It seems so incredibly intuitive? Like I've been trying to prove it by showing that if there weren't twin prime numbers at all, then you couldn't uniquely factor all of the imaginable composite numbers. You could still "create" an infinite number of composite numbers, and uniquely factor them with all existing numbers, but you just couldn't create ALL of them.

Like for example imagine the set of primes (2,3,7,11,17...)

You could still make an infinite number of composite numbers, you just couldn't factor ALL of the POSSIBLE ones. Again, the FTA doesn't break, and so there's no "issue" really, logically it still makes sense.

But even if you just add 1 more twin prime to the set (2,3,5,7,11,17...) you get more composite numbers than you had before. And then you just keep adding twin primes to the set of numbers and if you ever miss one then there will be composite numbers you couldn't "make up" or create. There would still be infinitely many, and the FTA wouldn't break, you'd just have... Less... Like 99.9999% instead of 100% but OBVIOUSLY the number line is 100% of the composite numbers + all the primes and so you have to add infinitely many twin primes.

That seems reasonable enough but why is a proof so hard?

The best example would be the derivative. By definition it is the instantenous rate of change. But how do you define instantanous? It is this specfic instant but that doesn't feel rigorous and surely there must be a concrete, precise definition. Or local extrema. How do they give the maximum or minimum shape of an area if you take them? What does ''local'' even mean?

I can solve problems without much difficulty, but I just don't feel like I have truly understood the material and this prevents me from moving forward.

Hey all. I am studying calculus 2 this summer and my professor has put forward a new criteria for getting an A and a quite unrealistic one at that. He said if using chatgpt or any assistance , we are able to create a new convergence test that has never been presented before (works even if it is applicable for even just a few types of series and fails for others) , we can get a good grade. Is it kinda possible that we do this? Has anybody got an idea. HELPPPP

p.s dont mind my english, its my 2nd language.

I don't know if this is the right sub(or flair), I don't know if this is math, physics or something else.

I asked my dad a question(it had something to do with points I think. I can't remember it right now) we then discussed and the answer i came to was that it depended on whether a vacuum has volume.

Imagine a separate dimension(or a pocket dimension), doesn't exist in this universe, that is empty has literally nothing does this space have a volume? And could you fit one vacuum in another?

I don't know if more context is needed if ot is just ask.

Again, I apologize if this is the wrong sub and thank you in advance.

Edit:clarification by vacuum I mean a lack of matter not dyson. Maybe nothing is a better word.

I'm in a discord server with some friends and we have a text /command game that we all play. /work every 10 minutes earns you money, /collect once a day gets you a paycheck dependant on your job, etc.

With the money, you can buy items and/or gamble. One of the games is chicken fights. You buy a chicken, place a bet, and then typing /cockfight generates a win or loss, either taking your money or doubling it- pretty simple. Well, its absolutely smoking me and I feel like it shouldn't be.

The chicken has a 49% win rate to start, every win increases the win rate of the next fight by 1% (so 49%, then 50%, then 51%, etc.). when you lose, the probability resets back to 49%.

Mathematically, I thought this would equal out to like a 51-52% win rate. But i am getting crushed. See my 3 bouts below.

43 wins - 66 losses

46 wins - 53 losses

3 wins - 11 losses

Is my sample size too small? Are the odds worse than what I estmated? Am i just super unlucky?

Does Anyone know what the exact estimated win rate should be for the equation mentioned?

Bear with me, this is a maths question. A less whimsical way of asking it would be: "How to determine whether a set of points in a 2D plane is randomly distributed?"

So, when I'm out walking in places like the Lake District, I often see fields full of sheep, from elevated viewpoints where I can discern the spatial distribution of the sheep. Like [this one](https://www.geograph.org.uk/photo/570884), say.

The other day I got to wondering whether the sheep were randomly distributed. They certainly weren't all flicking together. But you know how, when you ask people to place points randomly in an area, they tend to make them all quite evenly spaced? I wasn't seeing that either. There were a few small groups of sheep, and areas where there were none, and it all looked like it might be genuinely random.

I thought that's quite interesting. You might think that sheep might prefer to be near other sheep, or might prefer not to be near other sheep, but if the distribution is genuinely random it suggests that they're simply indifferent.

Anyway, how to tell?

I can use knowledge of my position and the terrain to morph the photo into what it would look like from a vantage point perpendicularly above the field. I can assign (x,y) coordinates to each sheep and calculate the distances between each pair of sheep, or the distance of each sheep from its nearest neighbour, or anything like that.

But what then? What metric should I use, and how do I test for randomness?

I was recently talking to someone about how every book can be stored as a point on a pole, through converting the text into some decimal number which could then be stored as a fraction, and therefore a point along the sticks length.

It got me thinking about how if there’s infinitely many points on a pole, and you go to hit the pole with an Infinitely thin stick then the probability of the stick hitting the pole is 0. Similarly, with a stick of finite length the same issue arises when you consider the leftmost point of the stick.

Intuitively i know this isn’t true, the stick will hit the pole, but is there a mathematical reason or something that I am missing?

So, the definition of a continuous function between topological manifolds is that the preimage of any open set in the image is also an open set.

I am not sure how this works in terms of intuition. Say I have the step function.

I don't think I can make an open set containing the discontinuity. All the open sets in the image I can think of will have open preimages.

It seems to me the definition should be the other way around. The image of an open set should also be an open set.

Or better yet. Since you can define convergence in a topological manifold without invoking metrics.

A function is continous at an element X of the pre-image if for any sequence, the image of the sequence is also convergent.

Can someone explain to me how the actual definition works?

In pure set theory, every element of a set is also a set. While I know this has useful applications for constructing mathematical objects, like building other sets. I'm curious about the negative consequences of abandoning it. What paradoxes or mathematical contradictions arise if we allow elements to exist that aren't sets?

Insertion Sort Algorithm:

The example with the solution:

This is wrong: If it actually belongs in position j, then k-j+1 comparisons will be used to move it, because there will be k-j+1 attempted iterations of the while loop...

Isn't it that there are k-j (not k-j+1) comparisons used to move the value a[k] to j position (it is correct that there are k-j+1 attempted iterations of the while loop)?

For example, if n=5 and we are inside the while loop and we want to move value in a[5] to a[1], we are going to need 4 comparisons (and 5 attempted iterations of the while loop).

So it looks like the solution (once again) computes the number of while loop iterations instead of the number of comparisons.

---

Of course, the rest of the solution is also wrong because we're making calculations with k-j+1 instead of k-j.

I thought I couldn't do math for as long as I can remember. For context on my learning difficulty, I have no idea how to "do math" the way my classmates do theirs. I'm really slow and that's me being generous to myself. I see them do fraction problems in multilinear algebra and my brain.exe stops working, the symbols appear and reappear like magic to me. That changed recently when I figured out how to study on my own, I just realised how my brain works, that it sees shapes. How I do calculations is like animations moving on slides, differential is a shape losing energy, integral is a shape gaining energy, convex and concave is like walking on terrain while blind so I need to guess whether the next step will incline or decline, and so on and so forth.

I've done the algebra study, calculus I–III, analytic geometry, and the like, but now I'm working toward studying real analysis and people say it's very notation heavy and gets more abstract. The problem is my mind can only do up to 4th dimensional graphs to help me with math, manifold dynamics and the like. So far the book that has helped me most is Stewart, but even for the Calculus version, I needed other books with a geometry-first intuition. LLMs sometimes help me translate what an equation is actually doing into shapes I can understand, but as we know, I can't rely on them for everything due to user bias, etc. The question is any tips for studying real analysis? Any recommended books or courses? Thanks in advance.

TL;DR: My brain is one useless GeoGebra graph generator minus the immediate equation translation and I need tips for studying real analysis.

Hi everyone! I could use some help. I am in the middle of reading Halmos's lectures on Boolean algebras. He makes a distinction between ∧, ∨, and ' meaning "and", "or", and "not" as representing the Boolean object but meet, join, and complement as functions in the Boolean algebra. However, in the lectures, he doesn't seem to explain the distinction or assumes the reader is already familiar. I don't think I am. At the same time, I'm thinking about union (∪), intersection (∩), and difference (∖). Are all of these supposed to represent similar operations but in a certain order of operation? So, if A is all Boolean objects, B is all Boolean algebras on those objects, and C is all finite set theoretic constructions on A and B, then I can think of it as something like C(B(A)))? Thanks for any assistance!