surreal meadow Jul 9, 2023, 3:10 AM

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eisenstein works

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and im sure you can device an algo out of it

median oar Jul 9, 2023, 3:11 AM

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That’ll just let you make polynomials that are irreducible

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But that’s only for Z

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And ℚ as well I think

surreal meadow Jul 9, 2023, 3:11 AM

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yeah by gauss's lemma

median oar Jul 9, 2023, 3:11 AM

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Honestly I can’t remember much from the class

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Ok are there any other algebraically closed fields other than ℂ and the field of all algebraic numbers

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It sounds like the field of all algebraic numbers is the smallest algebraically closed field to me

tardy stag Jul 9, 2023, 3:24 AM

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there are no algebraically closed fields with a lower cardinality than the field of rational algebraic numbers

lone heartBOT Jul 9, 2023, 3:31 AM

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@median oar Has your question been resolved?

median oar Jul 9, 2023, 3:31 AM

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Hmm

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And you can make it bigger without being C?

surreal meadow Jul 9, 2023, 3:33 AM

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well you can extend it finitely, but it loses algebraic closure

surreal meadow Jul 9, 2023, 3:37 AM

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tardy stag there are no algebraically closed fields with a lower cardinality than the field...

is this a countable field

tardy stag Jul 9, 2023, 3:43 AM

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surreal meadow is this a countable field

yes

surreal meadow Jul 9, 2023, 3:43 AM

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nice, that's kind of surprising

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so the closure is like much much much smaller than C

tardy stag Jul 9, 2023, 3:44 AM

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about as much smaller as you can get yes

tardy stag Jul 9, 2023, 3:45 AM

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surreal meadow nice, that's kind of surprising

at first? until you remember how polynomials work

surreal meadow Jul 9, 2023, 3:46 AM

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well it's like countably many finite extensions

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but it's still surprising given how many irrationals you can actually include

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(which is countably many lol)

tardy stag Jul 9, 2023, 3:48 AM

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the fact that the set of computable numbers is countable breaks me every so often

surreal meadow Jul 9, 2023, 3:48 AM

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what

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that's nuts

lone heartBOT Jul 9, 2023, 4:01 AM

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@median oar Has your question been resolved?

median oar Jul 9, 2023, 4:11 AM

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surreal meadow well you can extend it finitely, but it loses algebraic closure

Could you give it closure after extending it first?

surreal meadow Jul 9, 2023, 4:13 AM

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i take back what i said

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i don't think you can even extend it finitely

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since a finite extension is algebraic

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its an interesting question though, let me think about it for a minute

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im trying to think of what A(pi) looks like, where A is the algebraic numbers

tardy stag Jul 9, 2023, 4:18 AM

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we don't know if $e \in A[\pi]$ I think

ocean sealBOT Jul 9, 2023, 4:18 AM

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Hayley

median oar Jul 9, 2023, 4:18 AM

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Shouldn’t that be closed

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Since that’s what field extensions do

tardy stag Jul 9, 2023, 4:19 AM

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field extensions are closed under multiplication and addition

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but not algebraically

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(necessarily)

surreal meadow Jul 9, 2023, 4:20 AM

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now im confused myself a little

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A(pi) is not a thing right?

tardy stag Jul 9, 2023, 4:21 AM

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A being the algebraic closure of Q?

surreal meadow Jul 9, 2023, 4:21 AM

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yes

tardy stag Jul 9, 2023, 4:21 AM

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and A[pi] being a field extension of A?

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so you could do things like $x^2 - \pi x + 1 = 0$

ocean sealBOT Jul 9, 2023, 4:22 AM

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Hayley

surreal meadow Jul 9, 2023, 4:22 AM

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well im differentiating between A(pi) and A[pi]

tardy stag Jul 9, 2023, 4:22 AM

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oh wait what's A(pi)?

surreal meadow Jul 9, 2023, 4:23 AM

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oof i am less sure now that i read a more fleshed out explanation

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i know A(pi) should be the smallest field containing A and pi

tardy stag Jul 9, 2023, 4:24 AM

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yeah that's a field extension, generally written A[pi]

surreal meadow Jul 9, 2023, 4:25 AM

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well that's only necessarily true when pi is algebraic

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or the extending element is algebraic

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like A[pi] is the polynomials on pi with coefficients in A

tardy stag Jul 9, 2023, 4:27 AM

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so like... $3\pi^2 + \frac57\pi^7 \in A[\pi]$ you're saying?

ocean sealBOT Jul 9, 2023, 4:27 AM

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Hayley

surreal meadow Jul 9, 2023, 4:27 AM

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yes

tardy stag Jul 9, 2023, 4:28 AM

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yeah i mean that's the smallest field (not necessarily algebraically closed!) containing A and pi

surreal meadow Jul 9, 2023, 4:28 AM

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i don't think so

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at least given what i'm reading

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since pi is transcendental it seems that A(pi) would be rational functions of pi

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with coeffs in A

tardy stag Jul 9, 2023, 4:29 AM

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yeah it is, and your question is whether that new field is algebraically closed?

surreal meadow Jul 9, 2023, 4:29 AM

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well my concern is that this new extension should be algebraic since it is finite

tardy stag Jul 9, 2023, 4:30 AM

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A[pi] is very not finite...

surreal meadow Jul 9, 2023, 4:30 AM

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the extension is finite

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not the new field

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as in it has finite degree

tardy stag Jul 9, 2023, 4:30 AM

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what do you mean CH_Dead

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oh

surreal meadow Jul 9, 2023, 4:30 AM

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oh well i guess it's not finite

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since pi is transcendental lol

median oar Jul 9, 2023, 4:31 AM

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Well A isn’t finite either

surreal meadow Jul 9, 2023, 4:31 AM

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okok i retract my concern

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and go back to saying that either example of A[pi] or A(pi) should be, as you said, questionable examples of an extension that is not C

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i know you mentioned we don't know if e is in A[pi], but im not sure about other transcendetals

median oar Jul 9, 2023, 4:32 AM

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Well we don’t need e in A[pi] do we?

surreal meadow Jul 9, 2023, 4:32 AM

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perhaps we do

tardy stag Jul 9, 2023, 4:32 AM

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well i mean we'll never be able to hit every real number with any countable number of extenders

surreal meadow Jul 9, 2023, 4:33 AM

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ah fair enough yes

median oar Jul 9, 2023, 4:33 AM

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Well we didn’t need pi in A

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Why would we need e in A[pi]

surreal meadow Jul 9, 2023, 4:34 AM

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it's like asking why Q(sqrt2, sqrt3) should include sqrt6

tardy stag Jul 9, 2023, 4:34 AM

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$\sqrt\pi \not\in A[\pi]$ so... that won't work i think

ocean sealBOT Jul 9, 2023, 4:34 AM

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Hayley

median oar Jul 9, 2023, 4:34 AM

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That would mean we can make e from pi?

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Which…can we?

surreal meadow Jul 9, 2023, 4:34 AM

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median oar That would mean we can make e from pi?

from polynomials of pi

tardy stag Jul 9, 2023, 4:34 AM

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ocean seal **Hayley**

ie A[pi] is not alg closed

median oar Jul 9, 2023, 4:34 AM

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Yeah polynomials of pi

surreal meadow Jul 9, 2023, 4:34 AM

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ocean seal **Hayley**

hmm

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how come

surreal meadow Jul 9, 2023, 4:35 AM

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median oar Which…can we?

well my point was that we shouldn't just say "no"

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but we should question it a little

median oar Jul 9, 2023, 4:35 AM

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Isn’t x² - pi the polynomial with sqrt pi

surreal meadow Jul 9, 2023, 4:35 AM

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median oar Isn’t x² - pi the polynomial with sqrt pi

yes, but we need to know if sqrt pi is in A[pi]

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if it isn't then the field is not alg closed

tardy stag Jul 9, 2023, 4:36 AM

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yeah, and that's not in Q[pi] and I don't think it's in A[pi] but I haven't proven that...

median oar Jul 9, 2023, 4:36 AM

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Hmm

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If sqrt pi isn’t in A[pi] then you’re saying it’s not algebraically closed

tardy stag Jul 9, 2023, 4:36 AM

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correct

median oar Jul 9, 2023, 4:37 AM

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I feel like it should?

tardy stag Jul 9, 2023, 4:37 AM

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holothink

outer portal Jul 9, 2023, 4:38 AM

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Helloooo what’s going on here

surreal meadow Jul 9, 2023, 4:39 AM

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ocean seal **Hayley**

im inclined to say this is false

surreal meadow Jul 9, 2023, 4:39 AM

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outer portal Helloooo what’s going on here

we're playing around with fields

surreal meadow Jul 9, 2023, 4:39 AM

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surreal meadow im inclined to say this is false

since the extension of adjoining pi is not finite

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we can make arbitrarily large polynomials of pi

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though i can't seem to make the jump to a taylor series for sqrt(x)

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since we don't have any limits

tardy stag Jul 9, 2023, 4:40 AM

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taylor series won't help you because they have to be finite length

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we can get arbitrarily close to rootpi

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that's no problem

surreal meadow Jul 9, 2023, 4:40 AM

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the thing is

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this is an infinite dimensional vector space

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ok this is maybe a nonasnwer but there are field extensions of A larger than C if you just atke a set of variables that has cardinality larger than the continuum

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then you can just take the set of rational functions of those variables with coefficients in A

#

i am curious to see if we can do something with A[pi] or A(pi) though

#

actually, what im mostly wondering now is if A[pi] is a field at all

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does pi have an inverse in A[pi]?

median oar Jul 9, 2023, 4:53 AM

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It should for field extensions

surreal meadow Jul 9, 2023, 4:53 AM

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the thing is A[pi] just denotes the polynomials on pi with coefficients in A

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so it's a ring surely, but i don't know if it's a field extension

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A(pi) is for sure a field extension though

median oar Jul 9, 2023, 4:54 AM

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Oh right

surreal meadow Jul 9, 2023, 4:56 AM

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this has sparked my interest in algebra again, thank you

median oar Jul 9, 2023, 4:57 AM

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Hi how do I factorise 2x + 4

surreal meadow Jul 9, 2023, 4:57 AM

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i swore it off after my class on all this but these questions are pretty fun

median oar Jul 9, 2023, 4:58 AM

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Wikipedia is truly a black hole

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I was trying to read about diagonalisation of matrices and I ended up here

tardy stag Jul 9, 2023, 4:59 AM

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I always loved algebra but it's so hard

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oh thaaaat's why you were talking about finite extensions

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surreal meadow Jul 9, 2023, 5:04 AM

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oh ok

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if A[pi] was a field we would get that some polynomial of pi times pi is 1

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or that p(pi) * pi - 1 = 0

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for some polynomial p with coefficients in A, meaning pi is algebraic

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so we want to take a look at rational functions sadly

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also this holds for all fields adjoined with a transcendental element

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so it would seem F(x) = F[x] iff x is algebraic, and otherwise F[x] is not a field

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ooo so i may have a proof that sqrt(pi) is not in A(pi)

#

if sqrt(pi) was in A(pi), then it must mean we can write sqrt(pi) as a rational function of pi
then we have that we can express pi as sqrt(pi)^2 -> pi is expressible as the product of two rational functions on pi

tardy stag Jul 9, 2023, 5:17 AM

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ooooh

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and yesh the product of two rational functions is a rational function

#

but...

if sqrt(pi) was in A(pi), then it must mean we can write sqrt(pi) as a rational function of pi
is that true? A(pi) contains all elts that are roots of rational functions as well

#

eg root2

surreal meadow Jul 9, 2023, 5:17 AM

#

so 2 thing to note

surreal meadow Jul 9, 2023, 5:18 AM

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tardy stag and yesh the product of two rational functions is a rational function

we are thinking of rational functions with coefficients in A, but always evaluated at pi

#

so pi itself is a "rational function" here

#

and so is, say, 1/pi

surreal meadow Jul 9, 2023, 5:18 AM

#

tardy stag but... > if sqrt(pi) was in A(pi), then it must mean we can write sqrt(pi) as a...

A(pi), since pi is transcendental, turns into the rational functions with coefficients in A evaluated at pi

#

or is isomoprhic to it

surreal meadow Jul 9, 2023, 5:19 AM

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surreal meadow if sqrt(pi) was in A(pi), then it must mean we can write sqrt(pi) as a rational ...

i was going somewhere with this but i fear it's wrong now

median oar Jul 9, 2023, 5:19 AM

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I’m confused now

surreal meadow Jul 9, 2023, 5:19 AM

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i was going to say we can think of this product of rational functions, say
p(pi) * q(pi)
as an element of R

median oar Jul 9, 2023, 5:19 AM

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I thought A(pi) was the field extension of A with pi

surreal meadow Jul 9, 2023, 5:19 AM

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it's the smallest field containing every element of A and pi

median oar Jul 9, 2023, 5:19 AM

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And A[pi] are polynomials with coefficients from A of pi

surreal meadow Jul 9, 2023, 5:19 AM

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yes

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in general

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if x is algebraic over F

#

F(x) is isomoprhic to F[x]

median oar Jul 9, 2023, 5:20 AM

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surreal meadow A(pi), since pi is transcendental, turns into the rational functions with coeffi...

Should this be for A[pi]

surreal meadow Jul 9, 2023, 5:20 AM

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no

hoary girder Jul 9, 2023, 5:20 AM

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Hello!

median oar Jul 9, 2023, 5:20 AM

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Ok then I don’t understand what is happening

surreal meadow Jul 9, 2023, 5:20 AM

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when x is not algebraic over F, F(x) is different from F[x]

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and F(x) can be understood to be the rational functions with variable x

median oar Jul 9, 2023, 5:20 AM

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Well pi is not algebraic over A right?

surreal meadow Jul 9, 2023, 5:20 AM

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right

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so F[pi] is not a field here, and instead we want F(pi)

median oar Jul 9, 2023, 5:21 AM

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I thought field extensions are always fields

surreal meadow Jul 9, 2023, 5:21 AM

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they are

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F[pi] is not a field extension

#

the polynomials of pi with coefficients in A is not a field specifically

median oar Jul 9, 2023, 5:21 AM

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median oar I thought A(pi) was the field extension of A with pi

Fuck

#

() is field extensions

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Ok

#

dead

surreal meadow Jul 9, 2023, 5:22 AM

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yes, and when x is algebraic

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it just turns into F[x]

median oar Jul 9, 2023, 5:22 AM

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Right

median oar Jul 9, 2023, 5:22 AM

#

surreal meadow A(pi), since pi is transcendental, turns into the rational functions with coeffi...

Wait this is talking about A[pi] isn’t it?

#

Coefficient in A evaluated at pi?

surreal meadow Jul 9, 2023, 5:23 AM

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so

#

(\mathbb{A}(\pi) \simeq \left{ \frac{p(\pi)}{q(\pi)}\ \middle\vert\ p(x),q(x) \in A[x], q(x)\ne 0\right})

#

so the usual rational functions you know

#

but we restrict the coefficients to just be inthe algebraic numbers

#

and then we always evaluate at pi

ocean sealBOT Jul 9, 2023, 5:25 AM

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maximofs

surreal meadow Jul 9, 2023, 5:25 AM

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ok that's all the editing im doing haha sorry

olive mural Jul 9, 2023, 5:25 AM

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hey

#

i need help

surreal meadow Jul 9, 2023, 5:25 AM

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!help

lone heartBOT Jul 9, 2023, 5:25 AM

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Please read #❓how-to-get-help

olive mural Jul 9, 2023, 5:25 AM

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bro

#

like equation wise

#

tardy stag Jul 9, 2023, 5:26 AM

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hi please get your own channel we're talking here

surreal meadow Jul 9, 2023, 5:26 AM

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olive mural

!help

lone heartBOT Jul 9, 2023, 5:26 AM

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olive mural

Please read #❓how-to-get-help

surreal meadow Jul 9, 2023, 5:26 AM

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do you see the big bold letters

olive mural Jul 9, 2023, 5:26 AM

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hayley shut up

median oar Jul 9, 2023, 5:27 AM

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@surreal meadow I heard you like algebra

#

kekw

surreal meadow Jul 9, 2023, 5:28 AM

#

anyway, where i was going with the whole "pi is the product of two rational functions" is moot i think

#

maybe there is something there but i can't see it right now

#

we would have p(pi)^2/q(pi)^2 = pi

tardy stag Jul 9, 2023, 5:29 AM

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surreal meadow anyway, where i was going with the whole "pi is the product of two rational func...

i've been reading about something called the resultant and it's tough to say but it's a way of combining two polynomials into one that shares the common roots

surreal meadow Jul 9, 2023, 5:30 AM

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so

#

before i comment on the resultant thing

tardy stag Jul 9, 2023, 5:30 AM

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i'm not saying i've managed to understand the resultant to be clear

surreal meadow Jul 9, 2023, 5:30 AM

#

let me throw out there
p(pi)^2 - pi * q(pi)^2 = 0

#

which is a polynomial with coefficients in A[pi] which we may be able to work with

surreal meadow Jul 9, 2023, 5:31 AM

#

tardy stag i've been reading about something called the resultant and it's tough to say but...

are you suggesting this may help us here

#

i have no idea what a resultant is

#

also frosst im sorry im hijacking this channel

median oar Jul 9, 2023, 5:31 AM

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ocean seal **maximofs**

Is this how field extensions work in general?

median oar Jul 9, 2023, 5:32 AM

#

surreal meadow also frosst im sorry im hijacking this channel

No it’s completely fine I’m attempting to understand what’s happening and that’s completely fine by me

surreal meadow Jul 9, 2023, 5:32 AM

#

well it turns out to be isomorphic to that, and when you extend by an algebraic element i believe that field i wrote above just turns into the polynomials

#

you essentially multiply by conjugates

#

until the denominator goes back to an element in the original field

median oar Jul 9, 2023, 5:33 AM

#

What do you mean by being algebraic over a field

surreal meadow Jul 9, 2023, 5:33 AM

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ah

#

like

#

it is the solution to some polynomial with coefficients in that field

#

like i is algebraic over Q

#

because it is a solution to x^2 + 1 = 0

median oar Jul 9, 2023, 5:33 AM

#

Ah

surreal meadow Jul 9, 2023, 5:33 AM

#

but like

#

e and pi arent

median oar Jul 9, 2023, 5:34 AM

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So pi is not algebraic in A

surreal meadow Jul 9, 2023, 5:34 AM

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right

#

"transcendental," if not otherwise specified, usually means "not algebraic over Q"

#

in general it's just "not algebraic over the field you're working on"

median oar Jul 9, 2023, 5:34 AM

#

Are all things not algebraic in A transcendental?

#

Oh over Q

surreal meadow Jul 9, 2023, 5:34 AM

#

yep

#

so when you start with q and extend by an algebraic element it's like somewhat interesting

#

and there's a lot of theory (see: galois) about it

#

but it gets really tricky when you extend by a transcendental element like we are doing now

median oar Jul 9, 2023, 5:36 AM

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Just to be sure transcendental means it’s not algebraic over Q

#

Not A

surreal meadow Jul 9, 2023, 5:36 AM

#

right

#

if not otherwise specified it means over Q

#

but you'll notice since A is the algebraic closure of Q

median oar Jul 9, 2023, 5:37 AM

#

But if you’re not algebraic over A then you aren’t over Q either

surreal meadow Jul 9, 2023, 5:37 AM

#

they kind of mean the same thing

#

maybe i should take that back

#

we usually mean over Q, but yeah if you're not algebraic over A then def not algebraic over Q

tardy stag Jul 9, 2023, 5:40 AM

#

trying to figure out how to gain any information about p(pi) from p(rootpi) or vice versa

surreal meadow Jul 9, 2023, 5:40 AM

#

surreal meadow let me throw out there p(pi)^2 - pi * q(pi)^2 = 0

im mainly playing with this right now

median oar Jul 9, 2023, 5:41 AM

#

Oh right A comes from Q

#

Sorry I’m like at work atm

surreal meadow Jul 9, 2023, 5:41 AM

#

no worries frosst

lament forge Jul 9, 2023, 5:41 AM

#

i've read through the whole channel and i'm now kind of lost
what's the question?

median oar Jul 9, 2023, 5:41 AM

#

So I’m not often on my phone (I shouldn’t be at all hehe)

surreal meadow Jul 9, 2023, 5:41 AM

#

lament forge i've read through the whole channel and i'm now kind of lost what's the question...

is sqrt(pi) in A(pi)

#

where A is the algebraic numbers, and A(pi) is the field extension containing pi

#

also since this has stretched out pretty far, im thinking of making a very tiny writeup about this if thats ok with everyone involved

tardy stag Jul 9, 2023, 5:44 AM

#

yes pls

surreal meadow Jul 9, 2023, 5:44 AM

#

just for like a blog of mine

lament forge Jul 9, 2023, 5:44 AM

#

surreal meadow is sqrt(pi) in A(pi)

suppose it is, then it's p(pi)/q(pi) for some nonzero p,q with coefficients in A, and p(pi)^2/q(pi)^2 - pi = 0, therefore p(pi)^2 - pi*q(pi)^2 = 0, therefore pi is algebraic over A, and i think that's false (although i don't remember the proof)

median oar Jul 9, 2023, 5:44 AM

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Sick

#

Link pls

tardy stag Jul 9, 2023, 5:44 AM

#

don't mention the fact that I confused A(pi) and A[pi] for like ten minutes

median oar Jul 9, 2023, 5:45 AM

#

By what we just discussed pi isn’t algebraic over A

median oar Jul 9, 2023, 5:45 AM

#

tardy stag don't mention the fact that I confused A(pi) and A[pi] for like ten minutes

Same

surreal meadow Jul 9, 2023, 5:45 AM

#

yeah definitely not in A

#

because like its transcendental haha

#

so its not in the algebraics lol

median oar Jul 9, 2023, 5:45 AM

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That’s wack

surreal meadow Jul 9, 2023, 5:46 AM

#

ah

lament forge Jul 9, 2023, 5:46 AM

#

therefore p(pi)^2 - piq(pi)^2 = 0, therefore pi is algebraic over A
(my justification for this step is that p(pi)^2 only contains even power of pi and piq(pi)^2 only contains odd powers of pi, and they're both individually not zero)

tardy stag Jul 9, 2023, 5:46 AM

#

no i think that proof works

#

yeah

surreal meadow Jul 9, 2023, 5:46 AM

#

p(x)^2 - xq(x)^2 = 0

#

very nice

#

ok awesome

#

so we found an extension of A that is not C

#

very cool

tardy stag Jul 9, 2023, 5:47 AM

#

not algebraically closed :(

surreal meadow Jul 9, 2023, 5:47 AM

#

now we just need to prove the closure of this is C

lament forge Jul 9, 2023, 5:47 AM

#

...oh wait if you just wanted to check that this isn't C then it's way easier than that
A(pi) is countable and C isn't

median oar Jul 9, 2023, 5:47 AM

#

median oar And you can make it bigger without being C?

So the answer is yes

surreal meadow Jul 9, 2023, 5:47 AM

#

lament forge ...oh wait if you just wanted to check that this isn't C then it's way easier th...

that's too easy

surreal meadow Jul 9, 2023, 5:48 AM

#

median oar So the answer is yes

yup

#

and you can also make it bigger than C itself

#

which isn't surprising since C also has extensions

tardy stag Jul 9, 2023, 5:49 AM

#

wait @median oar i thought you wanted an algebraically closed field that contained A that wasn't C

median oar Jul 9, 2023, 5:49 AM

#

I did

surreal meadow Jul 9, 2023, 5:49 AM

#

i thought it was just an extension my bad

lament forge Jul 9, 2023, 5:49 AM

#

oh
hmm

median oar Jul 9, 2023, 5:49 AM

#

Oh it’s not closed is it

tardy stag Jul 9, 2023, 5:49 AM

#

well uh.. this isn't it lol

lament forge Jul 9, 2023, 5:49 AM

#

yeah we did just prove that A(pi) is not algebraically closed

tardy stag Jul 9, 2023, 5:49 AM

#

unlucky

median oar Jul 9, 2023, 5:49 AM

#

Rip

#

Well

surreal meadow Jul 9, 2023, 5:50 AM

#

if you extend any farther you may need to go to C to be closed

#

which is like

#

the original question lol

median oar Jul 9, 2023, 5:50 AM

#

We just need to stick someone that’s algebraic over A

surreal meadow Jul 9, 2023, 5:50 AM

#

there isn't anything

median oar Jul 9, 2023, 5:50 AM

#

Into A()

surreal meadow Jul 9, 2023, 5:50 AM

#

since A is closed

median oar Jul 9, 2023, 5:50 AM

#

But that would already be in A

#

Wouldn’t it

surreal meadow Jul 9, 2023, 5:50 AM

#

yeah

#

like A is alg closed so there's not (nontrivial) algebraic extensions

median oar Jul 9, 2023, 5:51 AM

#

So then how tf do you get from A to C

surreal meadow Jul 9, 2023, 5:51 AM

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you extend by a non algebraic extension

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and then close it lol

tardy stag Jul 9, 2023, 5:51 AM

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median oar So then how tf do you get from A to C

not with a finite number of extenders that's for sure

surreal meadow Jul 9, 2023, 5:52 AM

#

hm so what is the algebraic closure of A(pi)

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it looks a lot like Q the more you look at it

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A(pi) that is

median oar Jul 9, 2023, 5:59 AM

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Well since A looks like Q I’d hope A(pi) does too

surreal meadow Jul 9, 2023, 5:59 AM

#

it feels like closing A(pi) would be like closing Q in a way

#

but should be much easier

surreal meadow Jul 9, 2023, 6:00 AM

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median oar Well since A looks like Q I’d hope A(pi) does too

it's like adding sqrt(2) to Q

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hm never mind

#

i just keep thinking of x^2 - pi = 0 and compare pi to 2

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or any prime

tardy stag Jul 9, 2023, 6:04 AM

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i think if you want to describe A(pi) you might first try and describe A itself

median oar Jul 9, 2023, 6:05 AM

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Doesn’t it work?

median oar Jul 9, 2023, 6:05 AM

#

surreal meadow i just keep thinking of x^2 - pi = 0 and compare pi to 2

This

surreal meadow Jul 9, 2023, 6:05 AM

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work as in

#

a counterexample to A(pi) being closed

median oar Jul 9, 2023, 6:05 AM

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Well yeah

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Since we said sqrt(pi) isn’t in A(pi)

surreal meadow Jul 9, 2023, 6:06 AM

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tardy stag i think if you want to describe A(pi) you might first try and describe A itself

well it's just rational functions right

surreal meadow Jul 9, 2023, 6:06 AM

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median oar Since we said sqrt(pi) isn’t in A(pi)

yeah that was our idea and we proved it

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i just keep thinking of x^2 - pi = 0 cause it makes pi look like a sort of prime

#

none of x^n - pi = 0 have solutions in our field

tardy stag Jul 9, 2023, 6:07 AM

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surreal meadow well it's just rational functions right

yeah it's roots of rational (or, equivalently, integer) functions

surreal meadow Jul 9, 2023, 6:08 AM

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i mean specifically that A(pi) is rational functions with coefficients in A, evaluated at pi

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i don't know if that's what you meant

#

and that gives us a general descriptor for an element of A(pi)
now i wonder what a closure would look like. for starters we need all roots of pi, and what else?

lament forge Jul 9, 2023, 6:10 AM

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well you need roots of every polynomial where the coefficients are elements of A(pi)

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and i... think that should be enough...?

surreal meadow Jul 9, 2023, 6:10 AM

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yeah but i want to go bottom up

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rather than just write the definition

lament forge Jul 9, 2023, 6:11 AM

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...well now that we've written the definition, there are only countably many of those polynomials, so the algebraic closure of A(pi) is countable and therefore not C

tardy stag Jul 9, 2023, 6:11 AM

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don't think you'll be able to, I haven't found a bottom up defn of A even

median oar Jul 9, 2023, 6:11 AM

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I looked around and A just seemed weird to describe

surreal meadow Jul 9, 2023, 6:12 AM

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my question is if we can write this closure as A(pi)(U pi^(1/n))

median oar Jul 9, 2023, 6:12 AM

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What about x^5 ones

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Don’t you run into the same problem

surreal meadow Jul 9, 2023, 6:12 AM

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hm good point

surreal meadow Jul 9, 2023, 6:13 AM

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tardy stag don't think you'll be able to, I haven't found a bottom up defn of A even

right but here we take the definition of A for granted

median oar Jul 9, 2023, 6:13 AM

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So it’s just A but you stick pi onto every element of A

tardy stag Jul 9, 2023, 6:13 AM

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and pi^2

median oar Jul 9, 2023, 6:13 AM

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So now there’s 2 sets of it

surreal meadow Jul 9, 2023, 6:13 AM

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and pi^-1

lament forge Jul 9, 2023, 6:13 AM

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and pi^3

median oar Jul 9, 2023, 6:14 AM

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Ok now there’s countably infinite copies of A in A

lament forge Jul 9, 2023, 6:14 AM

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and 1/(pi+1)

surreal meadow Jul 9, 2023, 6:14 AM

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ocean seal **maximofs**

just this lol

lament forge Jul 9, 2023, 6:14 AM

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i would think there's only one copy of A in A

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A(pi) is definitely countably infinite though

median oar Jul 9, 2023, 6:14 AM

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I mean

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I mistyped

surreal meadow Jul 9, 2023, 6:14 AM

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yeah it's smaller than A[pi]xA[pi]

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so definitely countable

median oar Jul 9, 2023, 6:15 AM

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There’s countably infinite copies of A in A(pi) but it has a bunch of pi’s stuck onto the copies of A

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Very rigourous mathematics coming from me today

surreal meadow Jul 9, 2023, 6:15 AM

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like A, pi * A, pi^2 * A, etc.

median oar Jul 9, 2023, 6:16 AM

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Yes

surreal meadow Jul 9, 2023, 6:16 AM

#

then yeah sort of

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and then we can also do that in the denom

#

and we can add them

#

so it gets big, but still only countably big

median oar Jul 9, 2023, 6:16 AM

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So it’s like A with A

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But one of them has pi’s

surreal meadow Jul 9, 2023, 6:17 AM

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yeah it's like the direct sum of |N| copies of A

median oar Jul 9, 2023, 6:17 AM

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Isn’t it |Q| (I have no idea what I’m saying)

surreal meadow Jul 9, 2023, 6:17 AM

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they are equal haha

median oar Jul 9, 2023, 6:17 AM

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Oh

#

True

surreal meadow Jul 9, 2023, 6:18 AM

#

my point is that you basically have a vector space with basis
(1, x, x^2, x^3, ...)

#

which you might recognize

median oar Jul 9, 2023, 6:18 AM

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I’m really at the edge of what I can understand at this point

#

If not further

surreal meadow Jul 9, 2023, 6:18 AM

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i think this connection is a nice one to make

lament forge Jul 9, 2023, 6:18 AM

#

i have realised again that i don't actually know what question we're trying to answer

median oar Jul 9, 2023, 6:18 AM

#

surreal meadow my point is that you basically have a vector space with basis (1, x, x^2, x^3, ....

Rⁿ

#

Hey that’s a cool n

surreal meadow Jul 9, 2023, 6:18 AM

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R^infinity even

median oar Jul 9, 2023, 6:18 AM

#

Oh

surreal meadow Jul 9, 2023, 6:19 AM

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lament forge i have realised again that i don't actually know what question we're trying to a...

we are just talking at this point

median oar Jul 9, 2023, 6:19 AM

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Not finite

lament forge Jul 9, 2023, 6:19 AM

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fair enough

#

well i found a number that isn't algebraic over A(pi)

median oar Jul 9, 2023, 6:19 AM

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So there’s no algebraically closed field between A and C

surreal meadow Jul 9, 2023, 6:19 AM

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lament forge well i found a number that isn't algebraic over A(pi)

any new ones or just sqrtpi

surreal meadow Jul 9, 2023, 6:19 AM

#

median oar So there’s no algebraically closed field between A and C

there is

median oar Jul 9, 2023, 6:19 AM

#

Well e wouldn’t be in it either

lament forge Jul 9, 2023, 6:19 AM

#

sqrt(pi) is algebraic over A(pi)

median oar Jul 9, 2023, 6:19 AM

#

What

surreal meadow Jul 9, 2023, 6:19 AM

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lament forge sqrt(pi) is algebraic over A(pi)

oh

lament forge Jul 9, 2023, 6:19 AM

#

median oar Well e wouldn’t be in it either

are we sure that's true?

surreal meadow Jul 9, 2023, 6:19 AM

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sorry yes

#

i misread

lament forge Jul 9, 2023, 6:20 AM

#

lament forge sqrt(pi) is algebraic over A(pi)

it's a root of the polynomial x^2 - pi = 0 which has coefficients in A(pi)

median oar Jul 9, 2023, 6:20 AM

#

Well unless you know a way to make e from pi I guess

lament forge Jul 9, 2023, 6:20 AM

#

being in A(pi) and being algebraic over A(pi) are distinct because A(pi) is not algebraically closed

median oar Jul 9, 2023, 6:20 AM

#

Oh

surreal meadow Jul 9, 2023, 6:21 AM

#

median oar So there’s no algebraically closed field between A and C

the closure of A(pi) is surely smaller than C

lament forge Jul 9, 2023, 6:21 AM

#

lament forge are we sure that's true?

we're not

#

it is an open problem whether e+pi is rational or not

median oar Jul 9, 2023, 6:21 AM

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Oh that’s what we were saying

#

What’s the closure of A(pi)

surreal meadow Jul 9, 2023, 6:21 AM

#

yeah that's what we were asking, if we can describe it nicely

lament forge Jul 9, 2023, 6:21 AM

#

well chaitin's constant, or any uncomputable number, is not algebraic over A(pi)

#

because all numbers in the algebraic closure of A(pi) are computable

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(chaitin's constant is the probability that a random turing machine halts)

tardy stag Jul 9, 2023, 6:24 AM

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haha i was talking about computable numbers earlier

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those are weeeeird

median oar Jul 9, 2023, 6:25 AM

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This is what I got from my friend

surreal meadow Jul 9, 2023, 6:26 AM

#

summed up the whole convo in 2 texts

median oar Jul 9, 2023, 6:26 AM

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That’s the pi thing isn’t it

surreal meadow Jul 9, 2023, 6:26 AM

#

yeah

median oar Jul 9, 2023, 6:26 AM

#

Then the closure of A(pi)

#

Is between A and C

surreal meadow Jul 9, 2023, 6:26 AM

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yeah

median oar Jul 9, 2023, 6:27 AM

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Lol ok

tardy stag Jul 9, 2023, 6:27 AM

#

there is an algebraic closure of A(pi) and it's countable and a proper superset of A(pi) and a proper subset of C

median oar Jul 9, 2023, 6:29 AM

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(I have no idea how this links to diagonalisation of matrices but oh wel)

tacit arch Jul 9, 2023, 6:31 AM

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median oar (I have no idea how this links to diagonalisation of matrices but oh wel)

Guarantee if you ask this in the right advanced channel someone will answer

#

Someone might just answer with this though

lone heartBOT Jul 9, 2023, 7:29 AM

#

@median oar Has your question been resolved?

lone heartBOT Jul 9, 2023, 7:52 AM

#

Channel closed

Closed by @median oar

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#

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barren portal Jul 9, 2023, 7:56 AM

#

Find the least residues mod m that are invertable for m=5

lone heartBOT Jul 9, 2023, 7:56 AM

#

barren portal Find the least residues mod m that are invertable for m=5

barren portal Jul 9, 2023, 7:57 AM

#

is it the solution to congruence x $\equiv$ 1 (mod 5) ?

ocean sealBOT Jul 9, 2023, 7:57 AM

#

dotdoc.

barren portal Jul 9, 2023, 8:08 AM

#

i see since gcd of m any of its least residue is 1, then its follows they are invertable mod m

lone heartBOT Jul 9, 2023, 8:23 AM

#

@barren portal Has your question been resolved?

#

@barren portal Has your question been resolved?

lone heartBOT Jul 9, 2023, 8:53 AM

#

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

lone heartBOT Jul 9, 2023, 9:09 AM

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cedar dove Jul 9, 2023, 9:14 AM

#

I got a graph here that asks me to find "g(2)" This graph has two lines graphed on it. However, on x=2, the first line ends at y=-1 and the second line begins at y=1. the y=-1 is an "open circle". The circle is not bolded in on the graph. Would this be answered with y=1 or y=-1? Or is this undefined?

lone heartBOT Jul 9, 2023, 9:14 AM

#

cedar dove I got a graph here that asks me to find "g(2)" This graph has two lines graphed ...

vale wigeon Jul 9, 2023, 9:15 AM

#

show graph

cedar dove Jul 9, 2023, 9:15 AM

#

currently uploading from phone

#

vale wigeon Jul 9, 2023, 9:16 AM

#

cedar dove Jul 9, 2023, 9:17 AM

#

how come? Is it because its positive?

tardy stag Jul 9, 2023, 9:17 AM

#

it's because the circle is filled in

cedar dove Jul 9, 2023, 9:18 AM

#

I see, so we dont need to worry about the ones not filled in on this kind of question where it just wants to know what y is... got it, thanks!

tardy stag Jul 9, 2023, 9:19 AM

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the open circle means that curve is valid up to but not including that point, while the filled circle means it's valid including that point

#

so, yes that's right

cedar dove Jul 9, 2023, 9:19 AM

#

not including? I see. I was still pondering what it meant haha, thanks for pointing that out

#

.close

lone heartBOT Jul 9, 2023, 9:20 AM

#

Channel closed

Closed by @cedar dove

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#

Available help channel!

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rancid plover Jul 9, 2023, 9:30 AM

#

for this question, a person helpfully explained that the (–log_z(y))^2 becomes (–log_z(y))^2 due to x^2 = (-x)^2
would I have to then do anything to the numerator to compensate? I'd imagine not based off of the explanation, but just covering my bases here

lone heartBOT Jul 9, 2023, 9:30 AM

#

rancid plover for this question, a person helpfully explained that the (–log_z(y))^2 becomes (...

tardy stag Jul 9, 2023, 9:34 AM

#

nope, $(-\log_z(y))^2 = (\log_z(y))^2$ all on its own, whether it's in a fraction or not

ocean sealBOT Jul 9, 2023, 9:34 AM

#

Hayley

tardy stag Jul 9, 2023, 9:36 AM

#

indeed, this is really true for any number, $(-A)^2 = (A)^2$ no matter what $A$ is or where it's located in an expression

ocean sealBOT Jul 9, 2023, 9:36 AM

#

Hayley

tardy stag Jul 9, 2023, 9:36 AM

#

as long as it has that exact form

rancid plover Jul 9, 2023, 9:40 AM

#

righty, thank you

#

.close

lone heartBOT Jul 9, 2023, 9:40 AM

#

Channel closed

Closed by @rancid plover

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#

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barren portal Jul 9, 2023, 10:03 AM

#

How do I show $a^{\phi(b)} + b^{\phi(a)} \equiv 1 \pmod{ab}$

lone heartBOT Jul 9, 2023, 10:03 AM

#

barren portal How do I show $a^{\phi(b)} + b^{\phi(a)} \equiv 1 \pmod{ab}$

tardy stag Jul 9, 2023, 10:04 AM

#

\pmod{ab} for (mod ab) btw

barren portal Jul 9, 2023, 10:05 AM

#

tardy stag `\pmod{ab}` for (mod ab) btw

thank youu

#

<@&286206848099549185>

ocean sealBOT Jul 9, 2023, 10:11 AM

#

dotdoc.

vale crag Jul 9, 2023, 10:14 AM

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are a and b coprime here ? @barren portal

barren portal Jul 9, 2023, 10:14 AM

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yes

vale crag Jul 9, 2023, 10:15 AM

#

chinese remainder theorem seems like a good idea here then

barren portal Jul 9, 2023, 10:15 AM

#

oh

#

$a^{\phi(b)} \pmod{b}$, $b^{\phi(a)} \equiv 1 \pmod{a}$

ocean sealBOT Jul 9, 2023, 10:16 AM

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dotdoc.

barren portal Jul 9, 2023, 10:17 AM

#

These two beings the equations for the system, according to CRT , it has a solution mod ab, when (a,b)=1 right?

vale crag Jul 9, 2023, 10:17 AM

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alright

#

what can you say about a^phi(b) + b^phi(a) then

#

(looking mod a and mod b ofc)

barren portal Jul 9, 2023, 10:18 AM

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oh it satifies both

#

since , mod a, other one will b term vanish

#

and for mod b, a will vanish ?

vale crag Jul 9, 2023, 10:19 AM

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yeah so it's 1 mod a, 1 mod b

barren portal Jul 9, 2023, 10:28 AM

#

@vale crag

vale crag Jul 9, 2023, 10:29 AM

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uh yeah ?

barren portal Jul 9, 2023, 10:29 AM

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im sorry i didnt get the answer

barren portal Jul 9, 2023, 10:29 AM

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ocean seal **dotdoc.**

crt says there is a solution in ab right?

#

but i dont see any x's in here as in congruence equation

vale crag Jul 9, 2023, 10:30 AM

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the a^phi(b) + b^phi(a) is your x

#

it's the value we want to find mod ab

barren portal Jul 9, 2023, 10:31 AM

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vale crag the a^phi(b) + b^phi(a) is your x

i see this satifies both mod and mod b

#

but how this helps to write congruent to 1 mod ab?

vale crag Jul 9, 2023, 10:32 AM

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well do the usual thing

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(which is pretty much going back in the proof of CRT)

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a and b are coprime so there are bezout coefficients s and t such that as+bt = 1

barren portal Jul 9, 2023, 10:34 AM

#

$x \equiv a^{\phi(b)} + b^{\phi(a)} \pmod{ab}$

ocean sealBOT Jul 9, 2023, 10:34 AM

#

dotdoc.

barren portal Jul 9, 2023, 10:34 AM

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this is what I have now

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is it correct?

vale crag Jul 9, 2023, 10:35 AM

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ocean seal **dotdoc.**

i don't see where you're going here

barren portal Jul 9, 2023, 10:36 AM

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ocean seal **dotdoc.**

tbh this is where i am, i saw it as two equations

#

not sure where to lead

vale crag Jul 9, 2023, 10:36 AM

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we have a number $x$, which satisfies $$\begin{cases}x\equiv 1\pmod{a} \ x\equiv 1\pmod{b} \end{cases}$$

ocean sealBOT Jul 9, 2023, 10:36 AM

#

_aplatypus

barren portal Jul 9, 2023, 10:37 AM

#

okkie

vale crag Jul 9, 2023, 10:37 AM

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(and this x is in fact a^phi(b) + b^phi(a) but we don't really care tbh)

barren portal Jul 9, 2023, 10:38 AM

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alright

vale crag Jul 9, 2023, 10:38 AM

#

so here we have the perfect setup for CRT right

barren portal Jul 9, 2023, 10:39 AM

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(a,b)=1

#

i understand this system has a unique solution, mod ab according to CRT

vale crag Jul 9, 2023, 10:41 AM

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sure, do you remember how to compute that solution though ?

#

like say a=2, b=3, are you able to do it ?

barren portal Jul 9, 2023, 10:42 AM

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yes

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we construct an x

#

combining it's inverse and multiplication factors

barren portal Jul 9, 2023, 10:43 AM

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vale crag like say a=2, b=3, are you able to do it ?

let M=6, M1= 3, M2=2

#

then I'll solve inverses to multiply

#

so that taking modulo only those are left and other vanish

#

#

where ai are the solutions of each, yi, inverses, Mi, M/mi , where M is the product of modulii

vale crag Jul 9, 2023, 10:50 AM

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yeah that's the idea

barren portal Jul 9, 2023, 10:52 AM

#

ocean seal **\_aplatypus**

so do that here right?

#

then x=b b^-1 + a a^-1

vale crag Jul 9, 2023, 10:54 AM

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I just think it's a bit of pain to use the full computational formula here

#

that's why I was talking about the bézout coefficients earlier

#

it's pretty much the same thing (you're computing the modular inverses yada yada)

barren portal Jul 9, 2023, 10:55 AM

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sure

vale crag Jul 9, 2023, 10:55 AM

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works much better in this context though

barren portal Jul 9, 2023, 10:55 AM

#

sure

vale crag Jul 9, 2023, 10:56 AM

#

vale crag a and b are coprime so there are bezout coefficients s and t such that as+bt = 1

as+bt = 1, so as=0 mod a & as = 1 mod b

#

bt=1 mod a & bt = 0 mod b

vale crag Jul 9, 2023, 10:57 AM

#

barren portal

as and bt you have to see those as your M_iy_i from your pic

vale crag Jul 9, 2023, 10:58 AM

#

vale crag bt=1 mod a & bt = 0 mod b

can you construct a number which is 1 mod a and 1 mod b from as and bt then ?

barren portal Jul 9, 2023, 11:00 AM

#

as+bt=1, if (a,b)=1

vale crag Jul 9, 2023, 11:00 AM

#

yup as+bt is 1 mod a, 1 mod b here

barren portal Jul 9, 2023, 11:00 AM

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bezouts identity

vale crag Jul 9, 2023, 11:01 AM

#

turns out as+bt is straight up 1 from bézout's identity tho

barren portal Jul 9, 2023, 11:01 AM

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yes

vale crag Jul 9, 2023, 11:01 AM

#

therefore 1 is your unique solution (mod ab)

barren portal Jul 9, 2023, 11:01 AM

#

yes

vale crag Jul 9, 2023, 11:01 AM

#

and we won

#

x = 1 (mod ab)

#

and x was our original number a^phi(b) + b^phi(a)

barren portal Jul 9, 2023, 11:03 AM

#

I see

#

so you constructed the x using CRT, and verified that whenever (a,b)=1, there exist bezouts co-efficient such that as+bt=1

#

and clearly this satisfies both equations

#

which means the system must have 1 as it’s solution

vale crag Jul 9, 2023, 11:05 AM

#

well you know from CRT there's a unique solution mod ab, and you find it using that bézout stuff (which is almost the same thing as the computational formula you have, it's just easier to work directly with bezout here)

barren portal Jul 9, 2023, 11:05 AM

#

Makes sense now

#

Thanks, have a great day

lone heartBOT Jul 9, 2023, 11:20 AM

#

@barren portal Has your question been resolved?

#

Channel closed

Closed by @barren portal

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shy sluice Jul 9, 2023, 11:25 AM

#

I need help with some homework about factorising quadratic equations, this is where im at currently but I cant find anything that would go into the second bracket that matches the first bracket

H) 15a^2 - 14a - 8

15 x 8 = 120
1 120
2 60
3 40
4 30
5 24
6 20
8 15
10 12

6 - 20 = -14

15a^2 - 20a + 6a - 8

3a(5a - 5) + ()

lone heartBOT Jul 9, 2023, 11:25 AM

#

shy sluice I need help with some homework about factorising quadratic equations, this is wh...

gray isle Jul 9, 2023, 11:30 AM

#

you didn't factorise the first two terms properly

shy sluice Jul 9, 2023, 11:30 AM

#

oh really

gray isle Jul 9, 2023, 11:30 AM

#

3a(5a - 5) is 15a^2 - 15a
not 15a^2 - 20a

shy sluice Jul 9, 2023, 11:32 AM

#

I dont understand, isnt it 5^2 - 5 which would be 20? or have I got it all wrong

gray isle Jul 9, 2023, 11:32 AM

#

why are you doing 5^2 - 5

#

you've got it all wrong

shy sluice Jul 9, 2023, 11:32 AM

#

no clue lol

gray isle Jul 9, 2023, 11:32 AM

#

the goal is to factorise / manipulate to get something equivalent

#

15a^2 - 20a
can you identify the factor common in both
15a^2
and 20a

shy sluice Jul 9, 2023, 11:34 AM

#

5?

gray isle Jul 9, 2023, 11:34 AM

#

5 is a common factor,

#

are you able to factor out 5 from 15a^2 - 20a

shy sluice Jul 9, 2023, 11:35 AM

#

would it be 3 times?

gray isle Jul 9, 2023, 11:36 AM

#

try and show me what you get

shy sluice Jul 9, 2023, 11:37 AM

#

well 15 / 3 = 5 but 20 / 3 = 6.6 recuring

#

I dont understand

gray isle Jul 9, 2023, 11:37 AM

#

3 isn't a factor of 20

shy sluice Jul 9, 2023, 11:37 AM

#

right

gray isle Jul 9, 2023, 11:38 AM

#

5 is a common factor,

shy sluice Jul 9, 2023, 11:38 AM

#

OH

gray isle Jul 9, 2023, 11:38 AM

#

5 * what? = 15a^2

shy sluice Jul 9, 2023, 11:38 AM

#

I think I get it

shy sluice Jul 9, 2023, 11:39 AM

#

gray isle 5 * `what?` = 15a^2

45?

gray isle Jul 9, 2023, 11:39 AM

#

no

shy sluice Jul 9, 2023, 11:39 AM

#

3

gray isle Jul 9, 2023, 11:39 AM

#

no

shy sluice Jul 9, 2023, 11:39 AM

#

damn

gray isle Jul 9, 2023, 11:39 AM

#

none of those have any a in it

shy sluice Jul 9, 2023, 11:39 AM

#

ohh

#

so I said like 45a would that be better?

gray isle Jul 9, 2023, 11:39 AM

#

no

shy sluice Jul 9, 2023, 11:39 AM

#

ok

gray isle Jul 9, 2023, 11:40 AM

#

stop guessing

shy sluice Jul 9, 2023, 11:42 AM

#

gray isle 5 * `what?` = 15a^2

can you tell me the answer to this, I might understand it more then

gray isle Jul 9, 2023, 11:42 AM

#

if you want to factor 5 out of $15a^2 - 20a$ consider:
$$15a^2 - 20a = 5 \cdot \br{\red{\frac{15a^2 - 20a}{5}}}$$
and then simplify the part in red

ocean sealBOT Jul 9, 2023, 11:43 AM

#

ℝamonov

shy sluice Jul 9, 2023, 11:49 AM

#

Man I never had to do any of this for my other questions and now im all confused lol, if its ok can you tell me if this is right?

10x^2 + 19x + 7

10 * 7 = 70
1 70
2 35
5 14

10x^2 + 5x + 14x + 7

5x(2x + 1) + 7(2x + 1)

(5x + 7)(2x + 1)

gray isle Jul 9, 2023, 11:49 AM

#

yes

shy sluice Jul 9, 2023, 11:49 AM

#

ok good

gray isle Jul 9, 2023, 11:50 AM

#

i attempted to heavily break down the steps as you were making mistakes before

#

whether it be from mistakes in calculation or misunderstanding of factorisation itself

shy sluice Jul 9, 2023, 11:51 AM

#

yeah maybe I should go brush up on factorisation

#

.close

lone heartBOT Jul 9, 2023, 11:52 AM

#

Channel closed

Closed by @shy sluice

Use .reopen if this was a mistake.

shy sluice Jul 9, 2023, 11:52 AM

#

thank you!

lone heartBOT Jul 9, 2023, 11:53 AM

#

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

south crystal Jul 9, 2023, 12:11 PM

#

what value can $ab$ /times $cd$ get
if a,b,c and d are different from each other and ≠ 0

209
299
306
326
360

lone heartBOT Jul 9, 2023, 12:11 PM

#

south crystal what value can $ab$ /times $cd$ get if a,b,c and d are different from each othe...

ocean sealBOT Jul 9, 2023, 12:11 PM

#

ajax4074

south crystal Jul 9, 2023, 12:11 PM

#

oop

#

ahaha

mortal trellis Jul 9, 2023, 12:12 PM

#

are a,b,c,d supposed to be digits?

#

and ab, cd are the numbers formed by writing them next to each other?

tardy stag Jul 9, 2023, 12:12 PM

#

that's the only thing that makes sense to me

mortal trellis Jul 9, 2023, 12:12 PM

#

yeah same. still have to ask

south crystal Jul 9, 2023, 12:12 PM

#

yes

#

i'll write it wait a second

#

it says this basically

#

i think it should be simple but i cant write it like how we can when subtracting and adding

#

wair can i still say that

#

i'll show

#

like here

south crystal Jul 9, 2023, 12:16 PM

#

mortal trellis are a,b,c,d supposed to be digits?

ah yes yes ofc i forgot to mention that my bad

mortal trellis Jul 9, 2023, 12:17 PM

#

I would probably just try to play around a bit with the values

#

can a be 1? what would c have to be for ab*cd to be somewhere around 200-300

#

what if a=2

#

and so on

south crystal Jul 9, 2023, 12:19 PM

#

mortal trellis I would probably just try to play around a bit with the values

wouldn't that be a little long of an approach to this when we have options?

#

but okay

#

ah wait you said 300 bc of the options

tardy stag Jul 9, 2023, 12:20 PM

#

the options will guide your choices

south crystal Jul 9, 2023, 12:20 PM

#

i get it i get it

#

okay

#

let me try thay then

mortal trellis Jul 9, 2023, 12:20 PM

#

you could also do it from the other direction and factor all of the choices and try to play around with the factors you get

#

either way, it will involve playing around a bit

south crystal Jul 9, 2023, 12:20 PM

#

yeah i tried 360

#

but i didnt factorize it directly

#

since 60*60=360

#

i tried changing them and opening those 60's up

#

but i got overwhelmed with the amount of options we can rewrite it

#

as

mortal trellis Jul 9, 2023, 12:22 PM

#

60*60=3600

south crystal Jul 9, 2023, 12:22 PM

#

oh shit

mortal trellis Jul 9, 2023, 12:22 PM

#

yeah 360 has a lot of factors so for that its not as nice

south crystal Jul 9, 2023, 12:24 PM

#

mortal trellis yeah 360 has a lot of factors so for that its not as nice

hm yes i see

#

well i tried going on with 20 for ab and 21 again for (ab)

#

but i think i'm missing something

#

or do i really need to go on with 22 23 and so on

#

oh wait not 22

#

but you see what i mean i guess

south crystal Jul 9, 2023, 12:27 PM

#

mortal trellis can a be 1? what would c have to be for ab*cd to be somewhere around 200-300

this is what im trying right now

#

what i mean is that i've now eliminated 306 and 326 since i cant get them with 21 but i dont think thats right?

#

ah am i not supposed to eliminate in the first place maybe...?

south crystal Jul 9, 2023, 12:29 PM

#

south crystal ah am i not supposed to eliminate in the first place maybe...?

but to just find one of values given

tardy stag Jul 9, 2023, 12:29 PM

#

i would look at the last digit first

south crystal Jul 9, 2023, 12:29 PM

#

as in last of ab (b) ?

tardy stag Jul 9, 2023, 12:29 PM

#

as in last of the product

south crystal Jul 9, 2023, 12:29 PM

#

or the options' last?

tardy stag Jul 9, 2023, 12:29 PM

#

the options' last digits ye

south crystal Jul 9, 2023, 12:30 PM

#

oke

#

so 9 can only be

#

3*3

tardy stag Jul 9, 2023, 12:30 PM

#

there are some useful properties that will let you eliminate a lot of choices for a b c d

south crystal Jul 9, 2023, 12:30 PM

#

oh no 9*1 as well

tardy stag Jul 9, 2023, 12:30 PM

#

more than that

south crystal Jul 9, 2023, 12:31 PM

#

tardy stag more than that

are you saying that for 9??

tardy stag Jul 9, 2023, 12:31 PM

#

yes

south crystal Jul 9, 2023, 12:31 PM

#

1,3,9 tho

#

ah well

#

0

#

thats what you meant?

tardy stag Jul 9, 2023, 12:31 PM

#

no it couldn't be 0

south crystal Jul 9, 2023, 12:31 PM

#

oh shit what am i saying

south crystal Jul 9, 2023, 12:31 PM

#

tardy stag no it couldn't be 0

yeah yeah ofc

#

ooohhhh now i see

#

it could also be _9

tardy stag Jul 9, 2023, 12:31 PM

#

i think there's only one more possibility

south crystal Jul 9, 2023, 12:32 PM

#

now what times what gives _9...

tardy stag Jul 9, 2023, 12:32 PM

#

oh but that possibility is excluded by another constraint
you should still find it though

south crystal Jul 9, 2023, 12:32 PM

#

tardy stag i think there's only one more possibility

different from what i am saying right now?

tardy stag Jul 9, 2023, 12:33 PM

#

there's only one pair of one digit numbers that multiplies to _9

south crystal Jul 9, 2023, 12:33 PM

#

how is it 1 pair

mortal trellis Jul 9, 2023, 12:33 PM

#

go through the options for what _9 could be. 9, 19, 29, ...

south crystal Jul 9, 2023, 12:33 PM

#

3,3 and 9,1 is 2 pairs?

#

as in _3*_3=_9

mortal trellis Jul 9, 2023, 12:34 PM

#

or just multiply all single digit numbers

tardy stag Jul 9, 2023, 12:34 PM

#

okay... when i said _9 i meant non 0 in front

south crystal Jul 9, 2023, 12:34 PM

#

mortal trellis go through the options for what _9 could be. 9, 19, 29, ...

oooohh

#

there is no 19 29

#

butt

#

but ahah

#

39 can be= 13*3

#

hm it has the same digits tho

#

has two same digits*

mortal trellis Jul 9, 2023, 12:35 PM

#

keep going

south crystal Jul 9, 2023, 12:35 PM

#

not 49

mortal trellis Jul 9, 2023, 12:36 PM

#

why not

south crystal Jul 9, 2023, 12:36 PM

#

i tried factorizing it

#

am i not supposed to

#

factoring*

#

wait let me see

#

well its not in _3*3 form

#

so

#

_9*1

mortal trellis Jul 9, 2023, 12:38 PM

#

you are supposed to factorize

south crystal Jul 9, 2023, 12:38 PM

#

hm but i get a prime- ooohhh

#

no not ooh i dont see actually

#

i dont see what im missing

mortal trellis Jul 9, 2023, 12:39 PM

#

7*7

south crystal Jul 9, 2023, 12:39 PM

#

...

#

god damn it

#

7 is prime as well

#

i was trying 3

#

okay

#

but umm what are we doign rn?

#

bc we need 2 digits right?

#

2 2 digits in fact

mortal trellis Jul 9, 2023, 12:40 PM

#

this type of question rewards quick mental math and being familiar with small multiplications. you should work on that if you are gonna run into questions like this

#

we were checking what options we had for b and d

south crystal Jul 9, 2023, 12:40 PM

#

well it's not like i dont know 7*7 is 49

#

i tend to overlook or overthink a lot

crisp tangle Jul 9, 2023, 12:42 PM

#

could someone help me with this equation.I always found it hard to get my head around

mortal trellis Jul 9, 2023, 12:42 PM

#

so now you know that one of b and d is 9 and the other is 1

mortal trellis Jul 9, 2023, 12:42 PM

#

crisp tangle could someone help me with this equation.I always found it hard to get my head a...

read #❓how-to-get-help

south crystal Jul 9, 2023, 12:43 PM

#

crisp tangle could someone help me with this equation.I always found it hard to get my head a...

is it 6?

#

sorry i got distracted

south crystal Jul 9, 2023, 12:43 PM

#

mortal trellis so now you know that one of b and d is 9 and the other is 1

yep

mortal trellis Jul 9, 2023, 12:43 PM

#

dont encourage someone intruding your channel

south crystal Jul 9, 2023, 12:43 PM

#

well i didnt help him

#

okay

#

maybe that is considered help

#

anyway

south crystal Jul 9, 2023, 12:45 PM

#

mortal trellis so now you know that one of b and d is 9 and the other is 1

yes (b,d)=(9,1) or (1,9)

#

i2ll try replacing

mortal trellis Jul 9, 2023, 12:45 PM

#

so either ab=21 or ab=29. it should be easy to list the multiples of that

#

see if any of them are one of the options

south crystal Jul 9, 2023, 12:46 PM

#

hmm yeah seems logical

ripe ridge Jul 9, 2023, 12:46 PM

#

hi

#

so like

#

im going into 6th grade and i just need tips on math n stuff

south crystal Jul 9, 2023, 12:47 PM

#

why is everyone here ..

south crystal Jul 9, 2023, 12:48 PM

#

ripe ridge im going into 6th grade and i just need tips on math n stuff

read #❓how-to-get-help dont write here i own this channel rn

#

i eliminated 209 so far

#

and i got 399 by 21*19

#

which isnt one of the options but now i see the max limit i guess

#

also it cant end with 6 if i'm not miscalculating anything

#

well no it can

#

but only 336

#

which is not one of the options

worn fox Jul 9, 2023, 12:52 PM

#

south crystal why is everyone here ..

Because they think the picture in #❓how-to-get-help is literally telling them.to go to help-0 monkey

south crystal Jul 9, 2023, 12:54 PM

#

south crystal also it cant end with 6 if i'm not miscalculating anything

it cant end with 0 either.

mortal trellis Jul 9, 2023, 12:54 PM

#

south crystal also it cant end with 6 if i'm not miscalculating anything

well we started with assuming that the product will end in 9

south crystal Jul 9, 2023, 12:54 PM

#

oh yeahh

mortal trellis Jul 9, 2023, 12:55 PM

#

for ending in 6 or 0 you would have other options for b, d

south crystal Jul 9, 2023, 12:55 PM

#

299 then

south crystal Jul 9, 2023, 12:55 PM

#

mortal trellis for ending in 6 or 0 you would have other options for b, d

oh

#

so i need to see if i can really write 21*_9= 199

mortal trellis Jul 9, 2023, 12:56 PM

#

clearly 210 is a multiple

#

and 199 is only 11 away

#

so that cant work

south crystal Jul 9, 2023, 12:56 PM

#

south crystal so i need to see if i can really write 21*_9= 199

which i cant ?

mortal trellis Jul 9, 2023, 12:56 PM

#

this is the kind of things you can think about

south crystal Jul 9, 2023, 12:57 PM

#

well so do i need to look onto 306 326 and 360

#

because i eliminated 209 and 299

mortal trellis Jul 9, 2023, 12:58 PM

#

well could a=3 ?

#

or bigger?

south crystal Jul 9, 2023, 12:59 PM

#

hm no

#

then its 209

south crystal Jul 9, 2023, 1:01 PM

#

mortal trellis well could a=3 ?

a cant be = 3 but the answers can though now that i've tried

#

29*11=349 for example

#

i think from what you've shown

#

i need to try the same for the ones ending with 6

#

right?

#

as x6*y1 = _ 6

#

and
x3*y2

mortal trellis Jul 9, 2023, 1:04 PM

#

yes just continue like this

#

well for 6 and 0 there will be more options for b and d

south crystal Jul 9, 2023, 1:04 PM

#

south crystal and x3*y2

and the other way around does that count?

mortal trellis Jul 9, 2023, 1:04 PM

#

but you will get faster along the way

south crystal Jul 9, 2023, 1:04 PM

#

okay okay

#

i think i get it from here

#

i got it*

mortal trellis Jul 9, 2023, 1:06 PM

#

actually it will be nicer to factor these other numbers

#

just checked

#

maybe you can try that afterwards

south crystal Jul 9, 2023, 1:06 PM

#

well some of the possibilities eliminate themselves

#

cant i try

south crystal Jul 9, 2023, 1:07 PM

#

mortal trellis just checked

oh okay okay

mortal trellis Jul 9, 2023, 1:07 PM

#

try both approaches

#

they both have upsides and downsides

#

anyway, gtg

#

good luck

south crystal Jul 9, 2023, 1:08 PM

#

thanks

#

i think i can get it so im closing it here

#

thank you for your time and effort

#

.close

lone heartBOT Jul 9, 2023, 1:11 PM

#

Channel closed

Closed by @south crystal

Use .reopen if this was a mistake.

#

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

tepid radish Jul 9, 2023, 1:26 PM

#

um i need algebra help

lone heartBOT Jul 9, 2023, 1:26 PM

#

tepid radish um i need algebra help

tepid radish Jul 9, 2023, 1:26 PM

#

rlly bad

quiet vector Jul 9, 2023, 1:26 PM

#

JustAsk

tepid radish Jul 9, 2023, 1:27 PM

#

ok so

#

im doing this aops shit online shit

#

and im on unit 5 of 13

#

and i need it done by tmr

gray isle Jul 9, 2023, 1:28 PM

#

show t he specific question

tepid radish Jul 9, 2023, 1:29 PM

#

quiet vector Jul 9, 2023, 1:30 PM

#

thats some fine maths tho

#

cutethink

tepid radish Jul 9, 2023, 1:30 PM

#

what that even mean

quiet vector Jul 9, 2023, 1:30 PM

#

no shit in sight

#

KEK

tepid radish Jul 9, 2023, 1:31 PM

#

im done w this bru if somebody can finish all of these units today ill pay them 20 bucks

#

swear

quiet vector Jul 9, 2023, 1:31 PM

#

u gonna get banned before that tho

wary stream Jul 9, 2023, 1:31 PM

#

tepid radish im done w this bru if somebody can finish all of these units today ill pay them ...

We're not here to do your work

tepid radish Jul 9, 2023, 1:31 PM

#

damn

tepid radish Jul 9, 2023, 1:31 PM

#

quiet vector u gonna get banned before that tho

wait huh

#

anyways can yall help me w the question

#

please

quiet vector Jul 9, 2023, 1:32 PM

#

ofc

tepid radish Jul 9, 2023, 1:32 PM

#

thanks bro

quiet vector Jul 9, 2023, 1:32 PM

#

!status

lone heartBOT Jul 9, 2023, 1:32 PM

#

What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above

tepid radish Jul 9, 2023, 1:32 PM

#

3

lone heartBOT Jul 9, 2023, 1:32 PM

#

Show your work, and if possible, explain where you are stuck.

quiet vector Jul 9, 2023, 1:33 PM

#

well showing ur work would help

#

we could tell u wehre u went wrong if u alreadygot the concept down

tepid radish Jul 9, 2023, 1:33 PM

#

see i dont tho

quiet vector Jul 9, 2023, 1:33 PM

#

thats the problem lol

tepid radish Jul 9, 2023, 1:33 PM

#

im jus tryna speed past this class so i can take the geometry course which starts tmr and then the algebra 2 course that starts september

gray isle Jul 9, 2023, 1:33 PM

#

how'd you get an answer without the work

quiet vector Jul 9, 2023, 1:34 PM

#

he dont got the concept down ig

tepid radish Jul 9, 2023, 1:34 PM

#

i did do the work it jus got me 0.6939

#

somehow

gray isle Jul 9, 2023, 1:34 PM

#

show your work

tepid radish Jul 9, 2023, 1:35 PM

#

its in latex is that fine

gray isle Jul 9, 2023, 1:35 PM

#

as long as we can read it

tepid radish Jul 9, 2023, 1:35 PM

#

bet

#

First, let's rewrite the system of equations:

$\frac{3xy}{x + y} = 5$
$\frac{2xz}{x + z} = 3$
$\frac{yz}{y + z} = 4$
To eliminate the fractions, we can cross-multiply each equation.

$3xy = 5(x + y)$
$2xz = 3(x + z)$
$yz = 4(y + z)$
Now, let's solve the system by manipulating these equations.

From equation 1, we have:

$3xy = 5x + 5y$

Rearranging terms, we get:

$3xy - 5x - 5y = 0$

Factoring out common terms, we have:

$x(3y - 5) - 5y = 0$

Simplifying, we get:

$x(3y - 5) = 5y$

From equation 2, we have:

$2xz = 3x + 3z$

Rearranging terms, we get:

$2xz - 3x - 3z = 0$

Factoring out common terms, we have:

$x(2z - 3) - 3z = 0$

Simplifying, we get:

$x(2z - 3) = 3z$

From equation 3, we have:
$yz = 4y + 4z$
Rearranging terms, we get:
$yz - 4y - 4z = 0$
Factoring out common terms, we have:
$y(z - 4) - 4z = 0$
Simplifying, we get:
$y(z - 4) = 4z$
Now, let's eliminate the variable $x$ from equations 1 and 2.
We can do this by multiplying equation 1 by $(2z - 3)$ and multiplying equation 2 by $(3y - 5)$:
$(2z - 3)(3xy - 5x - 5y) = (2z - 3)(0)$
$(3y - 5)(2xz - 3x - 3z) = (3y - 5)(0)$
Expanding the products, we get:
$6xyz - 10xz - 10yz + 15x + 15y = 0$
$6xyz - 9xy - 9xz - 15yz + 10x + 10z = 0$
Now, let's combine like terms:
$6xyz - 10xz - 10yz + 15x + 15y = 6xyz - 9xy - 9xz - 15yz + 10x + 10z$
Subtracting the right side from the left side, we get:
$15x + 15y - 6xyz + 9xy + 9xz + 15yz - 10x - 10z = 0$
Combining like terms, we have:
$5x + 5y - 6xyz + 9xy + 9xz + 15yz - 10z = 0$
Rearranging terms, we get:
$5x + 5y + 9xy + 9xz + 15yz - 6xyz - 10z = 0$
Now, let's isolate $z$ in this equation:
$-6xyz + 9xz + 15yz - 10z = -5x - 5y - 9xy$
Factoring out $z$ on the left side, we have:
$z(-6xy + 9x + 15y - 10) = -5x - 5y - 9xy$
Now, we can divide both sides by $-6xy + 9x + 15y - 10$ to solve for $z$:
$z = \frac{-5x - 5y - 9xy}{-6xy + 9x + 15y - 10}$
Therefore, the value of $z$ in the ordered triple solution $(x, y, z)$ is $\frac{-5x - 5y - 9xy}{-6xy + 9x + 15y - 10}$.

ocean sealBOT Jul 9, 2023, 1:37 PM

#

gnawszn

wary stream Jul 9, 2023, 1:38 PM

#

tepid radish First, let's rewrite the system of equations: $\frac{3xy}{x + y} = 5$ $\frac{2x...

That is a chatgpt answer, isn't it?

tepid radish Jul 9, 2023, 1:38 PM

#

fax

wary stream Jul 9, 2023, 1:39 PM

#

You know chatgpt is wrong a lot, right?

tepid radish Jul 9, 2023, 1:39 PM

#

yea

tardy stag Jul 9, 2023, 1:40 PM

#

you also know gpt generated answers not allowed in this server, right? (especially unmarked)

tepid radish Jul 9, 2023, 1:40 PM

#

what the flip

#

i jus joined

#

i joiend for this one problem

wary stream Jul 9, 2023, 1:40 PM

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Then you should have read #❓how-to-get-help and #rules properly

tardy stag Jul 9, 2023, 1:40 PM

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then the rules should be fresh on your mind, seeing as you read them when you joined...

tepid radish Jul 9, 2023, 1:42 PM

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crazy

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so uh

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i cant be helped or nun

gray isle Jul 9, 2023, 1:42 PM

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and don't attempt to pass other entities' solutions as your own

wary stream Jul 9, 2023, 1:43 PM

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tepid radish i cant be helped or nun

You can be helped, if you properly follow the rules

tepid radish Jul 9, 2023, 1:43 PM

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alr cuddis ill come back when i got sumn legit

gray isle Jul 9, 2023, 1:43 PM

#

could've saved some time if you just said the solution you obtained was from gpt at which point we would've said DON'T use chatgpt for math

do you have an attempt you made youreself?

#

help is a two way street

#

requires some genuine effort on your part

tepid radish Jul 9, 2023, 1:46 PM

#

na cuddi

#

.close

lone heartBOT Jul 9, 2023, 1:46 PM

#

Channel closed

Closed by @tepid radish

Use .reopen if this was a mistake.

#

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

smoky arch Jul 9, 2023, 1:49 PM

#

How to change the green and orange formulas, so the blue marked intersection point (b) is at 0 on the y axis? thonkeyes

lone heartBOT Jul 9, 2023, 1:49 PM

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smoky arch How to change the green and orange formulas, so the blue marked intersection poi...

gray isle Jul 9, 2023, 1:49 PM

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horizontal shift to the right,

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look up graph transformations

smoky arch Jul 9, 2023, 1:49 PM

#

https://www.desmos.com/calculator/3uzeqv8h5s?lang=de

Desmos

Mod2

gray isle Jul 9, 2023, 1:50 PM

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and that'll give you an indication of what to do

smoky arch Jul 9, 2023, 1:53 PM

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gray isle and that'll give you an indication of what to do

I looked it up, but i have no idea how to do that with my formula, they give way simpler examples. NervousSweat

gray isle Jul 9, 2023, 1:54 PM

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same principle applies

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what were you looking at

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1 sec,

smoky arch Jul 9, 2023, 1:55 PM

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Am i entering it wrong? (Symbolab)

gray isle Jul 9, 2023, 1:55 PM

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may have misunderstood what you wanted

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do you want your region to go from
0 to 2
or 0 to 4

smoky arch Jul 9, 2023, 1:56 PM

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gray isle do you want your region to go from 0 to 2 or 0 to 4

What does that mean? 😅

gray isle Jul 9, 2023, 1:57 PM

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is this what you wanted

#

smoky arch Jul 9, 2023, 1:58 PM

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Yes! :D

gray isle Jul 9, 2023, 1:58 PM

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so yeh,
what were you looking at when i told you to look up graph transformations

smoky arch Jul 9, 2023, 1:58 PM

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A youtube video.

gray isle Jul 9, 2023, 1:59 PM

#

smoky arch Jul 9, 2023, 2:02 PM

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So the green graph i think is: 1-(b*x^d)/c^d

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How to shift that? 🥹

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@gray isle I dont really get the picture. :/

gray isle Jul 9, 2023, 2:05 PM

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well you're tryng to move your curves to the right?

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that would be under horizontal translation right c units

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how many units do you want to shift your curve to the right?

smoky arch Jul 9, 2023, 2:06 PM

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I think i got it, one moment.

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🤔

gray isle Jul 9, 2023, 2:08 PM

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the c in that table is not the same as the c you already had defined

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that would be under horizontal translation right c units
how many units do you want to shift your curve to the right?

smoky arch Jul 9, 2023, 2:09 PM

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gray isle the c in that table is not the same as the c you already had defined

Really? What else is it, as it somehow works. :o

#

gray isle Jul 9, 2023, 2:09 PM

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concidence

smoky arch Jul 9, 2023, 2:10 PM

#

So it it right or wrong?
https://www.desmos.com/calculator/dimhl1n1cz?lang=de

Desmos

Mod2

gray isle Jul 9, 2023, 2:10 PM

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the amount you wanted to translate just happened to be the same as the value of c you already had

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wherever that came from

smoky arch Jul 9, 2023, 2:11 PM

#

gray isle the amount you wanted to translate just happened to be the same as the value of ...

🥳 🎉 🍾 Thank you!

#

How do do the rep points thingy? Is it the medal or something else?

gray isle Jul 9, 2023, 2:12 PM

#

no rep system here

smoky arch Jul 9, 2023, 2:13 PM

#

.close

lone heartBOT Jul 9, 2023, 2:13 PM

#

Channel closed

Closed by @smoky arch

Use .reopen if this was a mistake.

#

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

mossy field Jul 9, 2023, 2:15 PM

#

Suppose a lottery exists in which your luck increases as you play. The probability of winning a prize starts at 1%. After every unsuccessful attempt, the probability of winning increases by 1% from your previous odds (so 1% becomes 2%). When a prize is won, the chance of winning is reset to 1%. What is the expected value of the number of prizes that you will receive after 100 uses?

lone heartBOT Jul 9, 2023, 2:15 PM

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mossy field Suppose a lottery exists in which your luck increases as you play. The probabili...

mossy field Jul 9, 2023, 2:15 PM

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We have tried recurrence formulas but failed to get a closed form answer.

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Via python scripts we know that the answer is 1.9095 ± 0.00015

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but not exactly sure what this resembles

old radish Jul 9, 2023, 2:17 PM

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Hi guys

lone heartBOT Jul 9, 2023, 2:19 PM

#

Show your work, and if possible, explain where you are stuck.

mossy field Jul 9, 2023, 2:20 PM

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old radish Hi guys

.?

mossy field Jul 9, 2023, 2:23 PM

#

mossy field Via python scripts we know that the answer is 1.9095 ± 0.00015

incorrect sorry

#

8.1918

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Hopefully some @ Helpers can help?

lone heartBOT Jul 9, 2023, 2:45 PM

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@mossy field Has your question been resolved?

mossy field Jul 9, 2023, 2:45 PM

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this bots needs to be better seriously

lament forge Jul 9, 2023, 2:46 PM

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i tried computing it by brute force (using a computer) and got an answer of exactly 786080126403229668198182908097394362384354574898764667238115242755521947164335122692038436352382356040222738777625219891364911130702970910977870565447613758476042568092806020045109899810694869084004791/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

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(that's about 7.86)

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i have no idea how to derive a nice recurrence for this

lone heartBOT Jul 9, 2023, 3:15 PM

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@mossy field Has your question been resolved?

elfin sierra Jul 9, 2023, 3:16 PM

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Hi, I have an exercise problem, that I would like someone to verify. This is about conditional expectations.

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This is my work.

📎 example-4.1.pdf

mossy field Jul 9, 2023, 3:26 PM

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lament forge i tried computing it by brute force (using a computer) and got an answer of exac...

uh ok

#

.close

lone heartBOT Jul 9, 2023, 3:26 PM

#

Channel closed

Closed by @mossy field

Use .reopen if this was a mistake.

lament forge Jul 9, 2023, 3:26 PM

#

...?

#

.reopen

lone heartBOT Jul 9, 2023, 3:27 PM

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✅

craggy cradle Jul 9, 2023, 3:27 PM

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@mossy field have u seen what Erric and I wrote earlier, we were kinda onto something but not sure

mossy field Jul 9, 2023, 3:27 PM

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where?

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@craggy cradle where

craggy cradle Jul 9, 2023, 3:27 PM

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I think help 12

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In there

mossy field Jul 9, 2023, 3:28 PM

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what was said though

craggy cradle Jul 9, 2023, 3:28 PM

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Ill find it

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2 sec

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And paste it in

mossy field Jul 9, 2023, 3:29 PM

#

given that the answer is exactly 7.86080126403229668198182908097394362384354574898764667238115242755521947164335122692038436352382356040222738777625219891364911130702970910977870565447613758476042568092806020045109899810694869084004791 I'll say the question is solved. 🙂 this is likely not something specific.

craggy cradle Jul 9, 2023, 3:31 PM

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This was our chat:

#

Firstly I tried the expectation formula but it gave me a completely unrealistic answer, but after some time I chose to approach the problem in a more simple way, without complicated formulas. So to calculate the chances of winning, I adopted a fractional perspective instead of considering percentages alone. By viewing 1% as 1/100, I added up the fractions: 1/100 + 2/100 + 3/100... + 14/100, which yielded a sum of 105/100. Based on this calculation, my probability of winning appears to exceed 100%. To determine the number of times this outcome would occur within 100 attempts, I divided 100 by 14. This calculation assumes that winning is expected in every 100% and that the percentages reset to 1% after each victory. It should in theory not be possible for the chance to exceed 100% but you get the gist of it. Im not sure if this makes sense, but my answer somehow ended up being pretty close to Hayley’s program

no this does make sense
since we are dealing with expectation
the only problem is, 14 is too high using the formula for the sum of the first n numbers you get 13.65

mossy field Jul 9, 2023, 3:32 PM

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but we dont need a sum of 105/100

craggy cradle Jul 9, 2023, 3:33 PM

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Ik, but u also cant get 7.8 wins thats where the problem lies

mossy field Jul 9, 2023, 3:33 PM

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why?

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I'm gonna computationally verify the result above as well

craggy cradle Jul 9, 2023, 3:33 PM

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mossy field why?

Either u win or u dont, cant be somewhere in between

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On average it can be 7.8 but not if you only spin 100 times once

mossy field Jul 9, 2023, 3:35 PM

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craggy cradle Either u win or u dont, cant be somewhere in between

its expectation though

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expectation definition is a weighted average

lament forge Jul 9, 2023, 3:35 PM

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lament forge i tried computing it by brute force (using a computer) and got an answer of exac...

here's my code ```py
from fractions import Fraction as F

probs = {}

def raw_prob(winnings, luck, n):
if n == 0:
return F(winnings == 0 and luck == 1)
if luck == 1:
if winnings == 0:
return F(0)
return sum(prob(winnings-1, i, n-1)*F(i, 100) for i in range(1, 101))
return prob(winnings, luck-1, n-1) * (1-F(luck-1, 100))

def prob(winnings, luck, n):
if (winnings, luck, n) in probs:
return probs[winnings, luck, n]
probs[winnings, luck, n] = raw_prob(winnings, luck, n)
return probs[winnings, luck, n]

print(sum(n*sum(prob(n, i, 100) for i in range(101)) for n in range(101)))```

mossy field Jul 9, 2023, 3:36 PM

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I'm gonna assume 7 is roughly correct and not 70

craggy cradle Jul 9, 2023, 3:36 PM

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I definitely also think the answer is somewhere in the vicinity of 7 or 8

mossy field Jul 9, 2023, 3:44 PM

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time for bug shooting

#

i got this:

#

8.272762261358117021221262743023522568438947036096886968992420305991112859521374678798456200358718664479445474886907388314467437407815104544672135616700214435254905728799469000716373294105484261117746

#


def cache(function):
    cache = {}
    def f(*args):
        if args in cache: return cache[args]
        result = function(*args)
        cache[args] = result
        return result
    return f

@cache
def wins_average(turns_completed=0, probability=1):
    if turns_completed == 100: return 0
    return probability * (1 + wins_average(turns_completed+1)) + \
        (100-probability) * wins_average(turns_completed+1, probability+1)

print(wins_average())

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@lament forge

craggy cradle Jul 9, 2023, 3:45 PM

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Im posting the problem on math.stackexchange

mossy field Jul 9, 2023, 3:46 PM

#

link

craggy cradle Jul 9, 2023, 3:46 PM

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Ill send u one when its posted

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👍

mossy field Jul 9, 2023, 3:46 PM

#

sure

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also just to keep active

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.close

lone heartBOT Jul 9, 2023, 3:46 PM

#

Channel closed

Closed by @mossy field

Use .reopen if this was a mistake.

mossy field Jul 9, 2023, 3:46 PM

#

.reopen

lone heartBOT Jul 9, 2023, 3:46 PM

#

✅

lament forge Jul 9, 2023, 3:54 PM

#

wins_average(98) gives 200 when the correct answer is 299 (1% chance you get it on the first try, 2% chance you get it on the second try, is roughly a total of 3%)

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the issue is that your 1 + counts more if it's further on because it will get multiplied by more copies of 100, which isn't correct

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i'd suggest using fractions instead of trying to write it directly in integer arithmetic

#

(fractions is rationals implemented using integers so they are exact)

#

(i'm going to go now but i'll hopefully be back in somewhere between 10 and 20 minutes)

mossy field Jul 9, 2023, 4:05 PM

#

After fixing I also get 7.86080126403229668198182908097394362384354574898764667238115242755521947164335122692038436352382356040222738777625219891364911130702970910977870565447613758476042568092806020045109899810694869084004791

#


def cache(function):
    cache = {}
    def f(*args):
        if args in cache: return cache[args]
        result = function(*args)
        cache[args] = result
        return result
    return f

@cache
def wins_average(turns_completed=0, probability=1):
    if turns_completed == 100: return 0
    return probability * (100**(100-turns_completed) + \
            wins_average(turns_completed+1)) + \
        (100-probability) * wins_average(turns_completed+1, probability+1)

print(wins_average())

craggy cradle Jul 9, 2023, 4:05 PM

#

https://math.stackexchange.com/questions/4733301/fair-lottery-suppose-a-lottery-exists-in-which-your-luck-increases-as-you-pla let me know if you have any suggestions, i rarely use stackexchange

Mathematics Stack Exchange

“Fair” lottery, suppose a lottery exists in which your luck increas...

Problem:
Suppose a lottery exists in which your luck increases as you play. The probability of winning a prize starts at 1%. After every unsuccessful attempt, the probability of winning increases b...

#

Ill gladly edit it

mossy field Jul 9, 2023, 4:06 PM

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ok sure

#

Edit out "fair", there's nothing fair about it lol

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also i would suggest not writing 8.1918 ± 0.00015, maybe my script has a problem

#

instead give the current exact estimate we have @craggy cradle

craggy cradle Jul 9, 2023, 4:08 PM

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Will do

#

@mossy field any ideas on what I should call the problem instead?

mossy field Jul 9, 2023, 4:13 PM

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Lottery in which your luck increases per turn but resets once you win

craggy cradle Jul 9, 2023, 4:14 PM

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Hurry

#

Look at new comment

#

They might be onto something

#

@mossy field

mossy field Jul 9, 2023, 4:16 PM

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ok sure

lament forge Jul 9, 2023, 4:19 PM

#

i don't think they are
i tried the computation they came up with and it gives an answer of 1058791184067875423835403125849552452564239501953125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000/129277986730885202106151856642773382549665465071825090615034369359178917814747670780594211447306244001059786961886915926297133393516168977984471526892507817 which is around 8.19 and not the same as the two scripts

mossy field Jul 9, 2023, 4:19 PM

#

lament forge i don't think they are i tried the computation they came up with and it gives an...

yah well see my comments

#

wait how is it an overestimate

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ah

#

no flooring

#

bye gtg, accept my friend requests

#help-0

Please read #❓how-to-get-help

Please read #❓how-to-get-help