help-49

[lyric charm]

good question, how do you know that (-2, 1, 7) doesnt work

let u = log(x), v = log(y), w = log(z) [all to base 2025]

you get:
u(u+v+w) = 48
v(u+v+w) = 12
w(u+v+w) = 84
and you added all three to get (u+v+w)^2 = 144 yes?

[last slate]

I did it in a different way

[lyric charm]

wait actually i am not getting the same values as you now

[last slate]

wait how did you get that?

wait wow

i didn't see that

[lyric charm]

i just rewrote the equations in terms of my new variables

[last slate]

i can form a perfect square!

[last slate]

yes, i follow

yk what i did?

[lyric charm]

show what you did then

anyway i was gonna follow with "therefore u+v+w = ±12"

[last slate]

i divided,
so u/v = 4 so, u = 4v
v/w = 1/7 , so w=7v
and then i put everything in terms of v in the first equation

but your way is undeniably neater

[last slate]

okay
wait! i found my mistake
log(x) = +-4

so mine matches u + v + w = +-12 as well!

how do i proceed?

[lyric charm]

well you've found that |u| = 4, |v| = 1 and |w| = 7 then

[last slate]

yes

[lyric charm]

with what signs can you add these to make 12?

[last slate]

if and only if they are of same sign

[lyric charm]

i.e. with what can each ± be replaced to make this true?

±4 ± 1 ± 7 = 12

yeah so

only (4,1,7) and (-4,-1,-7) work

[last slate]

okay so i was supposed to find value of a+b+c somehow, and make sure the sets i choose satisfy it?

wait this is pretty clever. i was supposed to check for sets that work and i've been given a factor in the question itself that includes all those variables, so i should pay attention to that to make sure my solution holds.

[midnight plank]

@last slate Has your question been resolved?

[midnight plank]

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[safe notch]

how many consecutive seats in a circle of 8
chairs

[lyric charm]

bro got 3 people confused wtf he means

[vivid yoke]

consecutive seat?

[lyric charm]

!xy

[midnight plank]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[scenic wyvern]

is this a question of philosophy or what

[austere blade]

it's probably a permutation question

[lyric charm]

@safe notch you there?

[safe notch]

yeah

[safe notch]

lol

hahaha

sorry

[austere blade]

could you show the original problem?

[safe notch]

say we gotta make the group of 2

my original prob

is

[safe notch]

bahahahhaahhaha

[scenic wyvern]

i remember you were taught how to find circular permutations given that two or more objects must be next to one another?

[vivid yoke]

Count when their seat's next to each other

[safe notch]

thats what im working on

[safe notch]

nooo

where tf

lmfao

[vivid yoke]

What have you tried?

[safe notch]

well

in the note

it has treated

their sitting positions as one block

and permuted along the rest

[sudden yacht]

Yes

[lyric charm]

well yeah subtractive counting

they count all possible arrangements, then the ones violating the IN-PK constraint and subtract those

[safe notch]

but previously i had done problems like this in linear case with the trick of consecutive positions

so i wanted to do with that

[lyric charm]

what trick?

[safe notch]

in a line how many consecutive seats there

are

like that

[sudden yacht]

You'll have to be more precise

[safe notch]

in a seat of seven find how many ways that 3 girls will always be together along 4 boys

it was the question

in the seat of 7
there are 7-3+1= 5 consecutive seats

on which the 3 girls can sit in 3! ways

so in 5 consecutive seats 5*3! ways

along boys in the rest of 4 seats

5*3!*4! ways

i wanted to do this like this

[lyric charm]

this seems to be nonsense

[safe notch]

no it isnt

[lyric charm]

or at best, it's the number of places where the girls could sit

but calling it "number of consecutive seats" is bs

brb

[safe notch]

ahmm

what should i say then

its a linear case

the boys and girls are to sit in a row

[lyric charm]

.

in a circular arrangement, the number of ways k people could sit together around a table of n seats is actually always n

regardless of k

[midnight plank]

@safe notch Has your question been resolved?

[midnight plank]

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[jade magnet]

how do you do the very last part

[jade magnet]

i did i and ii perfectly fine

but now idk if im as confident with d

oh wait

d is just the height

[lyric charm]

indeed it is

[jade magnet]

and i worked out earlier that the volume was abc/6

but it says hence

do i have to use the cos theta that i just worked out

i dont get how im gonna use cos theta to find 1/d^2

without massively overcomplicating the question

actually nvm

i see how

i have a diagram for ABC

so its just abc/6 = (A)d/3

where A is the area of ABC

ill simplify it down to abc/2 = Ad

now the question is what method should I use to find A

idk if i should use costheta to find the height of ABC or change it to sintheta and then use the formula 1/2(absinC)

[vivid yoke]

Holy...., there is a way to prove it without any of this

You should make a diagram first

[jade magnet]

prove cos theta or 1/d^2

[vivid yoke]

1/d^2

[jade magnet]

i did?

[vivid yoke]

a 3d one

[jade magnet]

how am i supposed to do a 3d diagram

bruh

[vivid yoke]

Hold on I will make one

[jade magnet]

i cant use 3d modelling software in a real exam

obviously

so uh i kinda just have to do diagrams of each of the sides on a plane

[vivid yoke]

No one teach you how to make one?

[jade magnet]

yeah?

what's with this

[vivid yoke]

what's the distance between O and AB?

can you calculate it using a and b ?

[jade magnet]

root(a^2 + b^2)

wait no

sorry

[vivid yoke]

no

[jade magnet]

root(AO^2 - (AB^2 / 4))

?

[vivid yoke]

Let say OM is perpendicular to AB what's AB

It's $\frac{1}{OM^2}=\frac{1}{OA^2}+\frac{1}{OB^2}$

[grand pond]

Alexis_Fx

[jade magnet]

oh yeah i see what i did wrong

i assumed that M was the midpoint between A and B

but it isn't

why 1/OM^2 though why didn't you write it as OM^2 at first

[vivid yoke]

The base should be like this

I have a reason to write 1/OM^2

Consider triangle OMC

distance between MC and O is exactly the distance from the origin to ABC

and OMC is also a right triangle with O is a right angle

[midnight plank]

@jade magnet Has your question been resolved?

[jade magnet]

oh okay i finally get it now

it took me forever to understand ts

[vivid yoke]

you should be able to solve it, no?

[vivid yoke]

ngl that cos(theta) look awful

[jade magnet]

yeah i think so

[jade magnet]

what the cos(theta) question?

that part was pretty okay

[vivid yoke]

I mean using that to solve for 1/d^2

Look awful

[jade magnet]

oh, yeah

it said "Hence"

so i felt i had to use both of my previous answers

to solve the last question

[midnight plank]

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[vivid yoke]

oops, okay you solved it anyway

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[harsh nebula]

is the only way trial and error?

[scenic wyvern]

question got cut off

[harsh nebula]

wait i think its good

[scenic wyvern]

determine the probability that:
...

nothing under that

[harsh nebula]

oo this one

[scenic wyvern]

oh ok

[harsh nebula]

stuck on d

[scenic wyvern]

oh you just wanted d), ok

[harsh nebula]

yep sorry

[scenic wyvern]

from my understanding it seems that trial and improvement seems to be the current best way for this

i might be wrong though

[harsh nebula]

ye i think it is

binomial distributions are discrertely distributed

there doesnt exist a n value that gives exactly 80%

[scenic wyvern]

definitely wouldn't be exact, but it's fine as long as it's > 80%

[harsh nebula]

alright thank you

i appreciate ur help! u have anice day

[scenic wyvern]

you too!

[harsh nebula]

.close

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[midnight plank]

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[lime oracle]

hey guyss this is a rlly simple question i know but i still dont understand how to differentiate W of S or S of W like i keep interchanging itttt

[lime oracle]

im making flashcards for formulas and stuff, this is about solving for direction btw for vectorsss

how do i know wheter its N of E or E of N if im solving with numbers and no arrows?

[lavish venture]

W of S means first start at south then go west. S of W means first start at west then go a bit south

[lime oracle]

waitt so an example im trying to understand is

10 newtons west and 15 newtons south

do i just follow the cartesian plan when plotting points?

but what would be an example if its W of S? 😭

[lavish venture]

you’re trying to find the resultant i assume?

[lime oracle]

i alreadyy have Rf and Angle i just keep messing up when solving when it cmoes to the direction 😭

my Rf is 18, angle is 56

(we do whole number round offs from our teacher)

like i swear my only problem is the direction 😔

[lavish venture]

you have to specify what the angle is being taken with respect to

because 56 usually means 56 ccw from (1, 0)

[lime oracle]

okayokay ill tryy this

[gentle hornet]

HOLY handwriting

[lavish venture]

https://tenor.com/view/spongebob-lets-see-how-it-looks-the-gif-7996276

[lime oracle]

WHATTT

[lime oracle]

im still confused icl 😭

[lavish venture]

your handwriting is dope

[lime oracle]

DANGG thank you 🙏 @?!??!@

[lavish venture]

have you drawn a picture

usually what angle results from typing it in to your calculator becomes clear when you have a diagram

[lyric charm]

so here's something that i'm wondering about:

in what cases has the distinction between "S of W" and "W of S" been important for problem-solving?

[lime oracle]

nonooo im visualizing onlyy

[lavish venture]

i’ll draw it

[lime oracle]

to know where the angle is situated at? 🥲

[lime oracle]

thank youuu

immm looking for more examples from my teacher to also understand in my enddd

[lyric charm]

hm

well

[lime oracle]

waittt in one of the pics i noticed she used quadrants with adherence w/ negative and positive signs?

[lyric charm]

angles that are "of south" are close to south

and ditto for the other 3 cardinal points

[lime oracle]

OOO

[lime oracle]

So this one would be south of west because its closer to westtt?

[pearl hull]

Castle short

[lavish venture]

sure but we could also describe it as 90 - theta degrees west of south

both would be correct

[lyric charm]

it's closer to south bc the southward component is stronger

[lime oracle]

ohhh okayy but if i used arc tan would it only be the south of west one? orrr

[lyric charm]

O-O-O is long castles!

short castles is O-O

[lime oracle]

ooooo

[lavish venture]

it would give you the one that "hugs" the x axis

no matter the quadrant

[lime oracle]

okaaayy, THANK UUU GUYSSSS 🙏 I GET IT LOWKEYYY 😭 HAHAHAAHAH

[lime oracle]

okayokayyy

[lavish venture]

🔥

[midnight plank]

@lime oracle Has your question been resolved?

[midnight plank]

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[trim current]

Can someone check this rearrangement for me. Just need some fresh eyes

[trim current]

  laguerre = scipy.special.genlaguerre(n-l-1 , 2*l+1)
  # Evaluate the Laguerre polynomial at 2*r/n
  laguerre_evaluated = laguerre(2 * r / n)
  R = ((2/n)**3 * fact(n - l - 1)/(2*n * (fact(n+l))**3))**(1/2) * (2 * r/n)**l * math.exp(-r/n) * laguerre_evaluated
  return R 

.close

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[keen saddle]

Can someone check my calculation
give me 2 ins ill share them

[keen saddle]

$f(x) = x - \cos^{-1}\left(\frac{2x}{1+x^2}\right)$

[lavish venture]

\cos

[grand pond]

Wumpus Man

[runic hamlet]

(for checking calculations you can always use WA)

[keen saddle]

Now let $x = \tan\theta$

[grand pond]

Wumpus Man

[keen saddle]

so $f(x) = \tan\theta - \cos^{-1} \sin(2\theta)$

[grand pond]

Wumpus Man

[keen saddle]

now i know that 2*theta is between pi/2 to pi

so i can simplify the arccos thingy to

$\cos^{-1} \sin(2\theta) = \pi/2 - (\pi - 2\theta) = 2\theta - \pi/2$

[grand pond]

Wumpus Man

[keen saddle]

so $f(x) = \tan\theta - 2\theta + \pi/2$

[grand pond]

Wumpus Man

[keen saddle]

pls check if this is correct

guess i will have to ping helpers

fml

i swear if someone does a 15m

[midnight plank]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[woeful turret]

!15m

[midnight plank]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[keen saddle]

<@&286206848099549185>

[astral canyon]

!15m

[midnight plank]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[keen saddle]

[astral canyon]

I think you're wrong though. Just try f(0) and you get different results

[pearl hull]

[keen saddle]

.

[woeful turret]

I don't think 0 is in the interval he was talking abt

[woeful turret]

It looks correct to me

[keen saddle]

nice

.close

[midnight plank]

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[keen saddle]

i swear inv trig needs to be banned

i spent way too much time on this q

[woeful turret]

No lol till my level it's nice

Maybe after that it gets harder

[vague citrus]

ping?

[keen saddle]

i did this wrong like 3-4 times

[vague citrus]

oh it was already resolved

[woeful turret]

With practice ull be able to do that almost instantly

[keen saddle]

yup but inv was done 4-5 mo ago

i dont think ill be getting any more practice lol

[woeful turret]

Ohh so maybe ur just a bit out of touch

[keen saddle]

yeahh

[midnight plank]

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[hoary hamlet]

MAT 2023

I understand the reasoning up to e). why is a number being a multiple of 1000 but not 10,000 mean that it can not be a square?

[unborn oracle]

If you square a number all it’s factors are raised to an even power

So what is the solution to 10^x=1000?

[lyric charm]

what are the exponents of 2 and 5 in the factorization of 1000?

[hoary hamlet]

what???

[hoary hamlet]

3

[hoary hamlet]

uhh

3

[lyric charm]

right

therefore at least one of these must be 3 in your number too

[hoary hamlet]

what do you mean?

[lyric charm]

if the last nonzero digit is 5, the number is divisible by 5 at least once more; if it's 2, 4, 6 or 8, the number is divisible by 2 at least once more

[hoary hamlet]

yeah

[unborn oracle]

Yes look at the factors of an arbitrary x. Now when we square it, every exponent of the prime factors is getting multiplied by 2, thus being even

[lyric charm]

xxx5000 is divisible by 2 exactly thrice
xxx2000, xxx4000, xxx6000, xxx8000 are divisible by 5 exactly thrice
xxxy000 for y = 1, 3, 7, 9 is divisible by 2 and by 5 exactly thrice each

in any case there is a prime whose exponent in the factorization is 3

[hoary hamlet]

i understand this

[hoary hamlet]

i do not understand this

[lyric charm]

said prime is either 2 or 5

[hoary hamlet]

yeah

[lyric charm]

you're familiar with the concept of prime factorization right

[hoary hamlet]

the way i visualisedd your explanation of this

is that every time you divide 5000 by 2

for example

it removes a 0

so it removes an even digiti

and adds an odd digit

so 5000

2500

1250

etc

therefore only divisible 3 times

[hoary hamlet]

i am

[lyric charm]

ok right

can you tell me then

what can be said about the prime factorization of a square

[hoary hamlet]

so lets say n^2, its the prime factors of n multiplied together, all squared

[lyric charm]

yes and what about the exponents on each prime

[hoary hamlet]

doubled

[lyric charm]

what will they be

ok and what's the word for a number that's double some natural

[hoary hamlet]

idk what youre looking for 😭

even?

2x?

[lyric charm]

even yes

in fact it's an iff

a number is a square iff all exponents on primes in its factorization are even

do you understand this y/n

[hoary hamlet]

ohh okay that makes sence

sense

is "iff" on purpose or a typo

[lyric charm]

it is on purpose.

it's math speak for "if and only if".

[hoary hamlet]

oh okay

didnt knwo that one

[lyric charm]

helps to know this word, bc it is very common in math.

as is the idea thereof!

[lyric charm]

anyway, continuing on that:

we on the other hand have established that for at least one the primes 2 or 5 (or perhaps both) the exponent will be 3

and 3, at least as of this morning, is an odd number

[hoary hamlet]

so how does this relate back to the question where if a number is a multiple of 1000 but not 10000 it cannot be a square

oh

[hoary hamlet]

prime factors of 1000?

[lyric charm]

of that number ending in exactly three zeroes, rather.

[hoary hamlet]

okay sure

well, if its a multiple of 1000 (number ending in exactly 3 zeroes), doesnt that mean the exponent of 2 and 5 will be at least 3, rather than be 3

[lyric charm]

ending in exactly 3 zeroes means multiple of 1000 and not of 10000

1000 | n implies v_2(n) ≥ 3 and v_5(n) ≥ 3

[hoary hamlet]

oh i think i get what youre trying to say

[lyric charm]

but n not divisible by 10,000 implies the stmt [v_2(n) ≥ 4 and v_5(n) ≥ 4] is false

thus at least one of v_2(n) and v_5(n) is less than 4 while also being ≥3

which means exactly 3

[hoary hamlet]

yeah i was thinking the possibilities are

2^3*5^3

or 2^4*5^3

or 2^3*5^4

which gives 1000 2000 5000

right?

which means both are not even at the same time

or none at all

which means not a square

i get it

or is this not the right way of thinking

[lyric charm]

overcomp but whatever

[hoary hamlet]

okay got it 😭

thanks

.close

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[last slate]

i need help on trig values

[last slate]

hello ?

[abstract birch]

what specifically

do you have a question you are working on

[last slate]

like sin 90 things like that

[abstract birch]

have you gone over the unit circle

[last slate]

no i havent

[abstract birch]

ok i recommend learning abt that since your question is pretty general

the unit circle helps you compute trig values easily

[last slate]

okay show me please

.close

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[bronze mason]

any thoughts on what "give the exact value if possible" means? because I'm pretty sure its always possible to find an exact value using complicated tricks but I don't know any

[wary trail]

just round to 2dp here

[bronze mason]

are you sure?

I'm pretty sure its possible to get an exact anwser here

[wary trail]

arcsin is a bit of a tight one to write in closed form, I don't know how it would be exact anyhow

unless you could write in surds?

even then might be a stretch

[scenic wyvern]

well the calculator already shows you the decimal approximation of arcsin(1/9)

this isn't any nice, common angle

so round it is

i don't think the answer field accepts surds as an answer anyway

[wary trail]

I don't even think one could write it as a surd actually

[bronze mason]

hmmm

I just really hate how this is worded

[scenic wyvern]

i mean it's p clear

if possible, give the exact value, but if the exact value is an infinite decimal, then just round it

[blissful pier]

(also if this is the application I think it is it should take arcsin(1/9) directly as the right answer)

if not, then the directions say round the two decimal places, no?

[sharp coral]

exact values would be expected only for values like [ 0, \frac 12, \frac{\sqrt{2}}2, \frac{\sqrt{3}}2, 1 ]

[bronze mason]

wait

[grand pond]

cloud

[bronze mason]

shouldn't there be 2 values?

that I submit?

[scenic wyvern]

there are

actually wait

yes there are

[bronze mason]

which is pi-angle right?

[scenic wyvern]

yes

[bronze mason]

would be 3.03

thank you guys

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[molten bay]

Any hack for such questions guys?

[steel crest]

just try them 1 by one?

[midnight plank]

@molten bay Has your question been resolved?

[potent stratus]

man thats the hardest question i have ever seen

from which test did u get that

[steel crest]

its super easy

just test the choices

[potent stratus]

yea i am studing the SAT so i am wonderring am i cooked or is this tha AP 😆

[steel crest]

this is not on the sat,this is abstract algebra

group theory

[midnight plank]

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[reef torrent]

Excuse me I don’t understand sequence A at all please help.

[lyric charm]

sequence A consists of the following transformations:

1. A translation: by 9 units left and 7 units down.
2. A reflection: over the line that passes through C and has slope -1.

A) "I understand both of these transformations now. Thank you."
B) "I understand transformation 1, but not 2."
C) "I understand transformation 2, but not 1."
D) "Both of these are unclear to me."
E) "Secret fifth option."

[reef torrent]

B and E

[lyric charm]

ok, explain yourself what you want to say extra

[reef torrent]

I don’t understand slope -1 it’s been awhile

[lyric charm]

are you familiar with slope as a concept

maybe you have heard it as "gradient" instead

[reef torrent]

Uh like

2 to the left up 4

[lyric charm]

it's probably easiest to simply show you the line they're talking about

[reef torrent]

That would be the slope?

[lyric charm]

https://en.wikipedia.org/wiki/Slope

[reef torrent]

Oh yes I do

[lyric charm]

https://www.youtube.com/watch?v=7DPWeBszNSM&list=PL0o_zxa4K1BVzbQjmhQxe5eQwuFQx2oV_

[reef torrent]

Wait so is the reason the line is on the c like that because it goes up by one and over one each time?

[lyric charm]

wdym "the line is on the c"

[reef torrent]

it passes through c

The slope is one because on the line it runs one and rises one on the boxes each time

I think I get it now thank you

[lyric charm]

the slope is -1 not 1

and the fact it goes through C is just explicitly given in the question

[reef torrent]

Yeah

Wait what’s your point

Besides the -1

[lyric charm]

the line's slope and its passing through C are not correlated in any way

[reef torrent]

I said like that meaning the line isn’t straight

[lyric charm]

?????????

the line is straight?

[reef torrent]

Straight as in running through it at a diagram ol

Diagonal

[lyric charm]

it... does run through the grid squares diagonally

i don't know what you're trying to say anymore

[reef torrent]

You made a comment on something I said and I’m trying to explain why I said it

I meant I understand why it goes through c like that now if that makes any sense at all

Thank you thougu.

.close

[midnight plank]

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[waxen shoal]

can someone walk me through this

[lyric charm]

you're supposed to use a 3D graphing utility

of some kind

[waxen shoal]

in the 1st one i know the curve will be on a sphere but unsure how to use it

[lyric charm]

what does this icon mean

[waxen shoal]

so i just plot it in a graphic calculator ?

[lyric charm]

yes

that's what they expect you to do

[waxen shoal]

damn

okay

.close

[midnight plank]

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[waxen shoal]

what about this one

[gaunt nimbus]

so

hows the sketch

[waxen shoal]

its

hard to draw

but can imagine it

cant imagine how the

curve looks like tho

[ocean orbit]

What it is (brief): the curve lies on the unit sphere and is a spherical curve with colatitude equal to twice the longitude. In spherical coordinates (azimuth θ and colatitude φ) it satisfies θ = t and φ = 2t. Using traces the full curve once on the sphere.

Why that is true (math):
Write with Then

x^2+y^2=\sin^2 2t(\cos^2 t+\sin^2 t)=\sin^2 2t,

x^2+y^2+z^2=\sin^2 2t+\cos^2 2t=1,

Parameter domain: use to get the whole curve (it is -periodic). If you take only you get a portion that runs once from north to south and back in a symmetric way, but to shows how the curve wraps around.

How to view it (recommended viewpoints): • Side view (e.g., elevation 30°, azimuth 45°) — shows the spiral/wrapped shape on the sphere.
• Top-down (elevation 90°) — shows the double-winding pattern in the xy-plane projection (you’ll see the figure with four lobes).
• Near-north-pole (low elevation, e.g., elev=5°) — shows behavior near the pole and how the curve departs from it.

[waxen shoal]

damn

i should really get familiar and used to with spherical coordinates

[lyric charm]

bad copypaste

[ocean orbit]

It's not for me to prove myself just a rough idea what the question wants and how to approach it and it's accurate

[lyric charm]

well, wherever you copy-pasted it from, it deleted all of the math notation.

[glossy garnet]

yes, generally if you're going to copy paste something it 1) better not be AI output and 2) could probably just be a link to the source

but also this is the help channel where users are provided help, not given cookie-cutter copied responses :)

[scenic wyvern]

that's AI, isn't it

[glossy garnet]

ok so it's AI output, which is not allowed here

[ocean orbit]

Really it takes a long time to write it

And I did not know about the rules I just joined it

[austere blade]

you can check the rules in #rules

[ocean orbit]

Found it

Do not spam, flood, or post unsolicited advertisements. Do not post responses written by ChatGPT or a similar Al tool.

[scenic wyvern]

there ya go

[ocean orbit]

Won't do it again

But the question was interesting

Been a while I solved such question

[midnight plank]

@waxen shoal Has your question been resolved?

[waxen shoal]

We all Hate ai in math community ???

[ocean orbit]

People have different takes on it some people want to use their own knowledge to solve all the difficult problems on the other hand some people use tools to assist them in order to get the same answer

For me I prefer efficiency and quality

If AI helps me with that I use it but I prefer to understand the concept behind it instead of just be happy with the answer I get

[midnight plank]

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[real star]

Jimmy thinks of 16 different positive integers, all less than 100. Prove or disprove that Jimmy is guaranteed to be thinking of 4 different numbers: a, b, c, d, such that a + b = c + d?

[real star]

I tried pigeonhole principle but got stuck...

[modest oriole]

what is ‘prove that disprove’

[real star]

prove or disprove 😭

mb

[modest oriole]

oh ok

[real star]

any ideas?

[modest oriole]

maybe try thinking 16 numbers such that they can’t make a+b=c+d true

[real star]

...

[midnight plank]

@real star Has your question been resolved?

[midnight plank]

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[upbeat gale]

Hi

[upbeat gale]

can someone try to explain to me why x,y,z are rank-1?

k=1

things like x^2,y^2, z^2 er k=0 ⊕ k=2, but why?

[runic hamlet]

that makes little sense

more context might help

[upbeat gale]

yeah, but i am little lost myself

i'll try

[lyric charm]

!xy

[midnight plank]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[upbeat gale]

I dont have an original problem, but i'll give context

I am dealing with selection rules for quantum physcis. In the book itself there are given questions like <nlm|z|n'l'm'> and then how Delta m or l changes will give you an idea of whether the answer is zero

however, in the actual exam, there will be things akin to z^2, x^2, r^2

instead of x,y,z

I have to realise that r^2 is a pure scalar with rank k=0

[midnight plank]

@upbeat gale Has your question been resolved?

[midnight plank]

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[frozen raft]

i need help with this question, i tried to firstly find the intersection points and i got 4 intersections, (2 pairs) and then i plotted it on mathematica, and i got like two regions which i assumed would give the largest area (you could also just have 1 i think), so i used the first pair of solutions as terminals to integrate the left most enclosed region and i get some random expression in terms of a, so i try to find the derivative and set it equal to zero but when i plot it, there is no stationary point, i do the same thing but change intervals to the second set of solutions which givese me an expression that does have a stationary point but its not a max, and it definitely isnt the correct answer if you plot it.

[midnight plank]

@frozen raft Has your question been resolved?

[frozen raft]

hell nah

but ill try it again

[midnight plank]

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[nova pike]

One have a complex number w, which can be expressed in the form of w = sqrt2 * cis(-2π/3) (in polar form). Now examine the sequence w, w^2, w^3, …, w^n for n is natural. I have to make a question about the sequence and answer it, but I want to find an interesting question about the sequence (other than finding the argument or modulus for any term for the sequence). Preferably a hard question. (Reposted from #math-discussion because I couldn’t find any questions)

[subtle blaze]

do any of these numbers land on an integer complex number? as in, a+ib where a and b are both integers

[midnight plank]

@nova pike Has your question been resolved?

[nova pike]

Is the answer, || yes, every 6th number in the sequence ||? And is there a way of formally proving it?

[vast cipher]

the argument can only be 3 values

[nova pike]

I know that, that’s how I did it

[vast cipher]

did you prove this yet?

[nova pike]

Oh wait I’m pretty sure it’s easy to prove. Like examine the real and imaginary parts separately for every 6 numbers, then one can show when both real and imaginary parts are integer

But I think that’s an easy question, maybe preferably something more difficult?

Also thanks for the question

[simple field]

you could maybe do some kind of intro to polar integration
smth like find the distances | w(n) - w(n-1) | and sum them
then introducing w(1; k) = [ sqrt(2) cis(-2pi/3) ]^(1/k) and w(n; k) = w(1; k)^n and then repeating, find the distances | w(n; k) - w(n-1; k) | and sum them - basically just cutting your arc into smaller pieces
then taking a limit to arrive at an integral
could also this set up as a general case

not that hard overall ig, maybe more just grindy, not that original either haha

[vast cipher]

find the angle between w^2, w, and w^3

easy with cosine law

to be fair i havent actually finished it yet

[simple field]

i dont really know that you can get that particularly hard of a question out of this

[vast cipher]

never mind you end up with a bad nested radical

hold on

woahhhhhhhhh

find the angle between w, w^2, and w^3

using cosine law you get that the answer is ||pi/3 ||

super cool

[nova pike]

I’m interested, how would we do this?

[nova pike]

I’m also interested in this. How would we do this, and can we generalise for any consecutive 3 numbers?

[carmine sigil]

How about considering the triangle formed by any three consecutive values in the sequence and finding the area of this triangle?

this is given by

[
A = \left| \frac12 \begin{vmatrix} 1 & 1 & 1 \ \Re{\omega_1} & \Re{\omega_2} & \Re{\omega_3} \ \Im{\omega_1} & \Im{\omega_2} & \Im{\omega_3} \ \end{vmatrix} \right|
]

[grand pond]

OmnipotentEntity

[midnight plank]

@nova pike Has your question been resolved?

[nova pike]

Is there any nice way to simplify this and make a generalisation?

[carmine sigil]

of course, I just wanted to leave this to you.

Notice that this is a determinant

and we have only column swaps

(which only affect the sign of the determinant)

and scalar multiplication

when moving from n -> n+1

so if the area starting from n is A_n then the area of A_{n+1} is 2 sqrt(2) A_n

[nova pike]

Ohhh okay, thanks

Okay I close the channel. Thanks for those that made contributions

.close

[midnight plank]

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[midnight plank]

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[viral dagger]

4. Some pieces are placed on an 8 × 8 table. There are exactly 4 pieces in each row and each
   column of the board. Show that there are 8 pieces among those pieces such that no two of
   them are in the same row or column.

[viral dagger]

how do i transform this into the halls marriage theorem?

my first thought was defining the family of sets where the family bit is like the columns and the sets are the location of where they are in the column (like the row)?

but im struggling to incorporate the fact that there are 4 pieces in each row and column using that

[runic hamlet]

not sure what you mean by family bit

but one side of the graph are the columns and the other side are the rows

I think that works

[viral dagger]

yeah thats what i meant

huh ok that clicks now visually

actually does it? how do you apply it then

does there being 4 pieces per column translate to the column being able to take 4?

oh yeah your finding if a transversal is possible right

[runic hamlet]

wdym take 4

you have the vertices of your graph

what are the edges

how many edges

[viral dagger]

like 4 arrows

[runic hamlet]

the degree of each vertex is 4, yes

[viral dagger]

are the arrows called edges?

[runic hamlet]

what kind of weird version of the marriage theorem have you seen

that doesnt mention the word edges

[viral dagger]

https://brilliant.org/wiki/hall-marriage-theorem

im using the first example as a visual guide and the algebraic thing for the definition

[runic hamlet]

see the part on bipartite graphs

[viral dagger]

oh

wait V1 and V2?

is it not V and W?

[runic hamlet]

who cares about names

[viral dagger]

hmm so on applying it here

say you make a subset of the verticies of the columns, call it V, it will always hold for |V|<=4 since each column has 4 edges

[runic hamlet]

uhh what?

[viral dagger]

would this be better?

actually is there like a direct proof that is way simpler than what im trying to do

[runic hamlet]

V_1 are the columns

V_2 are the rows

take a subset W of the columns

each vertex in V_1 has degree 4

but also each vertex in V_2 has degree 4

what does that mean for edges ending in W

[viral dagger]

im not too sure

[runic hamlet]

uhh the statement for the graph version doesnt seem to be correct

[viral dagger]

wdym?

[runic hamlet]

it should be for every W subset V_1 it holds that |W| <= |N(W)|

where N(W) is the neighborhood of W

the vertices that are connected to some vertex in W

[viral dagger]

neighbourhood?

is neighbourhood like the other edge endpoints for the ones that are ending in W

[runic hamlet]

yes

lets say |W|=7

how many edges go out of W

[viral dagger]

28

[runic hamlet]

could the neighborhood have <=6 elements?

[viral dagger]

no, since each elements have exactly 4 edges, whcih implies there are atleast 7 elements?

ohh

[runic hamlet]

each element has exactly 4 edges

[viral dagger]

oops

[runic hamlet]

but yes

follow up: after you have proven this, remove the 8 pieces from the board. can you find another set of 8 pieces without same columns/rows ?

[viral dagger]

so basically, after defining this, you write
let |W|=a, the total amount of edge going out of W is 4|W|=4a, since each element of V_2 has exactly 4 edges, then there are atleast a elements of V_2 that share an edge with an element in W, and by the hall marriage theorem, there exists a matching betweem V_1 and V_2 (or smth like this?)

[runic hamlet]

yes

[viral dagger]

oooh

graph theory is fun (if you understand it)

alrighty then, thank you!

.solved ❤️

[midnight plank]

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[runic hamlet]

I personally think the graph version is easier to understand

[viral dagger]

i agree

[midnight plank]

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[night gyro]

what to do

[gaunt nimbus]

f

o

i

l

[pearl hull]

Hi, what have you tried?

[gaunt nimbus]

oh nvm

[night gyro]

i dont know how to begin

[night gyro]

wdm

[gaunt nimbus]

${(a-b)(a+b) = a^2 - b^2}$

[grand pond]

k

[gaunt nimbus]

do u know this

[night gyro]

difference of 2 squares?

[gaunt nimbus]

ye

[pearl hull]

Hold up, let’s not give him any shortcut in the first place

[gaunt nimbus]

also is ${\sqrt{4b}}$ or ${\sqrt{4}b}$

[grand pond]

k

[left hatch]

Prove that the sequence (an = \frac{n!}{n^n}) converges, and find its limit.

[gaunt nimbus]

okieeeee

lets do foil then

:3

[night gyro]

what is foil

[pearl hull]

@night gyro By any chance, have you heard of “distribution law” ?

[night gyro]

ik this

no

[left hatch]

Evaluate the integral (\displaystyle \int{|z|=2} \frac{e^z}{z^2 (z-1)} , dz) using the residue theorem.

[pearl hull]

Cool, can you solve it with the given method?

[gaunt nimbus]

!occupied

[midnight plank]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[pearl hull]

Claim your own channel, check out #❓how-to-get-help

[night gyro]

root a times root a

what is it

idk

[pearl hull]

Ah, you’re having problem with this

[night gyro]

yh

see

[zenith snow]

Sqrt(a) ×sqrt(a)=(sqrt(a))²

[gaunt nimbus]

[ \sqrt{a} \times \sqrt{a} = (\sqrt{a})^2]

[night gyro]

oh ok

so is sqrt a squared

[zenith snow]

a ,yea

[night gyro]

is this right

[grand pond]

k

[zenith snow]

Do u know that sqrt(a) means a^(1/2)

[night gyro]

yes

[zenith snow]

So squaring over sqrt(a) gives u-?

[night gyro]

wait

how m i gna include tht square

[zenith snow]

Squaring over the whole thing

[zenith snow]

(a^(1/2))^2

[zenith snow]

Yea

[night gyro]

ok

then

[zenith snow]

U can cancel out 2

What do u get

[night gyro]

1/2 and 2

?

[zenith snow]

Uh no

[night gyro]

which

[zenith snow]

Yes so it's just a

[night gyro]

yh

[zenith snow]

Oh i got ur ques js now

[zenith snow]

Alr yes
And do u know that u can express two numbers with same power together
m²n²=(mn)²

[night gyro]

yes

[zenith snow]

So here b and a got same power in 2nd term right?

[night gyro]

yh

1/2

like this?

[zenith snow]

So can u write sqrt(a)sqrt(b)=sqrt(ab)

[zenith snow]

Yes and u can express 1/2 as sqrt

[night gyro]

like this

?

[zenith snow]

The brackets r not needed but yes

Dont forget the minus

[night gyro]

nice

[zenith snow]

Yea

Now do the next two

[night gyro]

ok now is other 1 turn

[zenith snow]

I think the sqrt is over only 4

But i cant be so sure

[night gyro]

ahh

les do over both if its easier bc my teacher did not mention anything

[zenith snow]

Alr doing over both

Uk sqrt(4)= ?

[night gyro]

yh 2

[night gyro]

which ever way is easier

so is 2b?

[zenith snow]

So u can write sqrt(4b)=sqrt(4)×sqrt(b)

[night gyro]

yh like this?

[zenith snow]

Both r easy ,the answer will say somthing

[zenith snow]

Yes

[night gyro]

ok

[zenith snow]

Sqrt(4)=2 so u can write it as 2sqrt(b)

[zenith snow]

Yes

[night gyro]

yh

[zenith snow]

And now u have a sqrt(a) in multiplied form with it

Follow the prev method and do the multiplication

[zenith snow]

This colour is nice btw

[night gyro]

like this?

[night gyro]

yes is blue my fav colour

[zenith snow]

Yes

[zenith snow]

Worthy colour

[night gyro]

yes

[zenith snow]

Alr next one now

U can simplify sqrt(4b) the same way first

[night gyro]

to

2b?

[zenith snow]

Nope 2sqrt(b)

Like the prev one

[night gyro]

like this?

[zenith snow]

Yes

Now u got 2sqrt(b) ×2 sqrt(b)

[night gyro]

4sqrt b squared

?

[zenith snow]

Yea

[zenith snow]

Yep

[night gyro]

ok

[zenith snow]

What does square over sqrt(b) give u

[night gyro]

what do you mean

[zenith snow]

(Sqrt(b))²

[night gyro]

b

[zenith snow]

Yea

[night gyro]

ok

[zenith snow]

So now add the both parts up

[night gyro]

so is 4b?

nice

next

[zenith snow]

Remember the minus in between

[night gyro]

now we need both red and blue?

[zenith snow]

Yes u add both of them up

[night gyro]

like this?

[zenith snow]

Yeb

See what cancels out

[night gyro]

final answer?

[zenith snow]

It's-

Minus

[night gyro]

how come - and - is +?

[zenith snow]

a-4b

[zenith snow]

Yes but the minus u r considering here belongs to the very first 2sqrt(ab)

[night gyro]

a ok

[zenith snow]

Yes

[night gyro]

done?

[zenith snow]

Yep

[night gyro]

awesome thank you

appereciate it

[zenith snow]

Np ,u r a smooth helpee

[night gyro]

.close

[midnight plank]

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[vivid yoke]

https://www.youtube.com/watch?v=5jkrXUpOzYU

[gaunt nimbus]

It’s simple

[midnight plank]

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[grave kernel]

sqrt(5-x) = 5 - (x^2)
number of solutions
just wanted to clarify, is the answer 2?
i did it with their graphs, but im having some confusion with a friend, hes saying its gonna be 0

[eternal pawn]

why does he say it's 0

[zenith snow]

He prolly meant one of them will be 0

[cobalt agate]

that's not true either

[zenith snow]

Oh wait i missed the sqrt

[floral ruin]

Still wrong

[zenith snow]

Yes cuz i missed the sqrt

[floral ruin]

I see. No problem.

[grave kernel]

no hes saying that number of solutions will be 0

so

[runic hamlet]

well he's wrong

[grave kernel]

thats wrong

yeah

cool

[floral ruin]

Yup

[grave kernel]

thank you!

[slender walrus]

tell him to justify his result

[grave kernel]

he did some, squaring and like dividing, idk, some algebraic stuff

[slender walrus]

can you show the work and we can point out the issues

[vivid yoke]

I would make a raw sketch of f(x)=5-x^2 and h(x)=sqrt(5-x) instead of doing algebra

[grave kernel]

yeah thats what i did

[grave kernel]

uuhhh, i dont really have that

i mean, its solved out

so

thank you so much tho

.close

[midnight plank]

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[jade magnet]

idk where to even get started on this

im stuck on i.a

actually i can see the weights of 1 ounce, 2 ounces and 4 ounces working

but idk how to show it

[shell wigeon]

i.a and i.b only ask for an example

I mean i.a asks to show that you can do 1 through 7 but you can just write each case

[midnight plank]

@jade magnet Has your question been resolved?

[jade magnet]

oh

okay

wait so it's brute force?

i just have to test each combo

that adds up to 7

[shell wigeon]

?

You found a set of weights already, just show how to use them to get 1 through 7

[midnight plank]

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[tidal turret]

how to prove it?

[tired osprey]

rewrite the traingle part ( I forgot what it's called)

also what have u tried

[tidal turret]

double inclusion

[lyric charm]

venn diagram but color the region for $A \cap C$ black to show it's empty (or something)

[grand pond]

Ann

[lyric charm]

this is what my suggestion would look like

[tidal turret]

what?

[tidal turret]

yeah

if AnC would not be empty then

[lyric charm]

we give 0 shits if A ∩ C were not empty

[tidal turret]

yes

[lyric charm]

yes LHS would otherwise look like this

[tidal turret]

yeah in that case rhs is (AnB)u(AnC)-(AnBnC) if AnC is not empty

venn diagrams surely are beautiful

or (AnB)u(AnC)-(BnC) more simply

[molten bay]

Z_49(+) z_7

Number of order 7 elements

[tidal turret]

.close

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[ebon atlas]

hi guys im sorry if the question is kinda silly and i did look up an explanation but i didnt get it.. anyways, i dont understand why the graphs of the gradient functions of sin(x) must look so different (the latter looks scaled down along the y axis) if they still represent the same quantity essentially?

[twilit jetty]

Im not sure what you mean by this, but the gradient is shifted to the left

for this picture at least

[ebon atlas]

likee why does f'(x) look so different for the two of them?

[twilit jetty]

consider that when youre doing degrees, the function has been horizontally stretched

[ebon atlas]

yeah i get that but im comparing the two graphs

[twilit jetty]

this would be how the function should behave

now if you stretch f(x) horizontally really far,

most of the graph is just this

so what youre seeing isnt just a regular function

[twilit jetty]

you can see here the gradient is a lot smaller

the gradient has been scaled inward by pi/180

[ebon atlas]

OHH wait thats kinda smart

ill think abt that

[twilit jetty]

now for reference, f(x) = sin(x) for the radians graph

and f(x) = sin(pi x / 180) for the degrees graph

try using chain rule on sin(pi x / 180) to find its derivative

remember that pi/180 is around 1/60, its a small number

[ebon atlas]

it alr looks like a smaller value so it makes sense tbh

but when you have the two sine graphs side by side in the image i originally shared

why do they still look the same D:

have they scaled it up for comparison

[twilit jetty]

because the only difference is a horizontal stretch?

you can see the x-axis

did you notice the numbers on the right compared to the numbers on the left?

[ebon atlas]

yes

pi is rlly small compared to 180 :p

[whole kayak]

x axis is clear

[mystic condor]

Radians and degrees, despite being different, are still proportional to each other

[ebon atlas]

yea nvm i understand that

that question was kinda impulsive

they represent the same value

[mystic condor]

They look similar, but because it takes more degrees for a full rotation the gradient is a lot smaller all the time for degrees

[twilit jetty]

you can think of a 1 degree difference compared to a 1 radian difference

[ebon atlas]

can u explain this bit a lil btw

[twilit jetty]

for starters, find the derivative of sin(pi x/180)

[ebon atlas]

okayy

im js learning abt differentiating trig functions rn btw

i might screw it up

[ebon atlas]

the two values r constant

so it should be pi/180*cos(x)?

💀

[twilit jetty]

youre forgetting something

should it be cos(x) or cos(something else)?

[ebon atlas]

um

yea idk

[mystic condor]

You just used the chain rule

[ebon atlas]

i thought it should be cosx

[twilit jetty]

lets remember what the chain rule looks like

,,(f(g(x)))'=f'(g(x))g'(x)

[grand pond]

mtt

[twilit jetty]

do you recognize this picture

[ebon atlas]

D:

why have i not studied this

[twilit jetty]

do you know what the chain rule is?

[ebon atlas]

maybe not...

BUT I THINK U ANSWERED MY QUESTION THO

ill learn that on my own