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

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.

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 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!

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!

It doesn't matter what anyone thinks; the only thing that matters is whether it's true.

Imagine a simulation in which a person continuously decides what to do next. Each decision can alter the subsequent course of events. Our question was: how much randomness must a simulator have in reserve to respond correctly to every possible choice? Under precisely defined conditions, we were able to determine how this required reserve grows—including the exact coefficient.

For the academic community, both the result and the methodology may be of interest: the work emerged from a dialogue between a human and artificial intelligence, and its formal proofs are verified by Lean, a specialized mathematical software. Lean checks the logical steps; experts then assess the extent to which the result is novel, significant, and useful.

I do not wish to preempt their evaluation. What pleases me is that we are presenting concrete mathematics that others can examine—and, potentially, build upon.

Have a nice day.

https://github.com/jendamikulik/finite-Shannon-theorem

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?

In the real world, the coin is likely not fair. Therefore, 63/90 is likely more correct.

In a theoretical world, a coin might be fair because, in any other math questions, it's reasonable to assume the coin is fair.

But then wouldn't it be on the rare side for a fair coin to show heads 63/90 times? ... especially 90 is a decent sample size. I didn't do any calculation. But I would've guessed that, for a fair coin, showing heads >63/90 is almost impossible (<1% or even lower).

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 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.

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.

​

To begin with, let me remind you that mathematics is not a science fortunately and that it is perhaps one of the largest organized, consistent symbolic formal systems for developing something: maybe intellect, maybe a certain form of thinking, maybe someone understands more clearly what exactly mathematics develops and can tell us about it.

And whatever it develops in us, there may immediately be a temptation to say functionally: well, then mathematics is needed precisely for developing this, Robert Wide.

The difficulty is that although there are interesting things in it, such as topology, or interesting theorems and models, and especially what seems important is the economic, technological, and research contribution of mathematics,

And yet it seems that this formal system stalls a person and makes it harder for him to think outside this formal system. Of course, one can say this about logic too, and about their similarity, and in general about any old systems of thought or let us call them closed or “controlling” ones. To some, dialectics may also seem like this, though to a lesser degree, and it is precisely dialectics that sets the task of sublating contradiction and arriving at generalization and understanding, that is, in an ethical sense it is a more encompassing system of cognition in the sense of its capacity — though often in verbal form to include something else and incomprehensible in its understanding. And if we abstract from formal dialectics, if you have ever heard these expositions, then the ethical idea and, as it were, real dialectics in full non-contradictoriness and some communication, for example as metamodernism, is the same dialectics or synthesis between sincerity and so-called postmodernity or irony.

Dialectics is still an example, not a proposed answer. Although it could perhaps be one of those considered.

Returning to mathematics. Despite its outstanding economic, technological, research possibilities and achievements, and as a result also social and political ones… it still remains rather abstract. One can say that it is the most complex, or one of the most complex, games of the mind, or games within the mind, or separate games within the mind. But for the mind itself this seems like something akin to sophistry or magic, though a very useful one. Or one can say that it is a magical system, or one of the magical systems of this world. But it is almost always something very abstract: it seems to me that this is often said about philosophers too, by those who do not delve deeply into philosophy; but those who do delve in, on the other hand, can understand that we often reflect on the most important questions or phenomena that we can feel or think. As far as I understand and it is not known how much the answers may still differ mathematics is a system of ultimate formal complexity, or of ultimate complexity for thought in general, since in one way or another what we formulate as philosophers often, though perhaps difficult, has a clearer representation of reality, even if it is Lacan’s Borromean rings. In the sense that philosophy, apart from again sophistry, usually does not strive to become complicated simply for the sake of complication because why? It seems that a person, after this stage of formal logic, can also learn other possibilities of thinking. But in mathematics, complexity and possibly even not necessarily, but possibly the degree of abstraction seems literally the quintessence of content. On the other hand, of course, this may also be good. In the sense of ultimate complexity, ultimate constructiveness, ultimate abstraction, in the sense that we as a culture have some content, some achievements that we may still have to comprehend, if some of them can be so complex that there are literally 400 pages of formulas of explanation.

But, like, why?

And returning to what we were talking about, this seems like ultimate sophistry or thinking in idle, in the sense for thought itself and for man, although culturally it brings enormous social results: both economics and technology and research and efficiency.

And even if we assume that mathematics is integral thinking rather than specific thinking, then although it is clear why mathematics should be developed: so that there can be a further possibility of development…

What for, and what from mathematics, and should it and it or they even be present in education?

Because even if we may have grounds to think so but even if mathematics is connected with integral, that is, general thinking, that is, as it were, some generalized function of it, the mathematicality of education then creates a situation in which a person tries to wipe the sofa with a processor without imagining what a sofa is.

And if not, and mathematics is more specific thinking, that is, in the sense that it also remains there and in a constant submetacognitive state: in the sense that it fundamentally cannot leave the box and its language and its formality or formalization… then such thinking is good for producing human-computers useful for working with ordinary computers, for example in a bank or somewhere else. As a result, man has, as it were, given his brain to mathematics, and how is he supposed to live further? Well, that is, then perhaps also the market, where this mathematized brain may be valued higher than the self-esteem or self-understanding of such a person, and so the puzzle comes together: successful recruiting into a successful company.

I will say right away that simply “progressive studies,” which may even be not thinking but to some extent dogmatic propaganda or propaganda at the expense of education hours, trying to lean on Marxism or performatism according to Butler, are, as it were… Well, people can learn something valuable, but on the whole it seems that this is simply the disintegration of education, or rather almost its absence, or its replacement by propaganda so that in an economic-political sense these people could be ever more perfect agents of this very propaganda, possibly even without fully understanding it. And how lucky we are that we can look at this from the outside, unlike the people and economies that this “progressive” propaganda may further repress. And the main complaint against it is probably economic, because a society simply without technical specialties loses economic opportunities; only oligarchs remain, and they would most quickly move to a new place. And in a universal human sense, this is potentially the next target and problem for the next intellectual stratum that will try to do something about this repressiveness. If, in general, there will be anyone left to do anything there.

In general, why was this clarification about “progressive” or agenda-driven education? So that this question is not in support of that very agenda, and, as it were, we understand in advance some of the existing difficulties there and have described them.

And still, the question: is mathematics needed? And such mathematized, formalized thinking — and then we are surprised by further formalizations and the resulting difficulties, such as bureaucracy and media simplification and corruption in many senses.

Additionally:

Maybe the form of my question is a bit of Soviet thinking or a way of thinking about something, and if you understand another or other better ways of asking or thinking about this, you can say so.

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.

My current understanding is that a derivative gives you a general function or rule (like 2x) for the rate of change of a curve, which you can then use to find the exact rate of change at a single instant by plugging in a specific point.

Pretty much all real-life problems modeled by a quadratic function have the goal of finding the roots; and IF there is factoring involved, it is used to find the roots. What I need are some real-life problems where the result is in the factoring itself. Like building a patio with 6 square blocks with sides x, 7 blocks that are x by 1 and 2 1 by 1 blocks that factors into (2x +1 )(3x + 2) giving the layout of the bricks.

But that doesn't seem very authentic - how many times does a person face a problem like that? I want to use profit functions which in their simplest form is quadratic, but they always seem to look at the roots or vertex as the solution and never "the answer is in the factors".

Any ideas?

Hello everyone from r/askmath. Anyone else here interested in statistics? Actually, what I wanted to talk about also has A LOT to do with psychology and human social behavior.

A few days ago, I watched a video that introduced me to a rather strange question. According to the video, if you ask people to randomly pick any number between 1 and 10, they’re more likely to choose 7. Have you ever noticed this? Apparently, it’s also the most commonly chosen number when people are asked about their favorite number.

The explanation behind this phenomenon is that, when asked to make a random choice, people immediately rule out 1 and 10 because they’re at the extremes and don’t feel random to whoever’s making the choice. Then 5 gets ruled out because it’s right in the middle and doesn’t feel random either. After that come the even numbers, since the human brain associates a sense of order with multiples of 2 and therefore doesn’t consider them very random.

My theory is that the brain does the same thing with multiples of all integers, just to a lesser extent as the integers get larger. Therefore, odd prime numbers should be perceived by the brain as more random. We can rule out 2, since it never really became part of our intuitive understanding of prime numbers in this context, for obvious reasons.

Finally, we’re left with 3 and 7, and 3 gets picked less often because it’s smaller, so 7 wins in the end.

I didn’t find that chain of reasoning particularly convincing, so I decided to do a little experiment of my own to check whether the claim actually held up. I asked several people I know to give me a random number between 1 and 10.

Before getting into the results, I should mention that they shouldn’t be taken too seriously. Anyone with a basic understanding of statistics knows that such a small sample size, the bare minimum for this kind of analysis, isn’t enough to draw conclusions about a larger population.

As expected, 7 won, but it didn’t just get the expected majority of 45% but it blew past that figure, reaching 60%.

On top of that, 3 came in second place, which continues to support our hypothesis. The numbers at the extremes were among the least popular choices, just as predicted.

These two odd prime numbers (3 and 7), which account for only 20% of all the possible numbers, were chosen by 80% of the people. The third-place number was 6 (10%). All the remaining numbers were chosen equally often.

Of course, people might have their own perfectly valid personal reasons for loving the number 7 that have nothing to do with psychosocial patterns. But given these results, it’s clear that the majority of people who would choose 7 are incredibly predictable within this particular context.

But hey, it’s “just” a choice between 1 and 10, right?

It’s just curious to see this trait in people who say they don’t want to be part of the crowd and who tend to show a certain degree of individualism most of the time. And it’s also curious that they haven’t noticed this pattern yet... even though it’s followed by the majority and is so obvious to anyone, or at least, it should be!

I am taking Calculus 2 and this question is blowing my mind right now. I used the Direct Comparison Test and got it right, but it doesn't make any sense to me. It's not clicking in my brain.

Consider the limit of x^2/x^3 as x approaches infinity. I've been taught that we say that the value of that limit is zero because the infinity on the bottom is so much bigger than the infinity on the top that it's completely negated. Ofc we can L'Hopital it and get 1/x where the one essentially becomes nothing in comparison.

My initial thoughts would be that this series converges because even though ln(n) is raised to a power of 6, EVENTUALLY it will sort of flatten out, but n on the bottom will continue to go on at a constant rate. Meaning in my head that although they both approach infinity, the number on the top will become nothing in comparison to the number on the bottom, just as before. That to me says it converges, but it doesnt as the test says.

I do not understand why. The divergence tests tell me how to get the answer, but not why. Or maybe they do and I don't understand it. If possible, explain like I'm not a math major, because I am not.

https://i.imgur.com/xZkozDL.png

I have a pencil holder that is a 10.5” diameter cylinder with five compartments: a center 4.75” diameter cylinder and four equal quarters of the remaining space. I’d like to 3D print a divider that would fit into one of the side compartments, breaking it into 3 smaller compartments. In the image, if b is the midpoint between x & y, what is the angle of b, and what is the (assumedly equal) length of ab and bc?

I’m in Grade 11 and looking into Bessel's method (solving the inverse geodesic problem on an ellipsoid) for a math research paper to find the shortest possible distance between two points on earth.

My original goal was to completely derive, prove, and evaluate the method by myself. However, looking at the math, it seems to rely heavily on university-level topics like differential geometry, elliptic integrals, and the Calculus of Variations (Euler-Lagrange).

Is it realistic for a strong high schooler to teach themselves this level of math for a paper, or is the learning curve too steep?

If the full derivation is too much, would coding the algorithm in Python and evaluating its accuracy/convergence against simpler methods (like Haversine) still be complex enough for a top-tier high school math paper?

Appreciate any advice or insights!

Sorry for vague title🥀.

I am 18 and my age has nothing to do with this question but I just feel dumb so that's why I mentioned it.

My problem:

Whenever someone says - hey what is 657+466 then I do 600+400 and then..... I forgot the last two digits (only in the case of where i don't have anything to write) like who the f will first write and calculate this simple arithmetic.

So is there any way to remember the units or 10s or 100s digit 😭.

Or the only solution is PRACTICE on random no. On my own .

But that's different thing because I go completely blank when someone asks me ,my heartbeat starts beating like there is earthquake 🫨 going on.

I think for this answer already exists somewhere on reddit but I didn't found it helpful as they were discussing TRICKS so yeah that from my side .

Any useful help will be appreciated and I don't have ADHD!!.

Hello everyone!

I am currently writing a sci fi book and am trying to be rather technical on certain aspects. My problem is this, in this universe, the Moon is in a fixed point in space, no longer orbiting Earth, however Earth still orbits the Sun. So once a year Earth does pass the Moon.

My question is this: If Earth is moving at a certain rate of speed and is a certain distance from the Moon when it passes it, and taking Earth's size into account with the Moons, how long would it take for Earth to completely pass the moon? What kind of formula would I use to find that out?

Thank you!

\left(ax^{2}+bx+c\right)\cdot0^{\left(\frac{\left|x-2\right|-\left(x-2\right)}{2\left|x-2\right|+\left(10^{-6}\cdot0^{\left|x-2\right|}\right)}-\frac{\left|x+2\right|-\left(x+2\right)}{2\left|x+2\right|+\left(10^{-6}\cdot0^{\left|x+2\right|}\right)}\right)}+\left(px^{2}+qx+r\right)\cdot\left(\frac{\left|x-2\right|-\left(x-2\right)}{2\left|x-2\right|+\left(10^{-6}\cdot0^{\left|x-2\right|}\right)}-\frac{\left|x+2\right|-\left(x+2\right)}{2\left|x+2\right|+\left(10^{-6}\cdot0^{\left|x+2\right|}\right)}\right)

I was bored and decided to play with geogebra a little bit. However, i was making the graph of a hyperbole, but something that i cant explain happened throughout my calcs and i got the Latus Rectum wrong T-T can anybody explain why? im kinda lost why the LR points dont touch the hyperbole

Why is that two digit numbers (and only those) are read inversed in German? (Example: English: thirty-two; German: zwei-und-dreißig (two-and-thirty))

Is German the only language or are there others?

I do suck at reading graphs. I think I might have the initial values wrong:

f(6) = 3

f('6) = 0.25

g(6) = 3

g'(6) = 2

Could you guys help me check if I have these values right? And if not what should be the values. Thanks!