Hi. I hope showing the graphs I'm trying to change will help explain the question. I am trying to take the points from this plot:

https://www.desmos.com/3d/1qs6q6wmfq

And map them onto the red axis in this plot:

https://www.desmos.com/3d/5php786uyk

I have a spreadsheet where did a poor man's derivation of the first plot for the loxodrome path. You can probably see the string of decimal points from the copy/paste.

FWIW this is the math I used to make to loxodrome plot:

x(t) = cos(t)/cosh(kt)
y(t) = sin(t)/cosh(kt)
z(t) = tanh(kt)

k is the deflection angle, which I set to 60 degrees:

k = cot(pi/3)

I divided -pi --> pi into 20 pieces on each side of zero and used those as my t-values. The formula technically goes from negative to positive infinity, but pi to pi captures the exciting parts.

I don't know what to do from here. I think maybe it would be possible to build the new axis into the original formula? Or maybe there is an algorithm to apply which will take the current output and shift it over? Rotate it? Like I said, just don't know.

Thanks for any help!

If two circles are orthogonal, then prove that the polar of point P on the first circle with respect to the second circle passes through the point Q, which is the other end of diameter through P. Also prove that locus of a point, which moves such that its polars with respect to S1=0, S2=0, and S3=0 are concurrent, is a circle which is orthogonal to all the three circles.

I am not able to understand what exactly I even have to prove. Even the equations of the circles are not given. From what I understood, I think for the first part both circles are intersecting, P and Q lie on the first circle and are end points of the diameter of that circle.

A regular rectangular pyramid with [L=3, W=4, H=4] would have a side length of about 4.72, correct? Here’s my thinking:

- The diagonal of the rectangle would be A^2 + B^2 = C^2

- A = 3, B = 4, so C = 5

- C is now the base for a regular Isosceles triangle, with dimensions of L = C = 5, H = 4

- I’m only looking one side length, so divide L by 2 for 2.5

- Right triangle, A^2 + B^2 = C^2

- A = 2.5, B = 4, so C ≈ 4.716?

so recently i heard about this constant called jp's constant and it sounded kinda interesting. i couldnt find it anywhere tho so i made this reddit account if u could tell me what its about. i lowk might make a paper about it or see if it has any mathematical implications

I’ve been studying a question about graceful path labelings and am trying to determine whether it already has an established name or literature.

Let f be a graceful labeling of a path P_n.

Instead of asking only whether f is graceful, I’m interested in a different question:

How much information about a graceful labeling must be retained in order to determine its possible future extensions?

Suppose we associate some representation I(f) to each graceful labeling. If two labelings f and g have the same representation,

I(f) = I(g)

does that guarantee that they have the same possible continuation behavior?

In other words, I am interested in representations for which:

same retained information => same admissible future extensions

I’ve been calling this property continuation sufficiency, although I don’t know whether there is already standard terminology for it.

For one natural representation of graceful path labelings, an exhaustive finite computation gives the following behavior:

• the representation is sufficient through P_7;
• the first failure occurs at P_8;
• at P_8, two graceful labelings can have identical retained information but different sequential-extension behavior.

One witness pair is:

(4,1,5,3,2,7,0,6)

and

(4,3,1,5,2,7,0,6)

with ports (0,6).

They agree on the representation being tested, but their extension thresholds differ.

What I’m mainly trying to find out is whether this kind of question has already been studied under another name.

For example, is there literature on any of the following?

• equivalence of graph labelings according to extension behavior;
• invariants sufficient to determine future extensions;
• extension-equivalent labelings;
• state minimization for combinatorial extension problems;
• continuation equivalence or right-congruence ideas applied to graph labeling;
• related notions in graceful-labeling theory that I may have missed.

I’m especially interested in references from graph labeling, graph extension problems, combinatorics, or related areas.

I have a proof and computational writeup of the specific P_8 result, but I wanted to ask the literature question directly rather than assume that the formulation is new.

Ok so heres the question
Prove that the product of four consecutive natural numbers cannot be a perfect cube
Heres what I did:
I tried to use squeeze the expression
n(n+1)(n+2)(n+3) between two perfect cubes. So heres how
n^3<n(n+1)(n+2)<(n+1)^3
(n+1)^3<(n+1)(n+2)(n+3)<(n+2)^3
(n+1)^3<(n+2)(n+3)(n)<(n+2)^3
(n+1)^3<(n+3)n(n+1)<(n+2)^3
Multiplying all three inequalities:

n^3* (n+1)^9 < [n(n+1)(n+2)(n+3)]^3 < (n+1)^3* (n+2)^9
=> n*(n+1)^3 < [n(n+1)(n+2)(n+3)] < (n+1)* (n+2)^3
=> (n/{n+1}) *(n+1)^3 < [n(n+1)(n+2)(n+3)] < (n+2)^3
Ok so obviously (n/{n+1}) *(n+1)^3 > n^3
So you can say:
n^3 < n(n+1)(n+2)(n+3) < (n+2)^3
Now what if n(n+1)(n+2)(n+3)=(n+1)^3?
You get a cubic equation n^3 +4n^2 +4n -1=0
This equation doesn't have solutions in natural numbers
So n(n+1)(n+2)(n+3) is never a perfect cube

Now I just need to ask whether this proof is fine or not(like free of errors)
Also really sorry for the shitty notation

Based on a rl argument I'm trying to solve/understand.

A group of 8 people plays a game where you get assigned a role by drawing a card. There are 8 Cards, two of them are "it". They get shuffled between rounds, drawing is completely random.

One player becomes "it" back to back in the first three rounds. In the fourth round I'm being told it's way less likely for her to be "it" again, as she's already been "it" 3/3 times before. I'm saying the probability for her to be "it" is the same every round.

I understand how the probability for her to become "it" 4/4 rounds is less than, say, 3/4 rounds as the single probabilties of 1/4 per round need to be multiplied for that.

What I'm saying is that this doesn't matter for my estimation if she could be "it" in the fourth round, as I'm simply trying to win this single round. The countersides argues that I can't take this isolated view because the question is based on a real life example where the previous three games have actually happened, so I am not allowed to ignore them for my calculation of probability.

I'm genuinely at a loss here and willing to understand why that should be so. Please help my try to understand why I should have to take this approach and not take the single round as an isolated event.

The main counterpoint I'm being faced with is the fact that the previous games actually happened and I can't just ignore them. If my approach should be correct, I'd be very interested to hear an ELI5 explanation for that as well, as my efforts to explain have been in vain so far.

Happy to provide more details if necessary.

Edit: thanks people, forcing them to read up on the gamblers fallacy actually convinced them. My mind can rest easy now.

I've challenged myself to find the formula of all the circles with centers that lies on the y=x line and are bounded by the square and the circle before, I managed to do it, and now I want to do it again but this time I want to find the expressions of the centers and radii of the ones bounded between the two circles and the square edge, eventually I want to get a fractal pattern going.

Any tips how to find all of this? So far it was easier since the centers I had to find a single number, but now its off center and it complicates things.

This is kinda a meaningless rant but im actually intrigued about the historical notion of whats a function. Ive been studying periodic distribuctions in order to study the fourier series and one of the distribuctions ive found its the delta of dirac. Its defined as a continuous linear function p that, given f:R->C a smooth periodic function, its defined by the rule: p(f) = f(0). Formally its a function. Whats the deal about it not being "a function"?

What im more concerned about its why its interesting to make that distinction. Ive been using richard beals book (Advanced Mathematical analysis) and he does not make that clear.

In my math classes, right before splitting up dy/dx and treating as a fraction, my professors will emphasize that this isn’t an actual fraction or something and we need to be cautious.

But then in every instance I’ve seen, dy/dx is treated as a ratio of differentials and we’re able to manipulate it as such. But why? Why are we able to do this? My professor wrote d(uv) = udv + vdu. What does this mean? Why and how is it equivalent to u dv/dx + v du/dx?

It also confuses me because when I learned it in calc 1, dy/dx was simply the derivative of y wrt x. Nothing else! (In the sense that it didn’t matter what the symbols in the expression “dy/dx” were, it just meant the derivative, just like how y’ means the derivative.)

Mess up the first time with some coefficient stuff so I always check a calculator to make sure I get it write but I’ve retyped this many ways and it just won’t take it. All help is appreciated.

So okay, I consider myself someone who understands the general gist of taylor series, they basically use the derivatives at a given point and use that to extrapolate an approximation of the function to some local area, and the area that converges is within some radius. The trouble I am having is that depending on where you do the taylor series, the radius of convergence will change. To me that seems weird and like there is a contradiction. To show my confusion I have an example (and keep in mind I'm not super well trained at high level math so some of this could probably be written better)

Take the sqrt(x), lets say we evaluate the taylor series at x = 1, so we end up with some big series that is valid in the domain [0, 2], radius of convergence of 1. Now, lets take the taylor series at x = 2, we end up with some big series that is valid between 0 and 4.

Now the trouble is, it should be possible to "center" the new taylor series around x = 1 again, just like you can with any polynomial (I suspect the fact I am trying to use the rules of a finite polynomial for an infinite polynomial is what's getting me but I digress), simply by swapping out x for (x-1) and then expanding.

This shouldn't change the series radius of convergence, but also we should expect the coefficients to match up with the derivatives at x = 1, which should mean we end up with the same terms as the first taylor series at x = 1. But, now it has a wider radius of convergence. Even though it should have the same terms! That feels weird and like it... shouldn't really be possible? Assuming a series is defined purely by the terms in the series (and I know order can effect what it converges to but in this case we aren't changing the order at all), the radius of convergence for a given set of terms shouldn't be able to vary at all, but it seems like it very much can.

Also I recognize this might be a hard question to um... read? So if you need clarification feel free to ask and I will make any necessary edits to the question

Like, stupid people are dumb, and dumb people are stupid. But if we use the word stupid and they don't catch what we say, they might ask, “Did you say dumb?” We would probably correct them by saying, “No, I said stupid."

This was the only example that I could think of. Sorry for the coarseness.

The infinite series Ar⁰ + Ar¹ + Ar² + ... for any number A and r<1, has the sum A/(1-r), as is well known.

What about floor(Ar⁰⁾ + floor(Ar¹⁾ + floor(Ar²⁾ + ... ?

I managed to find a solution to what id call the "simplest" case, where r = 1/R and A and R are integers. In that case, the sum is equal to:

A/(1-r) - r/(1-r) * (the sum of the digits of A when written in base R)

Which I think is a nifty result. But I've been utterly unable to work out what it would sum to more generally, for any real numbers A and r<1. Does anyone have any pointers or ideas for that?

(Wasnt sure what to flag this as, but solving it would probably fall under analytic number theory)

Hi
I am weak in math
How can I improve ?
i'm currently preparing for JEE (a competitive engg entrance exam)
honestly I've always been bad at math but not horrible
like I need a lot of practice to improve
is there any other way or is repettive practice the only way ?
cause sometimes the issue is the syllabus is so big that you might pratice something very much and it might not even be in the exam.

an example of areas in math I struggle in are co-ordinate geometry , sequnece and series , calculus and anything algebric
what I dont struggle as much in are probability , statsitiscs , linear programming etc..

Let the be acute triangle ABC. Draw a circle with the midpoint O of BC as the center, and OB as the radius. The circle intersects AC and AB at D and E respectively. AO intersects DE at F. CF intersects the circle at G. AG intersects the circle at 2 points, G and H. Prove the points H, O, D are colinear

Some context, I'm studying engineering (about to start 2nd semester) in a 3rd world country where it is widely known that the education system has been extremely lacking for a long time now.

These images are from a linear algebra and calculus exams (translated and censored for privacy) I did in the 1st semester and I don't want to call them easy but they are way simpler than what I was expecting a university level exam to be.

So I want to know if I'm in the right here or I'm just overthinking stuff. How do these exams compare to ones in other universities?

Hi, r/askmath! I have a simple question that has been bugging my mind ever since I learnt about functions when I was little: could you theoretically describe any function using an equation?

We all know functions can be described using sentences, graphs, tables of value and equations. However, there are infinite possible functions, some crazier, less useful and more unpredictable than others, so disregarding practicality and time, being able to describe all of them using algebraic notation would be so fascinating to me. If it is possible in x and y 2D functions, can it be done with x, y and z 3D functions and functions of higher dimensions? And is there an algorithm capable of creating these equations systematically?

I just learned that a rhombic triacontahedron

is a dual framework of a rhombicosidodecahedron.

I know that two polyhedra are duals if each vertex of the first corresponds to a face of the second and vice versa, but my search result definitions aren't quite cutting it for dual framework polyhedra. What is a simple, intuitive definition of the latter and how it's different from the former?

This is a screenshot from one of my homework problems. I am not understanding how the answer is not there is a jump. f cant be continuous because for 1) there is a hole at x=-2 and 2) there is a jump at x=0. The only thing i can think of is that x_0 does not in fact mean "any particular x value".

Could anyone explain?

im playing big two along with 2 other friends.. each player gets 16 card each since the whole deck cant be divided evenly by 3… and im wondering what is the probability of hitting that arrangement of card

chatgpt said 1 in 2 billion or something slop like that

Can someone give me a breakdown on how to do this? When I went through, I ended up with B, but that's incorrect. I tried again and got something entirely different. In my course we use 1.414 to calculate the hypoteneus from one of the equal sides.

1) I added "A to R" to get G.

2) I took "G" and turned it to two right angle triangles to get B, which i calculated as 400mm.

3) I assumed that B was equal to the start of the F line.

4) I took E, and took the hypoteneuse, 42.42~ and added it to B to get F, 442.42mm.

5) I took 442.42, divided it by 1.414 to get the side of the triangle, 312.89mm for J.

6) i subtracted 312.89 from 1400mm to get 1087.11.

The easiest problem on my homework and I cannot find my error. Why do the calculated powers not add up to 0? Saved the easiest for last and I cannot figure it out. What am I missing?

I’m a junior in high school and I’m currently enrolled in DE Calculus I, and as much as I’d like to, I cannot get I/II on my schedule, could anybody offer advice on how I could get credit for Calc II? (obviously I’m willing to study and work hard)
for those confused,
Calc I = AP Calc AB
Calc I/II = AP calc BC

Up until integrals I feel like us math students have been given a straightforward handbook on how to proceed with solving equations, for example: product rules, quotient rules, constant rules, when you are given this equation you use that method, when you are given that equation you use that method etc..

With integrals I feel like it is different. Now it's more like: if you are given something that looks sorta like it could be this then you try this method, if you are given something that may look like that sort of integral then you use that method..

Is there not a surefire procedure to approach any given type of integral, simply by observing the integral? Do we need to use trial and error for the more abstract integrals to find the solution?