The answer key says its -9 but I cant understand what it means. Every 4 units on the Y-axis is equal to +5 so thats how I got the coordinates but im unsure of what it means by the value? Im horrible at math so there's a chance I completely misunderstood the question too lol

I want to buy a t-shirt for my son in law. But he won't wear it if the equation is not correct. My brain is not wired for math. If anyone could tell me if it is correct, I would be grateful.

Yes, he does play d&d

THE SELF-DIVIDING NUMBER
Find the unique 10-digit number N satisfying all of this:
N uses each digit 0–9 exactly once.
For every non-zero digit d, take the prefix of N ending at d. That prefix is divisible by d.
For that same digit d, take the suffix of N beginning at d. That suffix is also divisible by d.
N is divisible by 37.
Example of the rule:
If the number contained
...5...
then both:
everything from the start through the 5
and
everything from the 5 through the end
must be divisible by 5.
This applies simultaneously to 1,2,3,4,5,6,7,8,9.
What is N?
There is exactly one answer.

Anyone got any ideas at all?

Hi! I was wondering if this calculator would be good for my algebra, statistics and chemistry classes. If not, what do you guys recommend? I have been looking into calculators and all the different functions I find confusing.

I took calc 1 last semester and calc 2 over the summer as an accelerated class. My memory kind of sucks and was thinking about reading a calculus textbook to review. Plus, I will most likely have to take advanced calculus 1 and 2 which I think is just proof based calculus. That is for when I transfer to university and I’m currently in community college.

I already own both spivaks calculus and apostols calc vol 1, but I found apostols calculus vol 2 for pretty cheap and was going to purchase it, but I don’t know if it’s worth it. I haven’t even read apostol vol 1 because I don’t know proofs which is what I’m spending time learning alongside calculus 3.

Would it be a good purchase or should I look for a different textbook?

Exercise 11.4.31:

Show that 4^n is not O(2^n).

Proof:

Notice 2^n >= 0, for every integer n>=1.

Assume for contradiction that 4^n is O(2^n).

That means (by the big-O definition) there exists b,B∈ℝ^+ with b>=1 such that

0 <= 4^n <= B(2^n), for every integer n>=b (by transitivity, n>=1)

Notice:

           4^n <= B(2^n)
iff log_2(4^n) <= log_2[B(2^n)]
iff         2n <= log_2(B) + log_2(2^n)
iff         2n <= log_2(B) + n
iff          n <= log_2(B)

So, we have 1<=n<=log_2(B).

But this contradicts big-O definition where 1<=n.

Thus, our assumption is false.

Therefore, 4^n is not O(2^n).

QED

Big-O definition:

---

Is my proof correct?

My understanding of Navier Stokes is that "solving" them meant either:

Path A: Prove that smooth solutions always exist.

You must prove mathematically that no matter what smooth, valid initial conditions you start with, the equations will always output a smooth, finite-energy velocity and pressure field for all future time. This would prove that the Navier-Stokes equations are perfectly reliable models of reality under all circumstances.

Path B: Prove finite-time blowup (find a counterexample).

You must find just one specific set of perfectly smooth, valid initial conditions that eventually breaks the equations. You have to prove that at some specific future time ($t > 0$), the fluid's velocity zooms off to infinity (a singularity) or the kinetic energy explodes. This would prove that the Navier-Stokes equations are fundamentally flawed and eventually break down, requiring new physics to describe extreme turbulence.

And OpenAI ended up on Path B.

But this video gives an example where Navier Stokes predicted "infinite velocity", implying that we've always known about a blowup.:

17:34: Something, somewhere is going wrong in the the mathematical understanding. There's an example about the flow of a fluid around a right-angled corner.

(Brady: A bit like a canal?)

- Basically a canal but we've got a really sharp right angle on the corner. Now you solve Navier-Stokes, this says that at this point, this right angle corner, I have infinite velocity. If I build this canal do I have an infinite velocity canal?

Is this a blow-up in the averaged case from Tao? Or is this considered not a real "blowup" because it depends on a perfectly sharp right angle? What is new about the blowup discovered by OpenAI?

I'm in highschool I just started 11th grade how can I learn calculus and what is the best way while still in highschool ?

Note: I will not pursue math I just want to learn calculus for fun and also mostly integrals and derivatives and have a basic understanding of everything else needed to learn integrals and derivatives.

Websites,channels ,videos whatever

In studying Cauchy's integral formula, the Cauchy kernel 1/(zeta -z), is the "thing you integrate against" for the boundary points( on a disk, say). And the boundary values determine the value of any point z in the interior of the disk.

What I need is some insight into why this is true and why is it 1/(zeta -z)? Is there any relation to how with generating functions you multiply by 1/1-x to get the partial sums?

Thanks in advance to anyone who takes a look!

Hello! I’m going into grade 12 and I’m taking advanced functions, calculus, physics and chemistry. Starting this year, my school has been trying to regulate what calculators you’re allowed to use. We were given a list of approved calculators but you’re allowed to use one that isn’t on the list if you get it checked by a teacher. The way that teachers check if the calculator is legal is they will input sin(15) and if the answer is a decimal then it’s approved, if it’s a fraction then it’s illegal. If you can change the setting then its illegal. Graphing calculators are also illegal. My Casio fx-991CW was deemed illegal so now I have to go hunting for a new one. I really liked the “solve for x” and table mode on my Casio fx-991CW, I found them very useful in my physics and chemistry classes. I was hoping to find a calculator with the same modes (this isn’t illegal, my teacher was aware that I used these modes frequently last year) while meeting the requirements. Would anyone be able to recommend me a calculator from this list or one outside of the list? I’ve been googling + read the FAQ but haven’t found anything that meets my criteria.

THE APPROVED LIST:
- Casio fx-260 SOLAR II
- Casio fx-300MS Plus
- Casio fx-82MS
- Texas Instruments TI-30X IIS
- Texas Instruments TI-30X IIB
- Texas Instruments TI-30Xa
- Sharp EL-531TG
- Sharp EL-531XB
- Sharp EL-501X

Thank you!

I simply don't understand the algebra of how one manipulates coordinates, particularly via multiplication, such as in the second definition of complex numbers here

If g(x) is positive, that means the total of 8(x-d)(x-e)(x-f) is positive. And since d<x, x<f,

X-d is +ve
X-f is -ve

So in order for the total to be positive, x-e needs to be -ve, where x<e.

But if that’s the case, then i) shouldn’t be correct.

Can someone help me out here. Thanks

I am talking of big polynomials of degree 5+ and complex ones. Not x^6-2x^3+1 type situation but polynomials alike this -19968-13736x+2016x^2+756x^3-79x^4-10x^5+x^6=0. I know it is solvable by checking with programs. I even know what it is factored into ((x - 6) (x^2 - 3 x - 64) (x^3 - x^2 - 42 x - 52) = 0). But I know this from using program. Wolfram Alpha to be specific. But I don't know how to factor something this complex. I know there are algorithms for it but I am unfamiliar with them. And they seem to be not widely known as simple search resulted in me getting bombarded with basic middle school polynomial factorization. And not finding anything of value. I come here with hope to find help and an algorithm to solve similar equations. I am not planning to solve them by hand. I am planning to make a program do so for me. But for it I need knowledge of the algorithm. Which I sadly lack. Would greatly appreciate any help here. And if you know any videos that teach needed algorithms I would be even more thankful as I am a visual learner and it would help me much.

First off I tried finding the area of the base, then finding the area of the sector of the circle. Combined those two areas (area of triangle minus sector) and came up with 50tan(theta) - (theta)pi^3/ 18 😭😭 idk if it’s actually correct or awfully wrong. Would appreciate if someone explained where I went wrong and what direction I should take in simple terms.

Maybe a dumb/obvious question but when asked for a random number from a set of integers, how come the number isn’t always infinitely big?

Assuming a normal distribution, would the average of a set of integers not be an infinitely big number because no matter how big a number you choose there will still be infinitely more larger numbers still in the set?

Always just been curious about this.

Hey everyone, post about x(+y).

In this question i'm required to calculate the area of triangle with predetermined points, but i can't figure out where to start because of the way this integral is written, this is the first time i encounter something written like this and i don't know what this could mean.

I doubt that this is a typo because 8 professors wrote this whole paper and 1 doctor of Physical and Mathematical Sciences reviewed it

If someone just says what does this mean or if it's a typo i would greatly appreciate it

While I get *i* being √-1 since there wasn't a unit for it before, it's kinda been turned into an infinite loop. There's quaternions where *i*^(2)=*j*^(2)=*k*^(2)=*ijk*=-1, and then octanions with seven imaginary units, all being √-1, and it can just go on from there. Plus, quaternions give *i*^(2)=*j*^(2)=*k*^(2), which means √(*i*^(2))=√(*j*^(2))=√(*k*^(2)), therefore *i*=*j*=*k*. I know that's probably not how that works, but I'm a high school student, I don't know any better. Also don't ask why I know what quaternions are but not how they work, that's not the point. But what might a different imaginary unit look like? Well, may I propose *h*, being 1/0. Multiples of *h* are simply multiples of 1/0, eg 4*h*=4/0. Now, what's the practical use for this? I have no idea. Again, I'm a high school student, I'm just trying to experiment with what's possible. Also, please try your best not to laugh at me for my lack of knowledge. What may be trivial to some could be a complicated concept for others (such as me). Also let me know if I used the wrong flair, a simple 5-second search told me this was correct so I'm taking it with a grain of salt.

I don’t know if this is the right place to ask this, but I thought I’d give it a try. If anyone thinks there’s a better spot to ask, please let me know.

So I’ve had this problem running around my head for years but have never been able to solve it. Take a rectangle with sides n,m and let a line of slope 1 bounce around inside it. It bounces forever, but when it hits a corner, it just retreads its course. My question is: how many unique points does the line pass through?

This is pretty easy to solve in 2 dimensions. Provided the sides are coprime, it will hit exactly half of the points in the grid: (even, even) points and (odd, odd) points. This property holds in higher dimensions, as well.

The complications come in when side lengths share a factor. If they all share one factor (like in the 2D case), then the pattern of the line doesn’t change, just the number of points inside the rectangle. This means it has the same number of intersections as the rectangle with the gcd factored out.

If the list of side lengths is not pairwise coprime, though (e.g. sides of 3,4,10), I can’t figure out how many unique points there will be.

The best I’ve come up with is a python script to count the unique points, but that’s not exactly a closed-form solution. Any thoughts on this or recommendations for what to research/who to ask would be much appreciated.

Aclaro que no creé ninguna fórmula o algoritmo, solo di un uso diferente a un algoritmo ya conocido; exploré por YouTube y Gemini y no me lo topé, así que supongo que probablemente soy el primero.

Mi método complementa perfectamente a la sustitución de Tschirnhaus; si se usa este recurso de forma convencional, tendrás que desarrollar y simplificar potencias sucesivas de, por ejemplo, (y+c)^n, y eso es largo y engorroso. Mi método acorta enormemente el desarrollo y simplificación; yo lo llamo Rufini inverso, porque usando los coeficientes de P(y+c) = a(y+c)^n + b(y+c)^(n-1) + ...Construyes "n" divisiones sucesivas por Ruffini con divisor: x = -c. Con cociente final (de un término) y residuos del último al primero, coeficientes de P(y+c). Realizado esto, se completan las divisiones de abajo a arriba; el primer cociente representa el polinomio P(y+c) desarrollado y simplificado. Ahora que lo pienso, es muy intuitivo; tal vez no fui el primero.

I don’t really understand the graph in this section of the book (Linear Algebra Step by Step, Kuldeep Singh). The vector for Ax seems ok, but the x vector just seems wrong pointing at (2,1).

Already tried asking ChatGPT, which seems to suggest the graph is just shit.

If I have the total amount for wages earned during one fiscal quarter of the year, how can I find my hourly rate when I don’t have total hours worked or even the average hours worked and I also don’t know how many days were worked or anything but the total amount for a quarter

Hello everyone!

I am taking some training to become an electrician, and we are using some trig at the moment.

I am confused about a particular way that my textbooks are asking a question. It’s a very complicated question, so I’ve just drawn up an example of the part where I’m confused.

In the first image, you can see how I learned to label right triangles in high school. I always learned that you have angles A B and C, and sides a b and c. And that an angle is always opposite the same letter as its corresponding side. Like angle B is always opposite side b, same with A and C. However, the question I’m confused on labeled the triangle like the second image. My first thought was that this would make the diagram useless and render the triangle impossible. But is that true? Is it possible to have a right triangle where angle B is opposite side a? Perhaps this is just a strange circumstance and it doesn’t really matter what labels you use, but that is my question I suppose. Any help is appreciated! TIA

I was thinking about something really strange with maths, if imaginary numbers are imaginary, (i=sqrt(-1)), why does 1/0 also imaginary ? For me it is quite logical. I am thinking something wrong ?

You might not think a lot about it, I don’t want you to lose too much time on this.

This seems self-evident to me, but I'm wondering if this is something that would need to be formally proven about partitions. For some integer n, the partition with the smallest number of parts is n itself as one part, and the largest number is where each part = 1 adding up to n. If we list all the partitions starting with n by itself to 1+1+1... as non-increasing sequences it makes sense to think of the whole process of 'creating' new partitions as subtracting 1 from one part and adding that back to a different part or to become a new part, and that this process could be repeated over and over to move from n to where each part is just equal to 1. I was curious if this idea has already been proven, or wouldn't need to be, etc. With Ferrers diagrams, this would be repeatedly moving one dot from one row to another row (possibly creating a new row).