it is! Go ahead, try if you want to. Or not. It's your life, your choice. But I'd be happy if you did try and talked to me about it.

Also a bit late but merry Christmas from Turkey(the country, not the bird)!

>>15087334
>monochromatic equilateral triangle
I assume, not including the interior of the triangle?

And may I ask, is it possible to show there is a monochromatic straight line (extending infinitely in both directions)?

>>15087334
What is your definition of a monochromatic triangle? Do only the three vertices have to be the same color, the vertices and the edges or the whole inside?

I don't think the statement will be true in the last case since you can partition the plane into two disjoint dense subset and give both a different color. The inside of the triangle will always contain a ball so it will always have points of both colors.

>>15087362
Only the three vertices have to be the same color, the edges and the interior are irrelevant. As for your question, I can not answer it right now unfortunately I have to attend class.
>>15087370
This answers your question as well I believe.


This is irrelevant to the discussion at hand but my friend who is next to me insists I include the n word in this reply so I shall.

Nigger.

>>15086559
>You don't really need calculus to do analysis
You do because a lot of book are written with the expectation that you know calculus.

>>15087089
nothing big that you can't pick up while studying calc

>>15087065
it’s 1/sqrt2 : 1/sqrt2, either it will or it won’t

>>15087334
>monochromatic equilateral triangle
This is ambiguous in that you might mean the entire perimeter or just three vertices. There is a short solution for three vertices:
>suppose have a counterexample
>wlog pick any two Blue points
>deduce that two other points must be Red
>deduce that two new points must be B; these are collinear with the original two B points
>like this we can make a regular-spaced grid of alternating lines of Bs and Rs. But that’s got equilateral triangles all over the fucking place
Not sure how to approach the full-perimeter version

>>15087565
I meant just the three vertices.
Yeah, what you said seems to make sense. Nice job! Good way to go about it.
As for the full perimeter version, would it even be true? I'm not sure but say if we labeled every point with both x and y coordinates irrational red, and the others green, would there be a monochromatic equilateral triangle with edges? There would be infinitely many green lines, for example y=0 but then the other two edges have a irrational slope so there would be red points in those lines, I think. Let me know what you think, I don't actually know but this is interesting.

>>15087723
>the other two edges have a irrational slope so there would be red points in those lines
There would be even if they had a rational slope. it's a fairly elementary proof in analysis that there's an irrational number between any two numbers, so the only way you'd end up with a completely green line would be if one of the coordinates was constant.
It would be possible in spherical geometry, then, but not Euclidean.

>>15087733
Right, so then it's not even possible when we consider the triangle with edges, let alone one with an interior. An interesting question would be to ask what if we had 3 colors instead, could we find a monochromatic equilateral triangle then(I mean just the vertices of course just to be clear)? If I recall correctly, I read something regarding that and it wasn't possible but I'm not really sure. Don't even remember where I saw it.

>>15087748
It's still true, but the proof is substantially more complex than the two-colour case.
Colour in the positive integers arbitrarily using three colours. van der Waerden's theorem says that, if we consider a large enough amount of them, we can find a monochromatic arithmetic progression of any size.
So we are forced to have such a progression of arbitrary length on the x-axis; assume without loss of generality that the points in this progression are red. This means that any point equidistant from any pair of these must be either green or blue; these points will form an equilateral triangle above the axis.
Now consider the bottom row of this smaller triangle. We can apply the theorem once again to show that there must be another monochromatic progression here, which we'll arbitrarily set to be blue. If we've coloured in enough points on the x-axis, this progression must have at least three points A, B, and C. Let D be equidistant from A and B, E equidistant from A and C, and F equidistant from B and C. Since A, B, and C are all blue, none of these three can be, so they must all be red or green. However, any of these three is also equidistant from a pair of points on the x-axis, so none of them can be red, either. And they can't be green, because they form an equilateral triangle with each other.
So we're left to conclude is that such a triangle will always exist, although the numbers involved are big enough that it's probably best to not brute-force it.

>>15087794
>It's still true, but the proof is substantially more complex than the two-colour case.
Do you know of any other things this is connected to in mathematics, especially elementary things or examples?

>>15087794
and realising that it's not the easiest thing to understand in text form I made a shitty diagram of it.
the main thing is that such a red sequence and such a blue sequence are guaranteed to exist if we go long enough (emphasis on long enough: the lower bound on the number of points needed to guarantee a six-member red sequence is over 11000)

>>15087794
Brilliant. Thanks a lot for your explanation. I must have misremembered what I read wherever I have read it. Grateful you corrected it. I have an exam tonight so I'll be quite busy however I'll try to learn more about this and read what you have written carefully and go over it myself. I'm glad I posted that problem. Hope we can talk more soon. If you're interested in combinatorics would you like to join a combinatorics discord server? I could post the invite link if you're interested.

>>15087723
>say if we labeled every point with both x and y coordinates irrational red, and the others green
You can make it easier: (x,y) is red iff x is irrational. Now you have these dense vertical stripes of two colours, and every triangle has to cross the stripes on at least two sides

need more science,

How do we generalise addition to topological spaces? What is the sum of two different topological spaces? Is there a useful appraoch that isn't just the cartesian product?

>>15087334
>>like this we can make a regular-spaced grid
Why is the grid regular-spaced? I don't think your proof works.

>>15091036
meant for >>15087565

foundations of mathematics - perfect, or flawed?

>>15081116
So I basically completed Spivak's Calculus and I feel like I have a pretty good grasp on single-variable calculus, not just theoretically, but also in application. It doesn't really seem complicated and the author was absolutely phenomenal in his writing.

Now I wanted to start a book on multi-variable calculus, so I tried going for Callahan's Geometric View, but I feel completely mogged by the first chapter alone. The notation is convoluted and the steps don't seem clear, as they were in Spivak's book. What's the issue here? Am I just too much of a brainlet for multivar calculus? Is this my limit? Or should I just keep reading even if I don't understand it yet with the hopes of eventually warming up to it?

>>15091012
For adding spaces together it's more common to use disjoint union as sum and cartesian product as multiplication, then you get distributivity. For pointed spaces you can do the wedge sum and smash product instead and that's distributive too. Also if you're interested in sums of elements of a single topological space, topological groups have that.

>>15091036
>>15091038
It's correct, but not that easy to express clearly and succinctly

>>15091515
Maybe this is the way to write it? I claim that you can't fill in the blanks without getting an equilateral triangle of Bs or an equilateral triangle of Rs.

I think I'm gonna drop the math degree and get a med degree. I just don't feel excited about math anymore.

>>15091720
Okay.

>>15081271
It’s kinda cringey he can prove a mapping to N exists without needing to show one

>>15091515
The three B points are not evenly spaced though. B1B3 is half the distance of B1B1.

>>15091830
>B1B3 is half the distance of B1B1.
No, it isn't. Maybe this drawing is clearer: >>15091548

Do you have a math gf?

wish i could get a degree in just logic without having to study all the other stuff

there are twice as many natural numbers, starting from 1, as there are even numbers

>>15092216
How so? What makes you say that?

If two vector subspaces of a finitely dimensional vector space V have the same dimension, then the do NOT necessarily have the same complimentary vector subspace.

Is this correct. I think it is, because consider U1 := <(1, 0, 0), (0, 1, 0)> and U2 := <(0, 1, 0), (0, 0, 1)>. These are vector subspaces of R^3 with the same dimension 2, but their complimentary spaces are not the same.
If you detached their "0-component", then the wouldn't be a subspaces of R^3 anymore; since it's a necessary condition that they are subsets of R^3 to be its subspaces.

>>15092224
i was wrong anon. the number of natural numbers is indeterminate between twice the number of even numbers, and twice the number of even numbers plus one.
So its cardinality doesn't have any definitive value. Just like you can't say that its cardinality is odd or even.

>>15092301
You can define a cardinality of a subset X of natural numbers N as follows
card(X) = limsup_n |{x in X : x < n}|/n

>>15092312
god, you're dense...

What exactly is the relationship between consistency of a set of axioms and the existence of the objects described by those axioms? Can you ever prove the existence of mathematical objects by proving consistency of axioms? And if so, how and why does this work?

>>15091898
No. All female math undergrads here are either:
>already taken
>lesbian

Bros how do you keep motivated to study maths? I'm in my last semester and i hate everything related to math, i just keep studying for obligation, i used to loved my career. But last week i had a pretty rough revelation. I can't do anything. Engineers can repair things, CS faggots can program, med fags can heal people, even physicist can explain how the things we see work, but i can't do anything useful, is like i've wasted all this years for nothing. Sorry for the faggot blog post.

>>15091898
no but i dream of giving a girl a womb tattoo of the snake lemma

I am trying to implement a little FEM program for solving the poisson problem in 2D.
To keep it simple I decided to use a uniform quadrilateral mesh. My shape functions v(x,y,i,j)=v_1D(x,i)*v_1D(y,j) are the product of two tent functions v_1D in x and y direction (with i and j being indices in x- and y-direction respectively where the function is 1, on all other indices it is 0 and it is linear in between).
Now I am currently thinking about how to construct the stiffness matrix. For the poisson problem it is defined as [math]A_{ij} = \int_\Omega \nabla v_i \cdot \nabla v_j \, ds [/math]
Now in all resources I looked at so far, this integral is actually calculated numerically. I've looked at an implementation for 1D though and t simply uses a case distinction to return an analytically calculated solution for each entry of A. This of course seems much more elegant and I am now wondering if I missed an reasons why it shouldn't be done in 2 dimensions as well, for uniform meshes anyway. For a quadrilateral mesh with different spacings in x/y-direction, there are only 9 possible cases where the integral doesn't trivially result in zero. Any ideas? I already tried it of course but due to some mistakes my results were wrong and I thought I'd ask some numerics anons before wasting a lot of time on something that won't work

how much of mathematics is work and how much is talent?

>>15092584
>Bros how do you keep motivated to study maths?
I learn it for its own sake, because I think it's fascinating. Even though I tell my family I do it for le job

https://www.youtube.com/watch?v=wD4xrnzKN1Y

>>15092907
"I suspect it has more to do with nature (talent) than nurture" - Halmos

How the fuck do I know if I like math?
It's been a long time since high school and now I'm wondering how the fuck do I know if I actually should study a degree in math or not.

>>15091898
No but I dream of having one

>>15094600
The one I encountered was in Chapter III (Modules), Section 10 (Direct and Inverse Limits). Quote:
>For further reading, I recommend at least two references. First, the self-contained short version of Chapter II in Hartshorne's Algebraic Geometry, Springer Verlag, 1977 (Do all the exercises of that section, concerning sheaves).

>>15094636
Haha, nice. I'm downloading on libgen rn to make a screencap. Thanks for mentioning this.

>>15086995
I like manifold theory and algebraic topology, I really like how abstract but visual can things be.

>>15092584
> I can't do anything
You can research math if you like it, almost no real world applications but if you like it you will be happy.

>1 hour reading and re-reading for each page
It's over? Is Allen Hatcher unnecessarily writing simple things in a complex way, or am i way too dumb to read this book?

I have prooved that there is a bijection between the set of whole numbers to the set real numbers if you order them like this

0,1,2,3,4,5,6,7,8,9 then
00, 01, 02, 03 ... 99, then
0.0, 0.1, 0.2 ... 9.9, then
.00, .01, .02 ... .99, then moving to three digits
000, 001, 002, ... 999;
00.0, 00.1, 00.2, ... 99.9 and
0.00, 0.01, ... 9.99,
.000, .001, .002 ... .999 and so on to four digits etc etc etc
thereby covering all numbers with any number of digits
hence the number of reals is countably infinute

QED

>>15094709
You showed a bijection between the naturals and the SUBSET of the reals of numbers with a finite amount of nonzero (decimal) digits.
A number like 1/9 = 0.1111111... does not show up at all in your list.

>>15091748
Found a Wildbergerist

>>15094722
How come, there's infinite N?

>>15085779
stop being a pussy and stop posting anime shit in math threads would be a good start imo

>>15094636
>>15094644
that's not even the meme lang exercise

>>15091748
He does though?

i recommend just self-studying mathematics and getting a degree in something that will get you a job that isn't finance or teaching
it's so easy to be auto-didactic with maths (which is so easy with maths given the vast supply of lecture notes and textbooks instantly accessibly online)

>>15095331
>and getting a degree in something that will get you a job that isn't finance or teaching
Like what?

thanks for the cyber beatdown.
i need to put in stuff to battle decks like that.

>>15095101
Lol what is this “Exercises” from?

>>15091898
I was courting this cute tomboy math girl
(Like perfect 4chan tomboy, dark but still white, ran track, clearly a virgin)
Then the pandemic happened after our first date and she went back home out of state to take classes remotely
I will never forgive the powers that be for this

I just made a thread but maybe I should have just posted here

How do I solve being mathematically braindead?

I managed to get my bachelor degree in biology without actually a fundamental understanding of anything related to mathematics. I simply learned formulas and how we were taught to use them in the most autistic, for-the-test way possible then braindumped after I stopped using it.

I blame the education system for not adequately giving context and emphasizing the importance of of WHY we learn stuff, I always was just thrown a formula and a few solved examples and then learned patterns. I specifically remember memorizing the quadratic equation and NEVER having an understanding of what it actually meant and now I feel like I'm too mathematically retarded to understand the WHY of things are important.

>>15095540
It might not be the most creative advice, but a good first step might be looking into the proofs and derivations of whatever you feel like you were made to memorise. Go through, step-by-step, trying to understand why and how each step works.

>>15094709
\left| \mathbb{N}\right|<\left| P\left ( \mathbb{N}\right )\right|=\left| \mathbb{R}\right|
The only way to go about it

>>15094709
[math]\left| \mathbb{N}\right|<\left| P\left ( \mathbb{N}\right )\right|=\left| \mathbb{R}\right|[/math]

>>15095601
I'm starting to do this but it's honestly so frustrating. Even most online resources jump straight to WHAT something is and how to solve it but never goes into theory or WHY something is.

>>15095057
Which natural number would correspond with 1/9 by this method?

>>15094709
Ok, now give me an explicit surjection and injection between these sets
Shouldn't be a problem, right?

why did my latex display equations suddenly change background color wtf

Wait, I'm feeling like a brainlet here, what's the difference between the ring [math]\mathbb Z/p\mathbb Z[/math] of integers modulo p and the ring [math]\mathbb Z_p[/math] of [math]p[/math]-adic integers? They are isomorphic, right?

>>15091486
Callahan's is a very weird book imo. I wouldn't recommend looking at it until your comfortable with multivariable calc already.

Also if you haven't studied linear algebra, do that before studying any multivariable calc.

What database do you guys use for math articles? So far in my life I've only ever used https://zbmath.org/ (alongside google) but I think it might be helpful to have another.

This proof is kinda fun bros:

For all natural numbers a,b: prove that if 2b divides 2a and 2b does not divide a, then a is congruent to b mod 2n.

>>15096014
Absolutely not!
For a prime p, the p-adic integers is an infinite ring of characteristic 0. It contains the integers, some rationals (the ones with a denominator not divisible by p), and even complex algebraic numbers.

Every p-adic integer has a unique representation as a (possibly infinite) string of numbers from the set {0, 1, ..., p-1}. If it's an infinite string, it goes off infinitely to the left.
If you start indexing the places from 0, then the nth place tells how many copies of p^n you have. So it's essentially base p representation.

For example, in the 2-adics, 10111 represents 23 (base 10), so basically binary. The infinite string ...111111 represents the number -1 (base 10). (Try to figure out how you can represent 1/3 in the 2-adics; it's just a geometric series)

For each n, the ring Z/(p^n)Z is a "rough approximation" of Z_p; for every p-adic integer, you can just take the first n digits from the right an get an element of Z/(p^n)Z (it's a homomorphism).

It's a very fun topic to learn about, and you can use it to prove Mosky's theorem (that no square can be split into an odd amount of equally-sized triangles).

>>15096049
What's n? It came out of nowhere.

>>15096043
If you're at a university, you should be able to access articles through them.

Otherwise just use scihub.

>>15096056
mod 2b i mean.

What the FUCK is the difference
>05C55 Generalized Ramsey theory
and
>05D10 Ramsey theory
????????????
I look at the Math Classification System and don't understand the difference.

>>15096051
Nice, clear explanation. Thanks. Now I'm mad why some beginner textbooks sort of lie (but not really) about the term "ring of integers modulo p".

I recently came to the understanding that people use [math]\mathbb Z_p[/math] for the [math]p-[/math]adic integers, but one of my old algebra textbooks (specifically, the one by Gilbert, “Elements of Modern Algebra”) uses that symbol for the ring of integers modulo [math]p[/math]. So for the longest time I thought [math]\mathbb Z_p[/math] and [math]\mathbb Z/p\mathbb Z[/math] were exactly the same thing but just with a different notation. Only now, many years later, do I discover this "p-adic" thing and it's melting my brain.

>>15096062
>>15096049
>if 2b divides 2a
So b divides a, or a = db.
>then a is congruent to b mod 2b.
That's equivalent to saying that a = k2b + b = (2k+1)b, i.e. a is an odd multiple of b.

Since 2b does not divide a, we have that a/(2b) = d/2 is not an integer. So d is not even, and must be odd.

>>15096092
It IS just different notation. That's why it's important to state what your notation means.
I've worked a bit with the p-adic integers, but even I use Z_p to mean the integers mod p, so long as it's clear from the context or if it was stated clearly.

>>15096113
Ahhhh okay, so it was just that reviewer being particularly meticulous. Which is not a bad thing, of course.

Reason this whole thing came about was because I'm trying to get my paper published (PhD student here) and the reviewer required me to make "major revisions" and one of his comments were about my notation of the ring of integers, saying that this notation is often used for p-adics instead. So, query: if I say "ring Z_n of integers", is that clear enough? I suppose the reviewer wants me to state clearly "ring Z_n of integers modulo n" in full.

In any case, thanks anon. This was honestly a great help.

>>15096108
impressive, very nice.

>>15096131
>if I say "ring Z_n of integers"
No, that would denote the ring of n-adic integers, if n is a general natural number and not necessarily a prime. (same as p-adics, but not necessarily an integral domain anymore)
Either say "let Z_n denote the ring of integers mod n", or just use "Z/nZ" which is always clear.
>my notation of the ring of integers,
That's a third, entirely different object from both "the ring of integers mod n" and "the ring of n-adic integers".


To summarize and make it clear:
>Z is the ring of integers, and is the set {..., -3, -2, -1, 0, 1, 2, 3, ...} which you learned in high school. It is countably infinite.
>Z/nZ, sometimes denoted Z_n, is the ring of integers modulo n, and it is the set {[0], [1], [2], ..., [n-1]} of equivalence classes of integers, modulo the relation of a ~ b if a-b is a multiple of n. It is a finite set with n elements.
>Z_n is the set of n-adic integers, which is the set of strings I described earlier. It is uncountably infinite.

Just be careful with your words.

>>15096166
Alright, cheers, I understand now.

>lose marks on homework
>answer is correct, but lost marks because "we didn't cover this in class"
>it's something that will be covered later on anyways
It's a fucking graduate course

>>15086361
It's still just a geometric series.

>>15083039
A different anon with similar problem with Abbott (wordiness), is Zorich any better? I'm reading his book and I'm able to digest it much more but I'm not sure if I should continue because I heard it is much harder and I think I'm a brainlet.

>>15096166
>ring of n-adic integers
No such thing

>>15096616
Let me elaborate on that: you can of course define [math]\mathbb{Z}_n[/math] to be the inverse limit of [math]\mathbb{Z}/n^r \mathbb{Z}[/math], but it's isomorphic to [math]\prod_{p|n} \mathbb{Z}_p[/math] as a profinite ring and, more importantly, nobody cares about that object. If needed, people would just use the profinite completion [math]\widehat{\mathbb{Z}}[/math] of [math]Z[/math].

>>15096616
>>15096632
So it exists. You just don't like it.

Bros please help.
I just saw someone denote the set [math]\{1,2, 3,\ldots,t\}[/math] with [math][1..t][/math]. Is this normal? From my earlier years of studying calculus, it was [math][1,t][/math]. Is this no longer the case? I literally have never seen [math][1..t][/math] before, I feel like I'm being gaslit into using a notation that is non-standard.

>>15096639
Ok sure you got me, I am being overly autistic about this.
So instead of shitting on [math]\mathbb{Z}_n[/math] for n not prime, let me turn this post into an appreciation post for [math]\mathbb{Z}_p[/math], for p prime.

There are good reasons why [math]\mathbb{Z}_p[/math] is important: Hensel's theorem holds (which sparks the "Henselian ring" direction), they are formal completions of [math]\mathbb{Z}[/math] along a prime (which sparks the "local fields and their rings of integers" direction, as well as the "formal scheme" direction), and they are a prime example of Witt vectors (which are interesting in their own's sake)

>>15096674
You can use whatever notation you want (or are forced) to use, as long as it's been stated clearly and you're using it consistently.
Don't get stuck with one way of thinking and doing things; math is a free country. You can make your own notations and definitions.

>>15096684
Okay, thanks, I'll keep that in mind.
But seriously have you seen [math][1..t][/math] before?

>>15096674
Simply accept that some people are subhuman and move on.
>>15096684
Just to spite you I'm going to use the notation [math]{t.1}[/math] now.

>>15096688
No.
But I've seen the notation [n] to denote the set {1, 2, 3, ..., n}

Which is the preferred grammar?
1 - Solution of an equation.
2 - Solution to an equation.
3 - Solution for an equation.
I genuinely do not know, /mg/. Do you?

>>15096787
>solution to
>solution space of

Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis
>https://vixra.org/abs/2111.0072
>http://gg762.net/d0cs/papers/Fractional_Distance_v6-20210521.pdf
Recent analysis has uncovered a broad swath of rarely considered real numbers called real numbers in the neighborhood of infinity. Here we extend the catalog of the rudimentary analytical properties of all real numbers by defining a set of fractional distance functions on the real number line and studying their behavior. The main results of are (1) to prove with modest axioms that some real numbers are greater than any natural number, (2) to develop a technique for taking a limit at infinity via the ordinary Cauchy definition reliant on the classical epsilon-delta formalism, and (3) to demonstrate an infinite number of non-trivial zeros of the Riemann zeta function in the neighborhood of infinity. We define numbers in the neighborhood of infinity as Cartesian products of Cauchy equivalence classes of rationals. We axiomatize the arithmetic of such numbers, prove all the operations are well-defined, and then make comparisons to the similar axioms of a complete ordered field. After developing the many underlying foundations, we present a basis for a topology.

>>15096805

>>15096806

>>15096808

picrel is from a French commutative algebra textbook/set of notes that i can't find anymore, does anyone recognise it?

If I'm mentioning the ring of integers modulo [math]n[/math] in passing, is it better to write as:
1 - ring of integers modulo [math]n[/math]
or
2 - ring of integers modulo a positive integer?
In the former case, it kinda feels incomplete if [math]n[/math] is left undefined, but on the other hand, it kinda feels like the [math]n[/math] can be `understood' intuitively to represent a positive integer despite not being formally defined.

hey

i'm trying to calculate what steroid dosage i need to take for a given result (approximately, the data is rough for obvious reasons). the formula which emerges from the research is

y = 29.1 log x -99
where y = lean body mass gain and x = total dosage required in mg over the length of the cycle.

visible here: https://www.desmos.com/calculator/smvwvyuqt2

currently i can plot "how much to dose to achieve a certain gain" but i can't work in the other direction, "how much you can gain on a given dose". it's not a huge problem because i can just consult this graph, but i have an excel spreadsheet that i use and i'd like it to work in both directions. thus, rearrange to solve for x.

problem is when i put this into wolfram alpha i get

x = e^((10 (y + 99))/291)

this doesn't work, unfortunately, and wolfram also graphs the initial formula differently to desmos. i assume it's something to do with order of operations? for reference, i had to drop out of normie maths and take dummy maths so i missed out on basically all formal math education past the very basics of rearranging algebra. can't deal with square roots or logs at all.

what am i missing here? apart from a brain.

>>15096828
https://pro.univ-lille.fr/fileadmin/user_upload/pages_pros/lorenzo_ramero/The-Grimoire-Project.html
here you go

>>15097198
If you're using "log" in both places with no extra qualifiers, Desmos will interpret it to mean base 10, while WA will interpret it to mean base e.

>>15097238
thanks

>>15097240
yep, that fixed it.

now if only i understood what it meant...

but i can plug it into my spreadsheet now which is the important thing. thanks for your help anon.

Can there be a partition of the Euclidean plane into two sets A and B , such that neither A nor B contains:

(a) a straight line (extending infinitely in both directions)?

(b) a (nonzero-length) straight line segment of positive length?

(c) any (nonzero-length) smooth curve?

>>15097427
>nonzero length
>positive length
Oops kind of repeated myself lel

>>15086995
Post your favorite book(s) on combinatory logic... do you mean infinitary combinatorics or something else entirely?

>>15097457
please, I should add. I'm genuinely interested to see what you're talking about because combinatorics is my field and I have a soft spot for logic too

>>15096910
If you are speaking out loud or writing chickenscratch on a board, it's always the first one. If you're writing formally I wouldn't write mod n without saying what n is

is mathematics independant from the observer

>>15097427
Ignore (a), it's a dumb question.

>>15097427
How about [math](\mathbb{R} - \mathbb{Q}) \times \mathbb{Q}[/math] and its complement?

>>15097530
i used to think so but now i believe it exists within the human mind and nowhere else. it's purely descriptive rather than fundamental still profound and important, just not fundamental like, say, philosophy or physics (and even these have their limits)

>>15097719
The complement has a bunch of lines like [math](0, t)[/math].

>>15097253
the log is the exponent you need to get the base somewhere.
log_10 (100) = 2 since 10^2 = 100
log_e (100) is something different since e^2 != 100
As a little exercise, ask yourself if the log_e 100 should then be large or smaller than 2

I need to relearn topology, I find it too hard and the notes our teacher gave us are absolute shit, plus I couldn't attend most of the classes since I had to go to work.
What is a good book on the topic? I mean, the basic concepts: topological and metrical spaces, continuity, homeomorphisms, connected spaces...

>>15097819
i just went through the course notes for this: https://www.math.toronto.edu/ivan/mat327/
didn't find it too difficult

>>15097427
Given choice this is easy. There's only continuum many smooth curves, well order them and the reals and build the partition via transfinite recursion.

>>15097427
>Can there be a partition of the Euclidean plane into two sets A and B , such that neither A nor B contains:
>(a) a straight line (extending infinitely in both directions)?
Make a checkerboard pattern.
The only issue that might arise is at the corners and the edges (lines that are 0, 45, 90, or 135 degrees off from the x-axis).
Too lazy to decide how to split those, but I came up with the bulk of the solution. Figure out the rest.

>>15097896
Well for (a) I think you can just take A = the union of the x and y axes , and B its complement

>>15098311
>the union of X and Y axes does not contain a straight line

>>15098315
Improving on >>15098311's idea

>>15081116
I'm a 5th grade teacher, and during the problems, I ask the kids questions like "What is 8 times 7?" for a problem, and I just trust them to get the right answer because I don't actually know off the top of my head. Especially for long division when it's not a math fact.

>>15098522
gtfo from this board midwit

>>15098527
stay mad, nerd

>take analysis courses
>one professor allows late homework submissions and even answering phonecalls during exams
>one checked on me during the final to see how I'm doing, then gave hints for the entire class for the problems I was stuck on. He did this FIVE times
>one bumped the grades of the entire class
>one gave me a high grade for the final even though I only solved like half the questions
>one assigned us homeworks and take even ask for them. Just gave us the grades

I'm going into algebra, but I gotta admit, analysis professors are the fucking best.

>>15098653
Your stories reminded me of the time I had a topology professor who showed up several minutes late to class on a test day, distributed the copies of the test, and then informed us that we weren't actually taking it and despite having printed it out he just decided to give everyone a 100% on it.
This was in a class where homework wasn't collected.
I still have no idea what the fuck that was about.

thinking about this as of late:
how would one go about putting conditions on a family $G_n$ of lie groups with embedding in $\mathbb{R}^n$ as a solution to an elliptic PDE where $SO(n)\unlhd G_n$ for $n>1$ such that $G_n/SO(n)\cong G_1$
Is there a rich theory for deformations if the related class of PDEs is nice but the isomorphism isn't there?

How do I get a % lower than a number
like 35% lower than 365
and how do i get a % higher than a number
like 42% higher than 285

Please give me recommendations on books that contain actual motivation (or are entirely motivation for the topic) for their content. Doesn't necessarily have to be applications obviously, but just texts that explain why work was done in creating theorems instead of simply going through the topic with very little motivation behind the content like picrel. Books like General Topology by Willard, Galois Theory by Edwards, and Analysis by Its History by Hairer and Wanner are the types of books I'm looking for across all topics.

>>15097819
Munkres

>>15097427
If you allow infinitesimals then you can easily translate every component of each ray y=ε, y=-ε, y=2ε, y=-2ε ... along the x-axis by 2ε, 3ε, 5ε, 7ε, 11ε ...
This seems useless but in combination with the aforementioned (R-Q)*Q and it's complement this has only points.

Suppose that [math]u(x,t)[/math] is a solution to
[math]\begin{cases}u_t-u_{xx}=0&\quad \text{in}\; (0,1)\times (0,\infty)\\ u(0,t))=1,\;u_{x}(1,t)=0&\quad \text{in}\; [0,\infty)\end{cases}[/math],
such that [math]u(1/2,1)=1[/math]. I'm asked to find the initial condition[math]u(x,0)[/math] from this info alone. However I don't think it is enough information. One possible solution obviously is [math]u\equiv 1[/math] however I wrote the general solution using separation of variables and tried to see if by using the condition [math]u(1/2,1)=1[/math] I could force the solution to be the constant [math]1[/math] but I believe I end up founding an infinite number of solutions that satisfy all this. It is possible that is wrong, the exercise sheet is filled with errors but I would like to see if maybe I'm doing something wrong. Any ideas?

What is a rigorous proof for the fact that permutations of iid continuous random variables are equally likely?

>>15101274
That won't work. There are aleph_0 primes and 2^(aleph_0) infinitesimals

>>15101335
iid?

>>15081116
logic or set theory, which one is the more important foundation of math?

>>15101340
I suppose he means independent and identically distributed

>>15101343
basic logic and basic set theory are absolutely fundamental to mathematical reasoning. All set theory after 1965 is just retards trying to prove that things are independent of ZFC for no real reason, most other mathematical logic is similarly gay

In ZFC the reals have a well-ordering, so do they have an analogue to large ordinals?

>>15101350
>for no real reason
the reason being AC is junk, nobody cares about anything it enables but delusional mathfags, it only complicates things and physishits not giving a flying fuck about any of the pathologies of AC is the reason why a physishit got fields medal instead of a mathfag
if you need a crutch for your imaginary nonconstructions, DC or CC is strictly superior

The Art and Craft of Problem Solving, Paul Zeitz

Can you find covariance from joint mgf?

>>15081116
I CANNOT GET THROUGH SEPTEMBERRR
WITHOUT A BATTLE

>>15082231
i took a calc 3 undergrad course over the course of one summer, but i feel like i could have done more than we covered if i wanted to. i even ended up teaching myself surface integrals and 3d taylor series which wasn't covered for some reason.

>>15082279
be familiar enough that if a topic comes up again later on, you are comfortable giving urself a quick refresher enough to move on without having to learn it all over again

>>15085779
in my experience, calc 3 hardly covers vector calculus. generalized line/surface integrals are really cool in higher dimensions. and it is definitely useful if u ever want to try to understand maxwell and gauss.

>>15101357
Yes. Assuming C you can just stack [math]\omega_{1}[/math] at the end of the well-ordered reals.

>>15081271
If you want a more formal modern treatment, read Amann/Escher or Zorich.
>>15096530
It's very comprehensive, so just continue. Yeah, it certainly is harder and strives for much more generality than most introductory RA books, but what of it? You can always try the problems in easier books, not to mention, problem books like Demodevich.
Note that a lot in Zorich's books is fully worked out examples and physical applications. None of this makes it wordy or unrigorous but you may want to consider Amann/Escher in case this annoys you (and you desire a bit more formality).
>>15082292
>>15083031
There's nothing wrong with symbols, no matter how much you hate them for whatever childish reason. Looks like A. R. D. Mathias was right about Bourbaki creating a generation of mathematicians that loathe formal logic and foundationalist questions, rather choosing to ignore Gödel's theorems and all the subsequent development in foundations.

>>15096530
Zorich is the wordiest of them all. Every paragraph is about some le heckin application wowzers!

>>15102207
Not at all. Every paragraph that's prefaced with the word "example" is. Math books aren't meant to be read like novels, undergrad. You decide what you want to read.

>>15097427
(b) A = sum of circles centered at the origin with rational radius

has anyone here actively used picrel and can testify to its quality (compared to other textbooks)? i'm reading it right now and although i'm liking it so far, it's kind of imprecise in places and not particularly smooth (the bits of Neukirch i've read were incredibly suave compared to this). also i've read somewhere that Janusz's commutative algebra is a little muddy in places

>>15102461
I did and I consider it time wasted. A lot of proofs are just wrong, especially when it comes to group cohomology. To find correct proofs you have to go to first edition. A sloppy book in many respects.

>>15102485
>A lot of proofs are just wrong, especially when it comes to group cohomology
well shit. you mean just plain wrong or just ineffective? because i've noticed his proofs are a little cumbersome. what about the first couple of chapters?

>>15102485
>>15102493
and do you have any better suggestions (besides something obvious like Neukirch)

anyone here use personal wiki software e.g obsidian for math stuff?

>>15102435
Violates 3

>>15091898
no, my dream as a physicist is to date a chemist

theoretical computing seems pretty interesting, especially the finite model theory and descriptive complexity stuff. modern research is applying category theory and cohomology and stuff there (essentially transferring ideas developed in computational semantics and type theory over to the complexity realm)

>>15101126
bump

>>15091898
I'd prefer a physics qt gf

the "10m chord" guy keeps posting on /pol/ for no reason
I drag the cringe back where it belongs

Is there a pair of relatively prime positive integers a,b such that a+b divides ab ?

Trying to understand the proof for 1) in pic related. I can follow the proof (bottom few lines of the page, note that 1 is the indicator function), but can't inf(n>k) 1An take on the empty set if w belongs to Aj but not Aj+1 (j > k)? Because then what we would have for inf(n>k) 1An = intersection{ {0}, {0}, ... {0}, {1}, {0}, ...} = empty. For the supremum, same thing but it would be a union which would yield {0, 1}. If An was increasing or decreasing it would be fine that wasn't stated. The only thing I can think of is that the reverse of the proof on the bottom of the page follow by the contrapositive.

>>15104056
[math]\frac{a+b}{ab}=k\to{}kab=a+b\to{}kab-a=a(kb-1)=b[/math]
Since a,b are coprime this implies [math](kb-1)|b[/math], which is obviously not going to work unless k=1, in which case a=b=2, contrary to assumption

>>15104056
[math]\frac{a+b}{ab}=k\to{}kab=a+b\to{}kab-a=a(kb-1)=b\to{}a|b[/math].
so, no.

>>15104079
You showed we can't have ab dividing a+b . But what I asked >>15104056 is whether a+b can divide ab .

>>15081116
How do i self teach myself math on my own?

>>15104084
Same: you showed we can't have ab dividing a+b when a,b are coprime, but what I asked is whether a+b can divide ab .

>>15104098
Oh, wait, I'm blind.
[math]\frac{ab}{a+b}=k\to{}ka+kb=ab\to{}ab-ka=a(k-b)=kb[/math]
Since a,b are coprime, all of a's prime factors must be contained within k, so [math]a|k[/math].
Similarly, all of b's prime factors must also be factors of [math](k-b)[/math], which implies that [math]b|(k-b)[/math], so [math]b|k[/math].
But this implies that [math]k\geq{a}[/math] and [math]k\geq{b}[/math], so [math]ka+kb\geq{}ba+ab=2ab=ab[/math], a contradiction unless either a=b=1 or one of them is 0, neither of which is valid under the initial assumption.

>>15104125
[math]ab=ka+kb\geq{ba+ab}=2ab[/math]
I hate [math] tags.

I have a question that I think it's quite simple, but I just can't figure out how to solve it. Say I have a square, symmetric matrix A with (orthogonal) eigenvectors [math] z_1, z_2, ..., z_n [/math]. Is it true that I can always find a orthonormal basis [math]x_1, ..., x_n [/math] of span{ [math]z_1, ..., z_n [/math]} such that, say, only [math]x_1 [/math] has non-zero first component? my intuition says that it is, but I just can't find a way to prove it

>>15104130
If you have n pairwise-orthogonal vectors z_i in R^n then their span is all of R^n , no?

So basically, your question is whether there exists an orthogonal nxn matrix whose first row has zeros in all but the first entry?

>>15104056
hint: if a+b=abc then b=a(bc-1)

>>15104125
Nice one, thanks anon

>>15104141
my bad, it should be an orthogonal basis [math] x_1, ..., x_t [\math] of span{[math] z_1, ..., z_t [\math] }, a subset of the eigenvectors. although the question is pretty much the same, yes, does there exist a n x t matrix whose only column with non-zero entry is the first?

>>15104056
if ac+bc=ab then bc=a(b-c) hence by coprimality and unique factorization* a=c but similarly b=c and that’s just stupid

*might be overkill, I don’t care. come arrest me cocksucker

>>15104155
I’m drunk btw

>>15104151
Sure. You just need to show there is an nxn orthogonal matrix B taking the first vector z_1 of your basis to e_1 , the first standard basis vector. Then BZ is the matrix you want , where Z = [ z_1 ... z_t ] .

>>15104186
> z_1
Technically should be z_1 divided by its length, since you didn't assume they were normalized ig.

>>15104186
>>15104195
hmm, sounds simple, but I'm not sure if e_1 is in span{z_1, ..., z_n}

>>15104211
Sorry, here's a correction. You need to find an orthogonal nxn B such that Z^T will send the first column of B^T to e_1 . You can achieve this by taking B^T to have first column Z^T^{-1} (e_1) , then extending this to an orthonormal basis.

Here Z is the nxn orthogonal matrix whose columns are all the z's , with z_1 through z_t first .

And ^T means transpose.

Now the first t columns of BZ should be the basis you want, since the first row of BZ should be e_1^T .

>>15104231
should have added, I am again assuming your z's are all normalized to unit length (otherwise Z would not be orthogonal)

>>15104065
help please

>>15104236
>>15104231
thank you, as I thought it was quite easy, I'm just a retard

>>15104065
If [math]\omega \in A_j[/math] and [math] \omega \not \in A_{j+1}[/math] for a [math]j \geq k[/math] then
[eqn]1_{\inf_{n \geq k} A_n} (\omega) = 0 = \underset{n \geq k}{\inf} 1_{A_n} (\omega)[/eqn]

>>15104231
>Z^T^{-1} (e_1) , then extending this to an orthonormal basis
Also, I should have said, you can just use the basis Z^T^{-1} (e_i)

>>15102493
>>15102495
Hi, sorry for not replying earlier. I think the book is readable, and it's not very long either. The first chapters are fine. One thing I remember being befuddled about is how he goes from reasoning about localizations at primes to results about the whole ring, so make sure you pay attention to that.
The thing about group cohomology is pretty easy to notice. You will see that something is very clearly missing or not right. There you should read the first edition of the book, which should largely fill up the gaps.
I haven't read Neukirch so I cannot recommend it to you. I have read a bit more than half of Marcus' Number Fields book, and highly recommend it to you. It is a little bit more elementary than Janusz, in that it mostly focuses on rational number fields.
Be sure to update us on what book you've chosen and how it's going for you :)

Are you ever afraid math will break while you're using it?

>>15104807
Kinda, mostly with high level stuff that relies heavily on obscure analytical results that most people haven't heard off. For example, in many fields you will sometimes find that to show existence of certain things you need to solve a PDE or depend that certain thing is well defined in weird situations and since this in itself is really difficult to solve it will be difficult to find a good source where you can find the result for the case you need. The worst is when you need a fucking Green's function or something ill behaved like that because no one likes to apply distribution theory correctly.

>>15104436
I've read all of Marcus, and I also find it a very good book, it was a light and pleseant read
But yeah, it doesn't cover too much stuff

>>15104839
Actually, I meant something more like the logic of math starting to contradict itself, rather than something like results being unreliable.

>>15104807
>>15105108
Not especially.
If it's because I'm fucking up, then it's entirely to be expected. Basic arithmetic is the hardest thing on the planet. I'd be more concerned if I didn't jot down something to the effect of [math](z+\frac{1}{z})(z-\frac{1}{z}=z^2+z^{-2}+2z^{-1}[/math] from time to time because it would mean I had suddenly become competent.
If it's for some other reason, then it would be a very, very exciting opportunity for discovery.
But mostly it's just going to be fucking up basic arithmetic and it's to be expected.

>>15104265
How is pic rel 0 and not empty?

>>15105198
is that not an intersection of singletons? Am I missing something?

>>15105108
If ZFC isn't consistent it only means ZFC is invalid but not that math itself is invalid. Rigorous arguments go back a long time and there are results we know are true in their own little math universe and well we are not going to throw that out because we haven't yet found a way to unify all math.

You can get a rational parametrization of the unit circle using complex numbers over Q.

z€ C(Q), z≡ a+bi, Q(z)≡ a^2+b^2

N(Z)≡ z^2/Q(z)= (a^2-b^2)/(a^2+b^2)+2abi/(a^2+b^2)

Q(N(z))= |N(z)|= 1, but We have avoided irrational numbers by using Q(z) instead of |z|, so to keep N(z) a unit complex number we needed to square the numerator z as well giving us z^2/Q(z).

Our next step is to make N a function with a single rational parameter, to do this we can use the property that N(𝛌z)=N(z), 𝛌≠0, we can divide z by a in N.

N(a+bi)= N(1+bi/a)= N(1+ti), t=b/a

e_{c}(t)≡ N(1+ti)= (1-t^2)/(1+t^2)+2ti/(1+t^2)

e_{c}(t) is the rational complex number parametrization of the unit circle, the c in the subscript is for complex. To get the rational parametrization for points we will need to define two functions.

C(t)≡ (1+t^2)/(1+t^2) and S(t)≡ 2t/(1+t^2)

e(t)≡ [C(t), S(t)]= [(1+t^2)/(1+t^2), 2t/(1+t^2)]

C(t) and S(t) are just the "real" and imaginary components of e_{c} respectively.

If you have e_{c}(t) and e_{c}(u) then the value of the perimeter e_{c}(t)·e_{c}(u) is t⊕u≡ (t+u)/(1-tu), ⊕ is circle addition.

the lammermeier product of two points A and B on a conic is defined to be the intersection of the conic and the line through a point O (which we have chosen) , which is parallel to the line through A and B; written as A*B. (A*B)O || AB, O is on the conic.

If we are looking at on the the lammermeier product of unit circle where O≡ [1, 0] then e(t)*e(u) is isomorphic to e_{c}(t)·e_{c}(u).

e(t)*e(u)= e(t⊕u)= e((t+u)/(1-tu)), e_{c}(t)·e_{c}(u)= e_{c}(t⊕u)=e_{c}((t+u)/(1-tu))

You can learn more in the attached document. I fuckin' hate English.

Let [math]g(X)[\math] be the min poly for [math]\gamma[\math], then
[eqn]\mathbb{Z}[X]/\langle g(X) \cong \mathbb{Z}[\gamma][/eqn]
Then the book states
[eqn]\mathbb{Z}[\gamma]/\langle p \rangle \cong \mathbb{Z}[X] / \langle g(X), p \rangle[/eqn]
I'm confused how they managed to combine the ideals. Is that a general rule of rings when you have [math]f : R/I \rightarrow S[/math] with ker [math]f = J[/math]
then
[eqn]R/(I + J) \cong S/J[/eqn]
Is there a term for this in category or a link where I can read a proof?

repost:

Let [math] g(X) [/math] be the min poly for [math]\gamma[/math], then
[eqn]\mathbb{Z}[X]/\langle g(X) \cong \mathbb{Z}[\gamma][/eqn]
Then the book states
[eqn]\mathbb{Z}[\gamma]/\langle p \rangle \cong \mathbb{Z}[X] / \langle g(X), p \rangle[/eqn]
I'm confused how they managed to combine the ideals. Is that a general rule of rings when you have [math]f : R/I \rightarrow S[/math] with ker [math]f = J[/math]
then
[eqn]R/(I + J) \cong S/J[/eqn]
Is there a term for this in category or a link where I can read a proof?

>>15104056
[math]
\begin{align*}
\frac{a + b}{ab} = c \in \mathbb{Z}& \iff\\ 1 + \frac{b}{a} = cb \in \mathbb{Z}& \iff\\ \frac{b}{a} = cb - 1 \in \mathbb{Z}&
\end{align*}
[/math]
Contradiction, because b and a are relatively prime.

Can you give me an answer to this question?

>>15105198
For each omega and each k you are taking an infimum of the set
[eqn] \left \{ 1_{A_k} (\omega) , 1_{A_{k+1}} (\omega) , \ldots \right \}[/eqn]
The set always has countable many elements and each element is either 0 or 1 so of course for each omega and k the infinimum is either 0 or 1. If you have a omega that isn't an element of all [math]A_n[/math] with [math]n \geq k[/math] then the infimum is 0 for that omega. Otherwise it is 1.

>>15106641
[math]
\begin{align*}
\frac{ab}{a + b} = c &\iff\\
ab = ac + bc &\iff\\
a = \frac{a}{b} \cdot c + c \quad \text{and} \quad b = c + \frac{b}{a} \cdot c
\end{align*}
[/math]
It is [math]a,\, b \text{ and } c \in \mathbb{Z}[/math]. Hence c has to be divisible by a and b. Note a and b are relatively prime. Let d be a natural number, such that [math]c = abd[/math]. So
[math]
\begin{align*}
\frac{ab}{a + b} = abd &\iff\\
\frac{1}{a + b} = d
\end{align*}
[/math]
This is a contradiction.

>>15106641
>>15106694
>positive integers

https://www.youtube.com/watch?v=OQ91S0dedWM
B
>O
O
>M
SHACK-A-LAK

>be 2nd year PhD student
>read textbook
>find it hard
>re-read Introduction
>"This book is aimed at the advanced undergraduate or beginning graduate student"
>feel shitty
Anyone else here with similar experiences? Man it sucks to be confronted with "lmao that book should be easy for you at this stage, you noob".

Does anyone have any experience with AMS' inquiry books? How are they?

A right triangle with slope 1 has an angle of pi/4 but for slope 2 the angle is pi*0.5ilog(1-2i)-0.5ilog(1+2i) which is a crazy transcendental number.
Why is atan(a)/atan(b) always irrational for a,b [math]\neq[/math]0,1,-1? Is it always transcendental?
Is there a branch of math that's devoted to questions about the rationality/irrationality/transcendentality of quotients f/g?

>>15108651
>pi*0.5ilog(1-2i)-0.5ilog(1+2i)
How the fuck did you come to that? That's not even a real number, it makes no sense as a length

>>15108651
algebra, see pinter

>>15107523
Most books are not meant to be fully absorbed through one pass.

>>15109513
Thanks for the reassurance.

>>15109880
x

>>15107523
I cannot believe you get away with this bs without posting the name of the book.
Fuck Russia.
Fuck Russians.
Fuck anti-Westerners.

Anons of /mg/, which set of letters is more often used to represent (group or ring) homomorphisms?
1. Latin letters [math]f,g,h,[/math] etc.
2. Greek letters [math]\phi,\psi,\theta,[/math] etc.

>>15110035
Latin letters are for the elements of the group/ring, and Greek letters for homomorphisms. I will fight anyone who disagrees.

>>15110055
I agree for cases like [math]a,b,c\in G[/math] or [math]x,y,z\in R[/math].
But the letters [math]f,g,h[/math] are often used for functions, surely those are fine for morphisms as well?

>>15110059
f and h are fine, but I'd still rather use Greek.
g is a little weirder to me personally just because my professors liked to use it to represent a generic group element, but even then it wouldn't be hard to discern from context.

>>15109929
2x

>>15110059
>But the letters f,g,h are often used for functions
Algebraic Geometry tells you that you should think of elements of any ring as functions, so f,g,h is good notation for elements of a ring.

>>15110534
Haven't reached that part yet.

What's the general consensus on Hungerford's "Algebra" textbook? How does it compare to Lang's "Algebra"?

Modular arithmetic notation feels so dissonant from every other notation in math. Why is it still used?

>>15111046
Do you have any other suggestions?
Besides, it's not even the most pressing candidate for replacement in number theory. Unless you want to argue that the Legendre symbol is somehow good.

>>15107772

>>15112410
We're still nearly 40 posts away from the bump limit you absolute mong

>>15111084
[math] a \mathrel{ \mathrm{mod}}_n b \\
a \equiv_n b [/math]
Actually we don't need special notations for everything so.
[math] n \mathrel{ \mathrm{div}} (a-b) [/math]

>>15112756
I mean being congruent mod n is just another way of saying being equal in [math]\mathbb{Z} /n\mathbb{Z}[/math], so I see nothing wrong with writing [math]a=b[/math] and then specifying in which ring, by adding [math]\text{mod } n[/math]

>>15110534
This anon gets it, but this only makes sense for rings
>>15110064
I like the notation Serre uses, where [math]g[/math] is usually a homomorphism, and generic group elements are usually called [math]s,t,u\in G[/math]

Math sucks
>t. Disillusioned 9 years after completing math PhD

>>15113404
What do you do now?
I'm just starting a Masters and feeling pretty disillusioned.

>>15113404
That's interesting. I once read of a mathematician that described his job as doing heroin for hours a day.

>>15113404
Why?
What did you specialize in?
How long have you been feeling this way?

>>15113443
That sounds amazing, I want to do that too.

Any good books on analytical geometry?

>>15081116
explain this mathfags
https://www.youtube.com/watch?v=T17mq1wjX_4

>>15113467
>i want to do that too
hate to be the one that tells you, but mathematicians can't afford to do h every day

>>15114048
What I meant was that that mathematician's job satisfaction was so high, he described it as doing h every day, not that he was literally doping daily. Seems some people are really meant for math, but I've also heard it's nontrivial to tell if you're one of them.

>>15113626
>mathfag
I'm a disgusting moron.
What the fuck is math?
>DUUUUUUUUHHHH
>DUH
>I DUNNO BOSS

>>15113626
Youtube became popular mostly through
- Jewish ownership
- hostile and possibly criminal manipulation of the RIAA and recording industry
Frankly, I think rap artists should collaborate on burning Youtube headquarters to the ground and making a big stink about jewish influence over the careers of negro recording artists
Frankly the negro terrorists are the good guys when it comes to the history of Youtube
ZAPP R.I.P.

Let [math]\gamma[/math] be the root of min poly [math]g(X)[/math]. then we know
[eqn]\mathbb{Z}[\gamma] \cong \mathbb{Z}[X] / \langle g(X) \rangle[/eqn]
Now can someone explain how that means
[eqn]\mathbb{Z}[\gamma]/I \cong \mathbb{Z}[X] / \langle g(X), I \rangle[/eqn]
?

also should that be
[eqn]\mathbb[\gamma]/I \cong \mathbb[X]/\langle g(X), \phi(I) \rangle[/eqn]
where [math]\phi: \mathbb{Z}[\gamma] \rightarrow \mathbb{Z}[X] / \langle g(X) \rangle[/math]
I'm looking for any proof of this. What is the name of this?

>>15115234
3rd isomorphism theorem

I don't think 3rd iso theorem works because it assumes the ideals (or normal subgroups) are contained in each other. Here they are distinct.

Here's a practical example:
[eqn]\gamma = \frac{ \sqrt{2} + \sqrt{6} }{ 2 }, g(X) = X^4 - 4X^2 + 1, p = 5[/eqn]

then the book states that
[eqn]\mathbb{Z}[\gamma] / \langle p \rangle \cong \mathbb{Z}[X] / \langle g(X), p \rangle[/eqn]

3rd iso theorem states that [math]I \subseteq J \subseteq R[/math] then
[eqn](R/I)/(J/I) \cong R/J[/eqn]
but here [math]\langle p \rangle[/math] is not contained in [math]g(X)[/math]

>>15115291
J = < p , g > which contains I = < g > idiot

you motherfuckers lied to me

told me all i needed was linear algebra to get through a quantum mechanics book

i'm on chapter 3 and it expects a ton of trig and analysis

>>15115622
>trig
That's what's known as a transitive requirement.
>analysis
Anons just forgot to tell you you need calculus.

>>15115644
i went through a few differentiation/integration exercises to refresh calculus, as well as the basic trig identities

the problem is that the book just introduces the schrodinger equations (time dependent and independent) and i just have no idea how they work even though the author moves on as though it were all self evident

Let [math]\mathfrak{p}[/math] be a prime ideal, and [math]\mathfrak{a}[/math] is also an ideal (not sure this is needed), then is it true that
[eqn]\mathfrak{a}/\mathfrak{a}\mathfrak{p} \cong \mathfrak{a}/\mathfrak{p}[/eqn]
and if so then why?

>>15115652
a/p is not well-defined since p need not be a submodule of a.

>>15114077
that much was clear, i made a joke because it sounded like anon wanted to do heroin every day. Clearly you are autistic enough for math, so no worries there

finished my master's thesis, and my advisor wants me to start cutting it down and doing more work to see about getting a publication. i kind of just want to kms instead.

There are indescribable countable ordinals between [math]\omega^{CK}_{1}[/math] and larger describable ordinals, but are there indescribable countable ordinals for which none of their suprema are describable?

do mathmeticians normally agree that 2+2=4 would be necessarily right in every possible universe?

>>15115234
>>15115240
>>15115291
are you reading Brian Conrad's ANT notes by any chance

which of the two should i get?

>>15081116
I don't understand this problem

>>15117145
I don't understand how I don't understand that problem, but I do this one.

>>15117145
[math]4x-(-4)=-4\to4x=-4-4=-8\tox=-2[/math]

>>15117145
>>15117155
4x-(-4)=-4
4x+4=-4
4x=-4-4=-8
x=-8/4=-2

>>15117191
And this?

>>15117222
-2x/5-(-5/4)=-2x/5+5/4=-3/2
-2x/5=-3/2-5/4
-8x/5=-6-5=-11
-8x=-55
x=-55/-8=55/8

>>15081271
I worked through Understanding Analysis by proving every theorem and solving every exercise. I would highly recommend although this is no means easy.

>>15117234
I can do most other of these problems in my head, but these fraction ones bewilder me.

Does anybody know of some sort of beginner quick reference guide.

>>15082231
A more realistic goal is to just get through calc 1 material over summer break. Doing calc 1, 2, and 3 is possible, but you'd probably have to study 8 hours a day

can someone explain how 17 times 29 is 493 thanks in advanced

-from a retard

>>15117464
17 x 29 = 17 (30 - 1) = 17 x 30 - 17

Then, 17 x 30 = (17 x 3) x 10 = 51 x 10 = 510

Then, 510 - 17 = 493

Next thread:

>>15107772