It all started with arguments about 0.(9)=1 on multiple random posts,

I assume everyone knows it already but other than the usual 3*1/3 and algebraic way, I also always backed it up with:

1. The sum: lim_{n->∞} of Σ (x=1 to n) of 9/10^(x), or 0.9+0.09+0.009+... converges/equal to 1

2. 0.(9) can be rewritten as lim_{n->∞} 1-0.1^(n)=1 with the ε-N proof of it as an overkill back up with N=log(ε^(-1))

3. ∄x ∈ ℚ that satisfies 0.(9)<x<1 ⟹ there's no difference in between 0.(9) and 1.

However, I always find a pattern from the people who disagree with those proofs in particular, their arguments are

1. Limits cannot be reached since it means approach to something therefore they're not equal, you can get infinitely close but never get there.

2. In hyperreals they're not equal.

3. It's an infinite approximation therefore not equal.

Now even though I consider myself to be familiar with/like the subject mathematics I have a bit of doubt in both of our claims, considering I'm still a high schooler and my school curriculum only covers up until basic limits, derivatives and integrals with not summation or Taylor series in the Calculus category, and I only recently self learned some Calculus courses.

But here are some of my counter arguments against them.

For the first one, limits work differently as it finds the value by looking at the behavior of the function or "neighborhood?", limits don't care what happens on that exact point based on limit definitions.

2nd, in hyperreals they're still equal by definition, it's from a simple google search I'mma be for real with you so I feel uncertain.

3rd, Infinite is not an approximation, it is equal, 0.(9) already includes infinite amount of 9's so it makes sense to use infinity and have the exact value.

Now my question is, are my counterpoints correct? In general I want to clear some minor parts of confusion about limits in order to get a better understanding of them so I would like to hear your thoughts/insights

The components could be complex numbers (or other real number extensions), functions, matrices, other vectors, or something else totally.

Say you are walking down the street next to a production lot when a back door opens

“Psst, over here” says a person dressed like a production assistant, “wanna be on TV and maybe win some money?”

You follow the PA inside, they inform you that there is a live game show being recorded right now, but the previous contestant was faced with a choice so unintuitive that they walked off the set. You have been chosen to replace them. The PA tells you simply that you have to pick a correct door to win a cash prize, because only one door has it.

You are shuffled on stage and behind a booth where the host introduces you and sets up your big choice, “so friend, which door will it be?”

You look at your options. There are three doors. Two are closed, but one is already open, containing a goat inside its room. At your contestant booth, there seems to be a dial set to select one of the three doors, but it’s unclear if the door it is set to was deliberately chosen, because there is no clear default/neutral setting.

You have no more info about the situation, other than what you see, and knowing that only one door contains the money. What are the odds of choosing the right door at this point for you?

My diffeq midterm has a problem where a tank leaks at a rate proportional to √h, but then gives V′(t) = −kpV(t). when I asked about this, my professor clarified that p > 0 is a constant of proportionality. Since the tank’s base area is 4, V = 4h, so I would expect V′(t) = −C√(V(t)/4). Can these descriptions be consistent with both k and p constant, or is there likely a typo?

Here’s the original question: “3. The rate at which water leaks from a small hole at the bottom of a tank is proportional to the square root of the height of the water surface above the bottom of the tank. Consider a tank with the shape of a rectangular prism with a square base of a side of length 2 feet and a height 10 feet. Suppose the tank is currently half full, and is leaking at a rate of 2 cubic feet per hour.
i) Let V(t) be the volume of water in the tank t hours from now. Explain why the leakage rate can be written as the differential equation V'(t)=-kpV(t), for some positive constant k.”

A line is many points next to each other, a plane is just many lines stacked next to each other, a space is just many planes stacked next to each other. But how on Earth do planes, which have 0 thickness, produce thickness when stacked on top of each other? No matter how many of them you stack, the resulting thing is still completely flat. 0 times infinity is still 0, right? If anyone can explain this, I would appreciate it. Plus, if the flair is wrong, let me know which is the right one and I'll change it.

P.S. As I am writing this, Google tells me zero times infinity is indeterminate, but it also tells me zero plus zero plus zero forever equals zero. I do not understand this either.

So here's the problem:

A Society needs to plan its production of P on a cycle of 4 MONTHS. The fixed cost of the initiation in the atelier of production of P is estimated at 300 MU. The direct cost depends on the available manpower. It's about 150 MU the unit in Normal Hour and 200 MU the unit in overtime.

The previsional demand of the end of the month of the product P, as well as the max capacity of production in Normal Hours and in Overtime are given in the following table:

We dispose of a limited capacity of storage of 3 units and the cost of storage is 15 MU per unit (stored per month). Stock shortage is prohibited, and the stock in the beginning and in the end of the cycle is null.

1.Give, for each month, the expression of cost of production C pk (x); k = 1,2,3 and 4.

1. Formulate the problem of planning of the production of P, in a way of a dynamic programming problem.

2. Determinate an optimal plan of production and of storing for the months 1 to 4.

3. Calculate the gain in applying the optimal solution compared to the solution which consists in producing for each month just to satisfy the demands

4. I've already highlighted in bold each terms and numbers I judged important. But it's in the model I should use where I'm blocked. So far, all the exercices we had in school were about solving optimal outcomes through linear programming, so this one comes out as new for me.

for month 1, I remarked that the number of stock should be 0 since we produced nothing yet. And that in Normal Hour, we could possibly produce the 4 P demanded for the month. So no need for overtime (?).

However C p1(x) should be in fonction of x, right?

Can someone guide me a bit just in the first step please?

Thanks beforehand.

The second image displays the commenter I am referring to, and OP's reply, where they tried to explain what they meant. All usernames are redacted to protect the users' privacy.

Am I missing something obvious/well-known? What is all this "universe" business about? What is "referencing" in this context? What do they mean when they say that 0.999... and its digits "grow limitlessly?"

I want to build a statistical algorithm that can be used to recognize basic patterns. For example, if I had a list of elements [triangle, square, circle, triangle, square], the algorithm could piece together that the next element would be circle. Of course, the sequences would be longer and more complex than that, but that's the basic premise.

I feel like there's probably some very simple answer to this, but currently I can't think of anything. I tried looking up some "pattern recognition algorithms" online, and some articles recommended autoregressive modelling, but that seems too complicated given how simple my idea is (or at least how simple I think it is). That's why decided to come ask here.

Is there some well-known algorithm for situations like these, and what is it?

I am unsure if this is possible to answer but something tells me it is.

If I want to buy a house, and there’s a field behind the house. I know that I plan on living here for 20 years. The owner of the field has announced that when he dies, the property will be sold and developed. The age of the owner is unknown, but it is known he’s at least 18 years old.

I want the rear property to stay being a field to maintain home value. What is the probability that the farm owner dies within that 20 year period?

I was doing advanced functions homework and I found a pattern in logarithms that I don't see mentioned in any rules, and that didn't seem obvious to me at first:

Or if rearranged one way using the power rule:

Which I thought was a neat trick! My class materials don't mention this. I've been using khan academy and random youtube videos as well for learning, and I don't see this trick/pattern/whatever come up anywhere.
I think it makes sense to me why this works, after thinking about it for a long time but, my questions are basically:

• Why isn't this one of the "log laws" (is it not true, not useful, or is it obvious?), and
• Can anyone point me to any material covering this? When I try to google it, all that comes up are explanations for the change of base rule, I can't find anything about this specifically.

Also if this is not the best place to ask this question, could you point me somewhere more appropriate. Thank you!

Definitions of Oval I've heard include:

-Any convex smooth shape

-Any closed smooth shape with reflectional symmetry

-A set of points with equal total distance to a number of Foci

-A continuous set of points where every point has exactly one tangent line & no line passes through more than 2 points

-A smooth curve that connects the corners of a convex Kite

Are these definitions somehow equivalent? I There's at least two groups that are different because a track shape would be definitely be included by definition 1, but definitely excluded by definition 4, but are any of them the same? & which one's correct?

Are there any numbers with exponents that the left and right side of the equal sign look the same?

Ultimately it would be in the following form

X^y = Xy

An example even though this is an incorrect statement

2^5 = 25

I don’t even know how to approach finding the answer to this random thought I had. If someone can find one, it’s just for the sake of neat math facts than any actual need of information. THANKS!

EDIT: I meant this in the form of the first value following the next like x then y. To the people in the comments who are smarter than me and realized what I was trying to say, THANK YOU!!!

I’m an above-average student in my geometry class. I asked my teacher to give me a problem that would challenge me, and this is the one he gave me. He said he would give me one hint each day and reveal the answer at the end of the fifth day.

Here are the hints I’ve received so far:

There is a solution involving drawing a hexagon or a pentagon around the figure. For at least one of these two polygons, one of its sides will not be a side of the original triangle, but rather a line segment derived from the triangle, such as a median or an angle bisector.

Today was the third day, and I told him I was giving up. However, he said he would still show me the solution in two days. If he reveals the answer, I won’t count that as having solved the problem myself, but if I can find a different solution from the one he shows me, I’d like to present that alternative solution as well.

And sorry for clickbaitish title i couldn't think anything else.

So our professor gave us these unclear questions that they take from their PowerPoints and change the wording. I believe our teachers answer key is wrong and they personally did the work wrong because they didn't realize the wording they changed on this. Can you please double check my answers, students got the exact same answers as me. If anything the teachers wording is unfair.

I do see where they got their answers but it’s not what the question is asking. (Flair?)

My aunt was trying to help her daughter with her homework but couldn’t come up with an answer for the fourth one. They just used 7 beads and ended up getting it wrong. Does this question even have a fourth answer considering there’s only 3 odd numbers between 0 and 6.

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Could arithmetic operations themselves have noninteger ranks, such as 1.5 between addition and multiplication or 2.5 between multiplication and exponentiation, while preserving the recursive structure of hyperoperations? If so, what would these fractional operations actually represent beyond simple interpolation? Given that ordinary integration corresponds to continuous summation and product integrals provide a multiplicative counterpart, could similar forms of differentiation and integration be constructed for higher or fractional hyperoperations? More generally, could this lead to an entire hierarchy of calculi in which the rank of the underlying arithmetic operation is itself a continuous variable?

This was one of my practice problems. Everything else had clear answers, but this I can't solve even with ai help. Maybe I am missing something? At least it gives us this cool fraction ladder that looks similar to chain fractions.

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I was thinking about this today, and i even managed to back this theory up: the lower the decimal you divide by, the higher the number.

(I'm not a mathematician in any way and I'm still in school. was just curious)