r/askmath

Consider the set R of all rationals with odd denominators in their lowest form, all elements r of the set R is such that it can be represented as r = p + q where p and q are non zero elements of a set S which itself is a subset of R.

Thus the set S contains all values of p and q which are required to satisfy r=p+q.

Now since S is defined on the basis of a criteria of inclusion, there could be more than one set which satisfy the criteria for S.

Let the set of all elements in R which have an odd numerator be RO and the set of all elements in R which have an even numerator be RE.

Let SO = S intersection RO and SE = S intersection RE. If any element is in S then double that must also be in S. And If an element is in SE then half that must also be in S.

What is the smallest such set S, both in terms of inclusion and cardinality?

I suddenly realized that I don't understand functions in spherical coordinates, nor, evidently, how circles work. What I want to do is tilt a circle (centered on the origin, in the xy-plane) in 3D space. In Cartesian coordinates it kinda sucks, but is doable, you rotate the plane intersecting the sphere that defines the center and radius by the angle you want. Taking that resulting equation into spherical coordinates gives me either 1 (if I include a z-term), or r = sqrt(cos(θ)^2 + sin(θ)^2 * cos(ɑ)^2) which gives me a volume bc of course it does, but I notice that this volume kinda looks like if I take the right slice of it, I'd get the circle I want.

I've tried approaching this from the other direction, letting θ vary freely, with r=R, and having φ=pi/2 + Xcos(θ)---but I can't quite figure out what the X should be.

Idk, I feel stupid, bc it can't be this hard to define a function that traces a circle of arbitrary orientation in spherical coordinates---not even an arbitrary circle, just getting a vector to trace a great circle of arbitrary polar angle!
But I can't get it to work and it's driving me mad. What is it that I'm failing to understand?

Here is the puzzle:

Ten prisoners are brought into a room and each is given a random number between 1 and 10. The same number may be given to more than one person. They can see the numbers of the others, but not their own. They are not allowed to communicate. After a certain amount of time, they must all say a number between 1 and 10 at the same time. They will survive if at least one of them says the number they were given. Before entering the room, they have a short time to think of a strategy. What is this strategy?

Solution:

>!Each person adds up all the numbers they can see that the others have. Let’s say the numbers are 2, 2, 3, 5, 5, 6, 7, 8, 8, 10. The first person has the number 2. The sum of the other numbers is 54. The first person now adds a number so that the last digit is 1. So 54 + 7 = 61. This means the number the first person says is 7. The second person does the same, except the last digit must be a 2. And so it goes on. Eventually, the correct number will be named. !< (Contains spoilers if you want to solve it yourself)

The question now is: Why does this work?

I was playing modded Minecraft and started wondering what the optimal dimensions for a multi block were; the multi block is basically a cube without corners (I'll attach a screenshot) and without the top surface with an empty interior, kinda like a bowl, and the value to optimize for Is the internal volume; what I was thinking of doing was get a constraint for the cost in blocks and use that to get the height in function of width and length, then derive by one of these 2 the volume (the number of blocks is a constant) and get the optimized length (or width) at a given width (or length), then substitute that so you get the volume in function of one variable and the constant max blocks and derive again to get the optimized value, but I feel like this isn't correct cause you are considering a variable like it is a constant and I don't think that would work, not sure though, what is the correct way to approach this and where is my approach wrong?

Is there a name for this pattern of primes, where a set of 4 numbers in a row can be ordered like this to get the 4th digit?

4 primes in a row: {2, 3, 5, 7}

Number 1 * (number 2.number3) = number 4

Notice number 3 becomes decimal of number 2 in the set

Does this happen elsewhere in the primes? what is it called?

x(9x-2a)+b(9x-2a)-(x+b)=0. In the given equation, a and b are positive constants. A solution to the equation is x=42. What is the value of a?

So I factored by grouping twice: (x+b)(9x-2a)-(x+b)=0 and then once again to get (x+b)(9x-2a-1)=0. Then I substituted 42 for x and got a to be 188.5 which I believe is the correct answer. However the question specified that both a and b are positive constants and if you were to solve for b you would get b=-42, right? This contradicts the question stem and I have never seen College Board make such a glaring mistake like this. Am I doing something wrong? I know the question is asking to solve for a but I was curious about the value of b. Thoughts?

There are ten numbers on a telephone keypad, arranged as a 3x3 grid with an additional number under the bottom centre.

To count the number of distinct triangles, I think I can follow the logic explained here to get 109. 10C3 = 120, minus 3 horizontal lines, 2 diagonal lines, and 6 vertical lines (2 on the sides and 4C3 in the centre).

But I'm more curious about how many types of triangles there are. Right now I'm just counting by hand and I see at least 13.

Four right triangles, with side lengths (i.e. how many buttons are covered by the non-hypotenuse sides) 2x2, 2x3, 3x3, and 4x2,

Three non-right isosceles triangles, all with a base length of three and with heights 2, 3, and 4.

Two non-right isosoles triangles that are tilted diagonally, e.g. [067] and [037].

One scalene triangle with a base of 4 and a height of 2.

The others are sort of skewed and not simple to describe but e.g. [068], [035], and [038].

Is this all of them? And more interestingly, is there a systematic way to count them?

https://preview.redd.it/10somo6clfbh1.png?width=1253&format=png&auto=webp&s=8e9047fefeeb3578e315285aa83ba4ac4f69417c

https://preview.redd.it/lbwrsj9hlfbh1.png?width=936&format=png&auto=webp&s=ca91e449659f40f873a3cff8f305e3107abdb6ef

https://preview.redd.it/6xi7w1dslfbh1.png?width=937&format=png&auto=webp&s=47634dc8a54ee647745b982564ca73b515304dda

https://preview.redd.it/aq5b3kbcmfbh1.png?width=935&format=png&auto=webp&s=8891d1b936cc773399b9317057d54b0a7fcaebe8

I'm an astrophysicist turned data scientist so bear with me as I butcher terms or make some assumptions while asking this question. I learned math for physics, so my courses were light on proofs. However I do enjoy reading about maths, solving the occasional problem for fun, or when I have time more rigorously revisiting the maths from gradschool. I chose the analysis flair because I had to choose a flair.

From my understanding while lurking on this sub, of how maths at your level works, for a certain field, it's basically based on certain axioms and all the theorems that derive from them, or from possibile conjectures. Then a body of knowledge of useful statements is built and so on.

From my side, I see a lot of mathematics that was discovered or invented (whichever term you prefer haha) to solve certain problems. Sometimes existing maths was sometimes coopted, for example imaginary numbers were discovered way before they were used in physics.

Now to get to the meat of my question. Let's take for example the equation of motion, the laws of gravity, and Newton's work. From what I've read so far, at that point in history, calculus was being developed to solve physical problems. Newton's Principia used this rather geometric approach to prove and develop his theory. Later on, however, the methods used, for example, to calculate motions were refined. We got the Lagrangian and the Hamiltonian. Then in GR or QM other powerful tools are used, vectors, matrices and tensors etc.

While looking at how certain problems are dealt with mathematically in physics I'm wondering, how many of the methods used exist the way they are because that is how they were first formulated, caught on, and remained unchanged, maybe because further developments were based on them, or because they just worked? And how often did a better more efficient way, maybe simpler and more elegant method, replaced it?

I guess my question can also be framed as, can the development and application of certain mathematical methods and tools be viewed as traveling along a path, to get to a set of working, useful solutions or methods? And how often were shortcuts or more efficient paths were discovered? How often does this happen nowadays?

This question I could also be valid for mathematics in general as well, I guess, as geometry was done in a certain way in ancient times, with a straight edge and compass, but later on algebra was introduced into geometry. Sorry for possible mistakes or wrong assumptions, but this is the clearest way I could formulate my questions so far.

I just wanted to know how did you start studying category theory and from what background did you come to study it and why was it necessary?

I am from CS and learning it for my thesis as it connects a lots of dots in categorical QM and type theory

Many moons ago, when I was in elementary school; I would get bored and add the numbers on my calculator.

One day I wanted to find the sum of all the numbers from through 100, and so I added them one by one. But then I thought "there's gotta be an easier way to do this" and stumbled upon a formula.

If you multiply the final number in the sequence by it's half and .5, you get the sum.So for example 1 through 10 would be 10x5.5, resulting in 55.

My question is, is this a known (albeit somewhat useless) formula in the world of math, and if so what's its name?

See above, late night stoned BC musings

Sorry for bad wording, am just an enthusiast soon to enter college.

Was reading the different proofs of the fact that there are infinitely many primes and also reading about how it is not known if there is also infinitely many Marsenne primes (and therefore infinitely many perfect numbers).

As rare as they are, Marsenne primes make my rookie mind think well why can’t we just assume there too also are infinitely many of them? I know, I know, one can’t presume without proof.

Which made me wonder about a more general thing which: what are some examples of these problems, mostly in number theory but all areas are welcome, where something was proven otherwise despite a general assumption that it was not the case? Most specific, in these examples that deal with number patterns and infinity, is there any interesting proof where a mathematician arrived at a conclusion that nope, this doesn’t go up to infinity, the Whatever Type Numbers end in number so and so.

Hope I made myself clear.

My question resides in trying to figure out whether or not I'm understanding this right. I've always been bad at math except the extended basics (passing trig, geo, calc, arithmetic. As well as probability/statistics)

I cannot do any of this on the fly in my head unless it's adding even numbers to other even numbers. I can do it if I sit down and do the math and carry the one and such.

I've looked at the wiki and read the entire thing about "coprimes" but it still really doesn't make sense to me unless it really is as simple as two values that are intrinsically linked together and changing one value from 10 to 5 or even 3 (I know) also forces the other value to reduce by the same ratio. Am I missing something or is that all there is?

I ask because I need an answer, but also because "prime numbers" scare the hell out of me and I hope this isn't a case where I need to delve into prime numbers.

Edit to add: I don't know what a prime is other than it is divisible by either 1 or itself. Not even sure if that's right since it's been many years since I last looked it up due to "Cube"

ETA: Might be the wrong place for this question. I'm getting a lot of algebra and I can't parse it. I was never great at any of this stuff, so I guess I'm ostensibly waiting for somebody who can put it in terms that doesn't require letters or extremes to fathom, to come along.

ETA (Edit To Add)

I am reading entirely.

For example the fact that 51 is divisible by 17 or that 199999 is a prime number. Do you guys have any other examples of this? I dont know if i explained it good enough or if this even is the right subreddit for this but i would love to hear your answers.

The general form of these series is that each term is a power of 10 (since moving the decimal takes no effort) multiplied or divided by a single-digit number (since single-digit multiplication/division takes far less effort than multi-digit multiplication/division).

It follows the basic principle of simple repeated fractions, but with only adding and subtracting error terms (no taking reciprocals and then cross-multiplying) and with having multiple options at each step from which to choose the best (rather than being given one automatically).

Taking π = 3.14159265, for example, we would start with either 3 (underestimating with 10^n times x), 4 (overestimating with 10^n times x), 10/4 (underestimating with 10^n divided by x), or 10/3 (overestimating with 10^n divided by x).

3 is the closest starting point (error of 0.14159265) and 3.33333333 is almost as close (error of 0.19170408), so we would throw 4 and 2.5 away and test 3 first, then 3.33333333.

The error “π – 3 = 0.14159265…” can be estimated as 1/10, as 2/10 (which could be re-written as 1/5, but that doesn’t matter here because we’re about to ignore it anyway), as 1/8, or as 1/7.

1/7 and 1/8 are the closest second-steps for π ≈ 3, so we throw away the 1/10 and the 2/10, and now we see what the closest second-steps would be for π ≈ 10/3.

The error “π – 3.33333333 = -0.1917408” would best be approximated as -0.2 (which could be written as -1/5 or -2/10 depending on the reader’s personal preference), but the two-step calculations 10/3 – 1/5 = 3.1333333333 and 3 + 1/8 = 3.125 are both less accurate than the two-step calculation 3 + 1/7 = 3.14285714, so we can commit to 3 + 1/7 now.

Calculating the new error “π – (3 + 1/7) = -0.00126449” creates a best new error term of -1/800, and so our new value is 3 + 1/7 – 1/800 = 3.14160714.

This approximation “πx ≈ 3x + x/7 – x/800” is accurate to within 1 part in 220,000, but it only takes about as much time and effort as “πx ≈ 3x + x/10 + 4x/100” (which is only accurate to within 1 part in 2,000).

Using continued fractions would take very little time to calculate “π ≈ 355/113” ahead of time (which is accurate to within 1 part in 12,000,000), but this takes more time and effort to use in the moment. If someone was multiplying πx ≈ 3x + x/7 – x/800 and if someone else was multiplying πx ≈ (300x + 50x + 5x)/113, then in the time it took the first person to get a final answer, the second person would only have finished calculating the numerator, and they would still need time to calculate the denominator.

During which time, the first person could be adding more error terms: 3 + 1/7 – 1/800 – 1/70,000 is accurate to within 1 part in 15,000,000 (already more accurate than 355/113 for less time and effort), and 3 + 1/7 – 1/800 – 1/70,000 – 1/5,000,000 is accurate to within 1 part in 880,000,000.

Is there a formula to know when the number x mark of inches and the number y of centimeters would correspond to each other on a definitely long, and I'd imagine just hypothetical, tape measure? I tried counting on my own, even longer than I'd confess but then got bored and realized I could've just checked a long tape. Internet tells me that they would supposedly never "touch" but doesn't explain me why, is that correct? If so how or why? I'm always fascinated by numbers but I've a hard time understanding them. Thank you all in advance. Edit: thank you all for your answers, just reading them seems simpler that what was in my mind. Thank you all once again.