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!

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.

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!

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!

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'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?

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?

I work in software so its relatively easy to understand why a base 2 system is preferred: because it can be represented by a simple on/off signal

I just don’t really understand the relevance of other bases. Like why is base 10 inherently good compared to base 12 or even base 60? Why is it seen as such a big deal in the history of math? Are there any cases where using a different base number is better?

Can you use anything for a base? Can you do math across 2 different bases? Like could I do math with a base 10 number system and a base 2 number system, or would that simply be translating one to the other?

(any admin can change the flair, I didn’t know what the appropriate discipline is)

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?

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?

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!

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

​

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.

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

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!

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!

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?

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?

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.

\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’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!

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 :)

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?

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.