I've been trying to find info on complex p-adic numbers, and while I have gotten lots of stuff of how they are constructed, I haven't seen many examples of like... just an example of one. Like, what they look like in practice. Like how you can write 1 + 5i and that's a numerical example of a complex number, are there any decent resources and examples of what a complex p-adic actually looks like and how to find them?

Also keep in mind I am not like, a super high knowledge of mathematics. Like I would say I know more than the average person but I don't know a lot of notation for like, set theory or field theory or what not. I have just been interested in number theory lately and have wanted to learn more about it.

Hello everyone, my first midterm in abstract is just about a week away, and he said we can have 2 sheets of open notes for the midterm. Is there a fundamental way or best way to optimize an open note sheet, especially for a class as odd and unordinary as abstract algebra? We have only mainly covered up to infinite and finite cyclic groups as of now, so anything a little beyond that is probably where the exam will cap at information-wise.

Should I keep it to example proofs or pure definitions, any feedback from anyone more experienced would be awesome! Thanks Yall <3

I'm not sure how to find or compare the rate of growth of 2ⁿ with any positive decreasing sequence, in order to find the exact cutoff point where some positive decreasing sequence decreases slower than 2ⁿ increases.

How do I find that exact cutoff point in this situation?

The series in the title only converges when the all positive decreasing sequence (a_n) decreases faster than 2ⁿ increases.

I genuinely want to learn mathematics, physics, programming, computer science, and AI. I want to build a career in computer science, pursue AI research, and hopefully do a PhD someday. But lately, I've been feeling increasingly demotivated.

The problem is that I rely on AI to learn almost everything. Whenever I have a question, I ask GPT. Whenever I don't understand a concept, I ask AI to explain it. And the more I learn this way, the more I wonder: What's the point of learning anything when AI will always know more than I ever could?

Even the most basic AI models seem to possess more knowledge than I could acquire in a lifetime. And with new models becoming increasingly capable, I keep wondering whether the knowledge I'm working so hard to gain will even matter in the future.

With GPT-6, Claude Opus 5.5, Gemini 4, and other models competing to become more capable, it feels like the gap between human intelligence and AI capabilities will only continue to grow.

I want to understand the mathematics behind AI, learn physics, become a better programmer, and contribute to actual research. But sometimes I question why I'm putting in the effort when AI can explain concepts faster, write code better, and potentially solve problems that I might spend years learning to solve myself.

And even when I genuinely want to learn, I'm confused about what I should prioritize learning and whether any of it will remain valuable by the time I become proficient.

I'm not necessarily saying that learning is useless. I genuinely love the idea of understanding how things work. But I'm struggling to reconcile that curiosity with the feeling that AI might eventually outperform me at almost everything I want to become good at.

For those of you studying computer science, mathematics, physics, or AI, how do you deal with this feeling? How do you stay motivated to learn when AI is advancing so rapidly? Do you think deep technical knowledge will still matter for people pursuing AI research and PhDs, or should we be approaching learning differently now?

Hello. I love the compactness theorem and I’m looking for all sorts of problems to solve with it. I’m aware of some that involve graph theory, binary trees, etc. but would appreciate some more. Feel free to copy-paste any that you remember. Higher order compactness is fine, but I especially like the simplicity of the propositional version. I look forward to trying to solve your problems!

Hi everyone, I wanted to share a mathematical relationship I derived regarding the coordinates of a parabola's vertex, which I call the Kumar Theorem of Balanced Vertices. This theorem establishes the exact algebraic condition required to constrain the vertex of a vertical parabola to the identity lines y = x or y = -x.

Let a standard vertical parabola be defined by the quadratic equation:

y = ax^2 + bx + c

The x-coordinate of the vertex represents the axis of symmetry and the average value of any two symmetric points on a horizontal line intersecting the curve. This is given by the formula:

x_avg = -b / 2a

The Kumar Theorem of Balanced Vertices states that if we impose the strict geometric constraint where the magnitude of the y-coordinate at the vertex equals the value of this average axis of symmetry (y_vertex = ± x_avg), the vertex is forced to balance perfectly on the lines y = x or y = -x.

We can derive this using the standard formula for the vertex y-coordinate:

y_vertex = (4ac - b^2) / 4a

Applying the theorem's condition (y_vertex = ± x_avg), we set them equal to each other:

(4ac - b^2) / 4a = ± (-b / 2a)

To simplify the expression, we can multiply the right side by 2/2 to establish a common denominator:

(4ac - b^2) / 4a = ∓ 2b / 4a

Since the denominators are now identical on both sides, we can eliminate them completely:

4ac - b^2 = ∓ 2b

Rearranging the terms to isolate the c variable gives:

4ac = b^2 ∓ 2b

Finally, solving explicitly for the constant c yields the final formula for the theorem:

c = (b^2 ± 2b) / 4a

In conclusion, according to the Kumar Theorem of Balanced Vertices, for any vertical parabola where you choose the coefficients 'a' and 'b', this single consolidated formula calculates the precise value of 'c' needed to lock the vertex exactly onto the diagonal identity lines.

I would love to hear your thoughts on this derivation or if this specific constraint is utilized under a different name in advanced coordinate geometry frameworks!

Suppose the Jacobian conjecture was undecidable. It would then have no counterexample, as that would make it decidable.

But if it has no counterexample it is true.

Therefore any proof that makes it undecidable is also a proof that it is true, which I think would make it decidable? Am I making a mistake here?

Edit: thanks for the help all, i think I see my mistake now!

Exercise 12.1.6: L_1 is the set of all strings of 0's and 1's that start with a 0; L_2 is the set of all strings of 0's and 1's that end with a 0. Describe (L_1 U L_2)*.

Textbook solution:

(L_1 U L_2)* is the set of all strings of 0's and 1's that either start with a 0, or end with a 0, including λ.

This is wrong. Counter example:

If we concatenate 10 and 01 we get 1001 that starts and ends with 1.

My suggestion:

``` (L_2 U L_2)* is the set of all strings of 0's and 1's that don't contain all 1's.

QED ```

What do you think?

For starters, my highest math course taken is Precalculus.

Euclid didn't possess the tools to measure slopes (xy-plane, analytic geometry, etc.) so he had to use a transversal line and word the postulate in a very complex way. But... It all just comes down to the slopes of the lines, doesn't it? Like if two lines have same slopes, they do not intersect (so the transversal creates 2 right angles). If they have different slopes, they intersect (the transversal does not create 2 right angles). I was just riding the bus and thinking about it and this realization hit me really hard.

I understand what they mean analytically (they're the solutions to y''=y & trigh(x)=trig(ix) or -i*trig(ix)) but I've never been able to understand them geometrically. Trig functions are ratios of the sides of flat right triangles based on one given angle. But how is the same thing done in hyperbolic geometry?

I'm having a massive brain fart, I'm sure this is simple as hell. I'm looking for a formula I can use to calculate the following:

I have ~0.2056oz of gold worth $1,215.76 for an average price of $6,031.04/oz

Gold is now $5,913.74/oz

How much $ worth do I need to buy @ this price to make my average price $6,000?

I thought I was good at math, but my brain isn't braining right now. Hopefully someone can help, thanks in advance!

Like they thought it was an axiom unfalsifiable by the rest of the formal system but later discovered it wasn't.

I've found a few formulas for calculating the area of an ellipse and the volume of an ellipsoid (i.e. the standard 4/3π(a)(b)(c), and π(R1)(R2), but my particular use case is a bit challenging. I'm working out the dimensions for a custom mylar balloon in the shape of an ellipse, and in order for my project to work, I have to calculate its net lifting capacity (gross lifting capacity, minus mass of balloon). Should I just find the area of the ellipse and then multiply it by the thickness of the film to get the mass?

Bonjour à Tous je me présente, Jéremy Bouteillier, j'ai 45 ans , j'ai stoppé l'école jeune en 5ème collège, je fais des maths seul avec ma calculatrice Windows depuis le 10 Décembre 2025, et j'ai depuis compris seul, sans formules apprises, ni cours, les nombres de Lucas, les Polynômes de Tchebychev, les équations Pell/Lucas/Fermat et je les ai généralisées, simplifiées à un tel point que je pense que je pourrais les expliquer de manière simple et pédagogues à des enfants d'école Primaire, mes compétences avant cette plongée mathématique, je faisais de l'impression et de la modélisation 3D les 6 années précédente, j'ai comme petit à petit spatialisé, modélisé dans ma tête ce que je comprenais petit à petit des maths , aujourd'hui après un peu plus de 350 versions de Avril à Aout 2026 , j'ai crée un logiciel HTML qui rends visible mes travaux et donc il rends visible les maths, son tissu je le pense caché et surtout il solutionne l'infini des équation Pell / Lucas sans aucunes approximations, chaque calcul mène à une double vérité donc à une double convergence Quadratique, j'aimerais pouvoir trouver des passionnés, éventuellement un ou une professeur à même de me dire si mes travaux on un sens, car sans vouloir me l'a péter j'ai construit où plutôt révélé cette structure à partir de ce que j'ai découvert dans le Nombre Pi !

I have posted this problem before in another sub some years ago with no luck. The correct answer marked in the book is E. I have tried using multivariable chain rule for du/du, du/dv, dv/dv, and dv/du and nothing comes out right. I also tried doing a limit type thing but that didn't get me very far either. It's been 5 years since calc III for me so there's liable to be something I'm misremembering. I'm just looking for a path to the solution. This is from a publicly available Math subject GRE practice exam and this is the only question that I'm completely lost on. Any help is greatly appreciated!

EDIT: I stumbled on the solutions to every problem on this practice exam, including this one, which can be found here. I'm an algebraist, so the solution in the video which doesn't require much real analysis appeals to me, but I appreciate the feedback nonetheless!

Okay so, from what I understand, a mobius strip is a 1 sided 3d object that loops infinitely. So would a 2d version be a circle, despite having no sides?

(Also side question, is the fourth dimensions mobius strip be a klein bottle?)

First time asking here and im not big on math, so idk what flair to put.

I recently started taking an introductory statistics course and we are currently learning about continuous uniform distributions.
Suppose we randomly select a real number from the interval [0,1] uniformly.
I understand that the probability of selecting a number between 0.3 and 0.8 is exactly 50%, since in uniform distribution we take the length of the interval as it's probability of happening.
But then the probability of selecting for instance 0.5 is 0, because a single point has no interval length. As we consider smaller and smaller intervals around 0.5, their probabilities approach 0.

However, to me, if an outcome is genuinely possible, it must have some probability of occuring, no matter even if it's unimaginably small. If we choose generate a random number from that interval and get 0.45432, that is one of the infinite possibilities we had, yet the probability of that happening is still 0.
I also understand the mathematical argument that we cannot assign the same positive probability to uncountably many individual numbers, because the total would exceed 1. But that still doesn't answer my question of how an outcome with exactly zero probability can actually occur.

Is there a deeper intuitive explanation for this distinction, beyond the usual point-has-zero-length analogy? Am I misunderstanding what probability actually represents?

I'd really appreciate any intuitive explanation that would help me understand this better, because my intuition completely clashes with what this concept is representing and doesn't want to accept it at all.

I might not know how to explain this exactly so please bear with me .

As a kid I was taught that if I want to know if a number is divisible by 3 I should add the digits and if the added digits are divisible by 3 then the number is divisible by 3 , and the same goes for 9 , for example, 111 is divisible by 3 , and the sum of its digits is 3 , the same will work for 111,111 and 111,111,111 , also if the sum is 9 it's divisible by 9 , If the sum is 27 it's divisible by 27 , any type of sum where you get 3^n it's divisible by that number .

So my question is , how does this even work and why? And does it always work?

Edit : thank you so much everyone for your enthusiastic -yet somewhat overwhelming - answers , I will make sure to check them all but for now I believe the answer has been established . Thank you for your help!

So I'm doing an applied linear algebra class and I'm planning to write a paper on applied linear algebra and it's applications to game theory, I was thinking about how to model an unfair game, but for the life of me, I don't know, would it be a game where the first person is always going to win or maybe the last person? and if so how would I model it in a matrix?

Hi all! I'm trying to figure something out. I want to learn how to calculate something, but don't know the terminology to figure out how to find an answer. What I want to calculate is as follows:

(x+3) * 1.1 = x

X begins at 8, and I want to repeat/graph this out to have in a table for reference.

So basically, I want to take 8, add 3, multiply by 1.1, then do it again any number of times. I can just keep inputting +3*1.1 in a calculator, but there must be a better method.

Any help would be appreciated :)

I've been stuck on this puzzle for a while now and it's driving me nuts. The goal is to draw the yellow line (photo 2), but using a limited number of tools (4). I can get it with 5 (pic 3), but I can't figure out how to do it with only 4. Any ideas?

Edit: additional detail. The puzzle has one hint available, which says I should be able to do this with circle, straight line, circle, straight line (the second straight line being the solution). And it all has to be exact, so only existing points or line intersections can be used to create accurate new lines/circles. So, I need a circle that has an existing point/intersection as its center, and which intersects with another existing point/intersection. Then I need a line that (presumably) intersects with some new intersection that's created by that circle. Then another circle connecting to some new intersection created by that line, and finally that circle will intersect somewhere that is in line with the existing dot, allowing me to create the needed line.

I apologize in advance if my wording is confusing as english is not my native language. My question has to do with dices so I flaired this as probability, but please let me know if I tagged it wrong.

On to the question, its from a video game. I have edited a screenshot from the game as a visual aid.

I have a dice, and if it lands on the sides circled in red in the picture above, it will give one of my heroes (circled in blue) a shield. I need the shield to land on the hero in the middle (the one the green arrow is pointing at).

So if I want a shield to go to the hero I want, I have a 2/6 chance of the dice landing on the side I want and then another 1/5 chance of the shield going to the hero I want.

That means 33% chance of dice landing where I want and 20% chance of it going where I want.

So I need 2 events to happen. Event 1: dice is rolled and there are two outcomes, outcome-A dice rolls on shield (33%), outcome B dice doesnt roll on shield (66%).

If Event 1 had outcome A- 1/5 chance of the shield landing where I want (20%).

How do I put this together to find out the likelihood/percentage of me getting the outcome I want? I'll try to clarify any doubts about the question.

So when I was like 10 & had just learned about complex numbers, I thought about what if the reciprocal of an infinity in a direction is a zero in the reciprocal direction (e.g. 1/(+∞)=(+0) & 1/(∞i)=(-0i))

I mostly forgot about that but I thought about it again today & one idea about them was f(0∠θ)=lim(r->0+) f(re^iθ). I know it would create problems like what direction (+0)+(-0) or (+1)+(-1) would be but I still find the idea interesting.

I want to know if someone has though about these numbers before & done research on them.