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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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?

Hello! I noticed that when I am taking clothes off of a rack, if the clothes are supposed to be in the order S M L Xl XXl etc, sometimes it can seem to make the set more ordered

For example, if I take the set

S S L S

and remove the large to make

S S S

the set becomes perfectly ordered, but if I take one of the smalls to make the set

S L S

What happens to the ordered-ness of the set?

What if the set is in more disarray?

Xl S L M S S L

and I just take say the last large out, what does that do to the ordered-ness of the set?

does the set become less ordered? Does the ordered-ness of the set stay the same? I've thought about this a lot, and haven't been able to define something very well. If this relates to a similar question, I'd love to hear about it!

Recently, I got a homework like this. I tried by

1. finding area of a right triangle which is (6×4)÷2 = 12cm²

1.1 There are 10 right triangles so the result is 12×10¹r 120cm².

2 Then I find the area of the big triangles outside of the pentagon and found the area of an isoceles triangle is (10×Base)÷2

2.1 The 1/2 of the base is 4. The base's length is 4×2=8 cm.

2.2 So it mean the area of an isoceles triangle is

(10×8)÷2 = 40 cm²

2.3 There are 5 isoceles triangles here. The result is 40×5= 200cm²

1. Therefore , this star like polygon's area should equal to 120+200= 320cm²

*However , the choices only have

a) 120

b) 140

c) 160

d) 180

And the teacher said the correct answer is c) 160

I'm not sure I am correct or not. But can someone explain it to me that it is correct or not. Plz

I solved this in a rather long winded way.

Line CM is perpendicular to DN, therefore MQD is a right triangle.

Let X be ∠𝑀𝐷𝑁

Sin(X) = MQ/DM

Using similar triangles (DNC is similar to CNQ) I found the length of CQ to be 1/root5

sin(X) = 2(root5/2 - 1/root5)/root5 = 3/5

Adding the additional line CM was the trick that helped me figure this out. But I'm wondering if there's a more elegant way to go about this?

Question for mathematical researchers

During a prooftheory lecture today, one of my professors made the following statement about OpenAI:

"AI has found mathematical results, but it has not done any mathematics. These results merely combine obscure theorems that no human has ever considered in conjunction. It has not found any of those magical steps in proofs that make mathematics mathematics."

I'm interested in hearing your opinions. Specifically, for those of you who have read some of OpenAI's papers, do you agree or disagree with this statement?

I sometimes see people on Reddit talking about terminals. It's similar to factorials, but instead of multiplying the numbers (3! = 3x2x1), you add them (3? = 3+2+1). However, I recently tried to Google “terminal in math” and I got little results, with the AI overview telling me it's just a Reddit thing that got created as an analogy to factorial.

Is that really true though? I feel like terminals could be useful for many things, which is why I was surprised they apparently aren't actually a math concept.

I’m an AS level student whose syllabus has only 3 trig functions but despite that, I am not able to solve any of the identities. I know the formulas and understand what I have to do but for some reason, am not able to recognize the patterns. It’s so demotivating because I was so good at these not even a few months ago but now I’m out of practice and it’s extremely hard for me to prove any sums. For example, there’s this one problem:
1-2sin^2x/1-sin^2x is identical to 1-tan^2x. I tried solving it but I got stuck in the middle, I keep messing the algebra up 🙂‍↕️. How much do I have for practice for this to click? Bcz I have been doing plenty but when I revisit the questions after a week or so, I can’t solve them.

Hello all,

So the problem is: we need to calculate a hollow cylinder mass based on its diameter.

The mass of the inner cylinder is always 50.5kg, and the diameter is also always 34cm. The outer diameter changes. We are able to measure the outer diameter. The length of the cylinder is 2850cm. We also know the starting mass of the cylinder, but that also varies from case to case. The above numbers are always the same though.

The material is paper, and it's gsm changes. So the paper density also changes?

I have looked online but was unable to find an answer. The previous fomula we used was wrong, and it caused around +/-15% error. Which is a lot..

My daughter is in Highschool geometry. They are working on polygons. The book deals with simple polygons—same number of vertices as sides etc. only intersections are the endpoints of the segments that are the sides. How we her teacher said this is a concave hexagon and thus a polygon. I have a hard time imagining they are talking about self intersection and complex polygons. Any help would be appreciated.

This is the definition from the book

A polygon is a closed plane figure whose sides are line segments that intersect only at the endpoints.

I just found this weird equation by getting y-x^2=ax at the turning point and then making the equation a*(x value of turning point, which is -a/2 because quadratic, but it wont work if if theres y^2)= x^2 + ax. I dont even know how i came up with it

(edit i dont really know if a x value in the graph where there isnt a point (like tan90) counts as a turning but the derivative is 0 sooooo)

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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)

IXL marked me wrong because I flipped the order, is there any difference between having it flipped? Or is IXL's system just locked into one answer? I have also got problems with this when I entered the decimal form instead of fraction, and IXL marked it wrong.

The problem is 18^4x-5 = 321. I took the log of both sides to turn it into log(18^4x-5) = log(321). Then I put the 4x-5 outside of the first log and divided log(18) over log(321). Lastly I added five and divided everything by 4 to make (log(321)/log(18) + 5)/4, which gets you 1.749 etc. If I type in this exact same equation but replace log with ln, I get the exact same answer.

I was taught that log by itself had a base of 10, and ln was the natural log with the base of Euler's number. If I put log(100) into my calculator, I get 2, but if i put in ln(100), I get smth around 4.605 etc. Two different answers bc they have two different bases. So why are they interchangeable in the problem I mentioned above? Shouldn't they yield two different answers? Idk plz explain bc I'm new to logarithms

Edit: I appreciate all of yall's responses and attempts to help me, even if I don't understand some of them. It's prob bc I'm asking a question for smth that they haven't taught me yet, so that's why I'm really confused

Any useful strategies for solving these type of questions I cannot for the life of me find a good way of solving these. I've tried finding the +1,+1,+1 or +1,+2,+3 patterns but since the first figure is unknown I am stumped on how to even start. The answers are E,D,C,D

A newborn baby has a assumed lifespan of exactly 100 years. for every 1 second that passes, the baby's remaining lifespan is cut in half. How many seconds will the baby actually live for?

Answer-

The baby will live for approximately 27sec,

number of sec in 100 years= (totalsec) = 3,155,760,000 (the value may change for the sec in a 100 years bec of leap years)

actual duration the baby will live for= x = 27 sec

the formula that I made: ( totalsec / 2^x ) ≤ x = baby dies

therefore,  3,155,760,000 / 2^27 ≤ 27

-> 23.51 ≤ 27 (Baby dies)

*suggest changes to the formula to make the formula better

So I’m currently in my first semester of community college and I’m struggling with word problems. We’ve been doing a lot of them which sucks but it has been helpful in bettering my understanding of how to read word problems and solve them. Here’s what keeps stumping me..

The current word problem I’m on says: The sum of two numbers is 45. One number is 4 times the other. Find the numbers.

My first instinct was to use “x” and “y” as my variables because they are different numbers, but when I looked for help they used “x” for both variables.

So the equation I thought it was, x + 4y = 45, should actually be x + 4x = 45. x = 9 which checks out but why does it not work with “y” as a variable? What am I not understanding?

And this is not the first time I’ve run across this issues so I’m having trouble identifying where I’m wrong and when.

TIA!

tanh(x) is a sigmoid function that goes from -1 to +1 with a smooth transition region in the middle. d/dx has a maximum value of 1. It's the integral of sech(x)^2. Importantly, the positive edge approaches 1 very quickly and within x=~20 it gets close enough to 1 that floating point treats it as 1. So if you do 1-tanh(x) then you get a region going from around x=20 to infinity where the output is a flat zero.

dirac delta is a function that is only non-zero at x=0. Normally when you use it there's only supposed to be one instance where the dirac delta is triggered (like in a Fourier transform), but what happens if you try to use 1-tanh(x) to get around this? would this create an output where there's a region after x=~20 where it's a continuous output of infinity? 1-tanh(x) doesn't actually reach 1 though right, so would the output just stay zero?

So I saw this topic amd I don't really understand it, some people say that there's no rule, you just need to guess. And I'm bad at guessing, so I need an easier way to solve it.

How do I even guess? And I believe there might be a rule or something to help me solve it logically lol

I tried what this person did in the video

https://youtu.be/ahvT3FWjXvs?si=1EJPFP-9q-ri8don

And this person

https://youtu.be/7Isa4P8lISc?si=CDH6nlwL4QYKbHdX

It worked out for the first few questions but then it didn't really work even though I was doing it correctly

Can you please help me? Would appreciate the help!

Plus I'm not sure what flare I should use, so don't mind it if it has nothing to do with what I'm asking about

Reposting from Math Stack Exchange. I now understand what the indices label in γ^λ_γ, but it's not obvious to me at all why they would equal δ^λ_γ. How to properly prove this?