help-27

[wheat umbra]

somebody give me an intuition for adjoint operators

[devout snow]

@wheat umbra Has your question been resolved?

[raven plaza]

No.

[wheat umbra]

@raven plaza wtf?

then why answer

[devout snow]

@wheat umbra Has your question been resolved?

[devout snow]

@wheat umbra Has your question been resolved?

[wheat umbra]

<@&286206848099549185>

[tawny pewter]

i had to endure this in class last semester

but tbh i don’t get it

[devout snow]

@wheat umbra Has your question been resolved?

[tawny pewter]

https://www.youtube.com/watch?v=L1gd5PrTcUY This seems not too bad too

[devout snow]

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[sick valley]

$\lim_{z\to \pi n} (z-\pi n) \left[\frac{\cos(z)}{\sin(z)} - \frac{1}{z}\right]$

[woven radish]

jandro0103

jandro0103

[sick valley]

can i do this ?

sorry i messed

wait...

[woven radish]

jandro0103
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[sick valley]

can i do this ?

also : the limit on the right approach 0

can i just evaluate the limit on the left side by using de l'Hospital ?

i mean (if i apply the limit on the right side) :

$\lim_{z\to \pi n} (z-\pi n) \frac{\cos(z)}{\sin(z)} - 0$

[woven radish]

jandro0103

[sick valley]

pls help

[viral heath]

isn't (z- pi*n) just 0, so the ans is 0?

[sick valley]

sin(pi n) = 0 also

so its 0/0

but my question is another

[viral heath]

right

[sick valley]

.

[viral heath]

idk

[sick valley]

<@&286206848099549185>

[sand dove]

remember that sin(z) = (-1)^n sin(z - pin)

so then recall sin(t)/t when t-> 0

[sick valley]

but

im not asking on how to evaluate it

[sick valley]

im asking if i can do this

[sand dove]

you're asking whether you can do lim(f+g) = lim(f) + lim(g)

if you know one of f or g converges

[sick valley]

anyway thanks for the tips

[sand dove]

then yes

if lim(g) exists and is finite

[sick valley]

i know that lim(g) is approaching 0

[sand dove]

lim(g) = 0

here

[sick valley]

yes

[sand dove]

so you're allowed

because if f diverges

then f+g diverges too

and if f converges

then f+g converges too

[sick valley]

ok so when i cant do this ?

[sand dove]

if you have convergence of neither f nor g

[sick valley]

so if both diverges?

[sand dove]

you need to make sure at least one converges

otherwise it's false in general

[sick valley]

okok

[sick valley]

so this is correct ?

[sand dove]

yes

[sick valley]

that i writed -0

[sand dove]

(when n is not 0)

(otherwise it's -1)

[sick valley]

oh yes

[sand dove]

also did a little oopsie

forgot a (-1)^n

anyways

that's about it

[sick valley]

but cant i just apply the l'hospital since its 0/0 ?

[sick valley]

here

sin(pi n) = 0

[sand dove]

uh

with which functions

[sick valley]

i mean when i get to this point

$\lim_{z\to \pi n} (z-\pi n) \frac{\cos(z)}{\sin(z)}$

[woven radish]

jandro0103

[sick valley]

i can apply the l'hospital

[sand dove]

how would you apply it here then

with function is your "numerator"

and which function is your "denominator"

(there's also easier than l'hôpital)

[sick valley]

$\lim_{z\to \pi n} \frac{(z-\pi n) \cos(z)}{\sin(z)} = 0/0$

[woven radish]

jandro0103

[sand dove]

that's gonna cause you unnecessarily problems imo

[sick valley]

why?

[sand dove]

idk bringing in the derivative of cos(z)

harder computations

[sick valley]

N) cos(z) + (z- pi n) * -sin (z)

D) cos(z)

[sand dove]

don't you know the limit of sin(x)/x?

[sick valley]

so its (-1)^n / (-1)^n = 1

[sand dove]

when x -> 0

[sick valley]

isnt this enough ?

[sand dove]

(z-pin)(sin(z))

just to show we make a lot of mistakes

with unnecessary computations

[sick valley]

sorry

[sand dove]

at least get the cos(z) out of the fraction

cos(z) -> (-1)^n

[sick valley]

(-1)^n / (-1)^n

[sand dove]

yes that seems easier

[sick valley]

so its 1

anyway

what were you suggesting?

[sand dove]

well

sin(x)/x -> 1 when x -> 0

(x complex)

[sick valley]

ok

[sand dove]

$\lim_{z\to \pi n} \cos(z)\frac{(z-\pi n)}{\sin(z)} = \lim_{t\to 0} (-1)^n\cos(t)\frac{t}{(-1)^n\sin(t)} = 1$

[woven radish]

rafilou is not not born in 2003

[sick valley]

mm

so you sub t = z - pi n?

oh no

[sand dove]

yeah

[sick valley]

ok i got here

[woven radish]

jandro0103

[sick valley]

mm now ?

[sand dove]

cos(t+pin) = (-1)^n cos(t)

same with sin

[sick valley]

ok

[sick valley]

so we use lim(f*g) = lim f x lim g ?

[devout snow]

@sick valley Has your question been resolved?

[devout snow]

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[devout snow]

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[lavish patio]

Given a n x n array with unique entries, with an equivalence relation of a possible rotation by 90k and then a possible reflection across a line that is 45k (k is any natural), how many, and what, are the equivalence classes

[lavish patio]

i solved this problem for 2x2, using differences in diagonal entries, but the general case probably needs linear algebra or smth

no idea where to start with the general solution

most likely needs a case for even and odd n, as odd n results in n times more classes just due to the center

another wording of the question is, how many different initial states would you need if you needed to get all n! possibilities given the transformations

and how would you obtain one of every initial state

[honest lion]

If I understand your question correctly I believe if you are given the location of the 4 corners of your array you can figure out the position of all the other elements. Thus, considering 2x2 should be enough

[lavish patio]

how come

oh i figured it out, you only have to consider quadrant amount and n mod2

ty
.close

.close

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[zenith viper]

wouldnt tan pi/6 be sqrt3?

[pseudo basin]

no

[faint zinc]

1/sqrt(3)

[pseudo basin]

pi/6 is 30°

[sand quarry]

Try to use the 30-60-90 triangle I mentioned to you in the previous help channel

Again don't memorise those

[zenith viper]

wait can i use the circle?

this one?

[pseudo basin]

sure

[honest meadow]

Yes!! One way is to memorize the triangles, the other is to just use sin / cos to find tan

[pseudo basin]

remember the coordinates are (cos, sin) in that order

[zenith viper]

wait so

OH i got it now

we cant have a sqrt for denominator

thanks

[pseudo basin]

eh.

[honest meadow]

You can but generally we rationalize the denominator

[pseudo basin]

yeah like you shouldn't treat the presence of roots in denom as something inherently illegal lol

[zenith viper]

sorry I have another question

how do u graoh this?

i find graphing very confusing

im guessing its based off of the angle so it would probably be d

but how can i be so sure? (The answer was d im just checking if there is something else I can look at incase)

[pseudo basin]

y = sqrt(3)*x goes thru the origin (as does any line of the form theta=something)

a and c are out on that alone

and it's got positive slope greater than 1

[zenith viper]

oh i see

thanks

[devout snow]

@zenith viper Has your question been resolved?

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[zenith viper]

.reopen

[devout snow]

✅

[zenith viper]

how is this done?

[jagged harbor]

how have you been doing polar-to-rectangular conversions normally?

[zenith viper]

idk i usually find the polar coordinates

[plucky maple]

I recommend visualizing this

Then it will be easy to figure out

[zenith viper]

so i should graph it?

[jagged harbor]

you know how an angle corresponds to a point on the unit circle?

[plucky maple]

Yeah sure

[zenith viper]

sort of

[jagged harbor]

you take this standard correspondence with your given angle, then scale both coordinates by the given radius

[zenith viper]

the words are kinda confusing sorry

[jagged harbor]

i.e. an angle theta corresponds to the point (cos theta, sin theta)

then scale everything by the radius

[plucky maple]

Just make the relevant sketch, it’ll be easy to understand then

[jagged harbor]

the sketch makes more sense visually

[knotty sage]

this might help

[zenith viper]

oh ill try that

[knotty sage]

forgot the right angle but still the idea is there

[zenith viper]

wait 2pi/3 is r or y for me?

[knotty sage]

2pi/3 is the theta polar coords are always (r,theta)

[zenith viper]

wait but its in place of y in (x,y) form

so like do i use -9 as x and 2pi/3 as y?

[knotty sage]

doesn't matter it represent something differently

[knotty sage]

no

polar coord use a distance (radius) and an angle to show position
where cartesian only show position

[zenith viper]

then what do i use to insert into the formulas?

sorry im just super confused

[knotty sage]

the point given is in polar form (r,theta) and you want to convert to rectangular (cartesian)

[knotty sage]

you can use this

ofc don't forget the rectangular is represented as (x,y)

[zenith viper]

so i find y using 4sin theta and x using rcos theta?

then the answer i get for those is the coordinates for the rectangular

[knotty sage]

it'll be -9sin(2pi/3)

for y

[zenith viper]

oh ok

and -9cos(2pi/3) for x?

[knotty sage]

yep

[zenith viper]

or do we find that some other way?

ok thanks

sorry is it ok if u kinda show how ud solve it to end up with a fraction? when i use a calculator i get a decimal

and im pretty sure the question is looking for fractions

[knotty sage]

ok

[knotty sage]

are you familiar with reference angles and special angles ?

[zenith viper]

yes

[knotty sage]

what is the reference angle of 2pi/3 ?

[zenith viper]

120?

[knotty sage]

still not the reference angle (that's just the same angle in degrees)

[zenith viper]

oh sorry i meant the angle was 240

so 4pi/3

i accidentally subtracted 360 from 240 after getting it as an answer when subtracting 120

[knotty sage]

the reference angle is (pi-2pi/3)=pi/3
but you still need to consider quadrants
the angle is in quadrant to (so the cosine value should be negative and the sine will be positive)
-9cos(2pi/3)=((-1)(-9)cos(pi/3),-9sin(pi/3))

[zenith viper]

oh ok is there a step after that?

because in the answer key it says that the answer is:

[knotty sage]

ofc you need to find the values of sin(pi/3) and cos(pi/3)
pi/3=60deg

[zenith viper]

oh i got it im so dumb i forgot u can get sin and cos from the coordinates on the circle

[devout snow]

@zenith viper Has your question been resolved?

[devout snow]

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[tawdry harbor]

not sure how to simplify it

[knotty sage]

factor out the 1/3 and it's just in form of af(x)
integral of af(x) is aF(x)

[tawdry harbor]

Thank you!

.close

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[remote sun]

How to check for c and d

[oak bane]

what happens to f(x), f'(x) as x approaches infinity

[remote sun]

f'(x) becomes a value bw 0 and 1

f(x) diverges to infinity

ohhhh

I see so after some point f(x)>f'(x)

[oak bane]

yeah

[remote sun]

right and since f(x) diverges the sum must diverge too

so no beta exists

[oak bane]

yea

[remote sun]

thank you very much!

.close

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[junior jewel]

How do I solve for perimeter of region

[light fractal]

try connecting the transition points between the segments and the circular arcs to the centers

in order to figure out the measures of the circular arcs

[junior jewel]

so then each arc would be 60?

[light fractal]

not quite

[junior jewel]

or 120

[light fractal]

yeah its 120

360 - 90 - 90 - 60

[junior jewel]

or js 360/3

but nvm

[light fractal]

why would it have to be 360/3

[junior jewel]

idk if this is right but each circle is 360

[junior jewel]

didnt really indertsnad that

[ebon coyote]

It's more that, if you connect the centres to each other, you get an equalateral triangle

[junior jewel]

oh

[light fractal]

you subtract the blue angles from 360 to get the orange angle

[ebon coyote]

^

[junior jewel]

ah

yeah i didnt even learn that yet

[ebon coyote]

That angles around a point sum to 360 degrees?

[junior jewel]

i said that wrong

[junior jewel]

i knew that

[junior jewel]

but i didnt think to solve it like he did

[ebon coyote]

Ah

This is less of something that is "taught"

This is general problem-solving skills

[junior jewel]

thanks

i think i got it

[ebon coyote]

In this case, explicitly:
- if we connect the centres, since they have the same radii, we get an equilateral, so its internal angles are all 60 degrees
- if we connect the centres to the point where the "perimeter" diverges into a straight line by leaving the curve of the circles, then these lines are connecting to tangents of the circles
- this gives us rectangles drawn connected to the main equilateral triangle
- so the other blue angles are each 90 degrees

[junior jewel]

oh

okay

[ebon coyote]

You can sorta see that you do know these individually, but the main trick is to force it out of the question and convince yourself (and a reader - in this case just drawing the lines and right-angles will probs be enough) that what information you're drawing out is logical

And then bringing those things together

[junior jewel]

oka

.close

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[somber garnet]

this is so confusing

[clear tree]

Im guessing he wants you to fjnd the hypotenuse that the 2 distances of the trains make

[frozen aurora]

first find how far away train A is at 11 am and how far away train B is at 10 am

[devout snow]

@somber garnet Has your question been resolved?

[clear tree]

Yeah you can find the distance of train A at 11 am and the distance of train B at 10 am and then try to find the wanted time from there

[devout snow]

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[somber garnet]

how

[somber garnet]

can someone help

[frozen aurora]

you know the time and you know the speed

thus you can find distance

[somber garnet]

like how much they traveled in 1 hour?

[somber garnet]

how do i find B's distance?

[frozen aurora]

yes

[frozen aurora]

it traveled for 1 hour at 45 km/h

[somber garnet]

towards the station right

[frozen aurora]

yes but that doesn't matter

well it does, but not for the distance

it was at some point an hour before arriving to the station

[somber garnet]

wait so basically

i find the distance of the 2 train

[frozen aurora]

it's the same way you did it for train A

[somber garnet]

then the hypotenuse?

[frozen aurora]

not sure what you mean by "the hypothenuse" so can't say if it's correct or not

rn you are finding the "boundary values"

[somber garnet]

huh

well

what am i supposed to do after finding the distances

[frozen aurora]

is this a calculus or no calculus question

[somber garnet]

calculus

[frozen aurora]

great

[frozen aurora]

what are they btw

as in what did you get

[somber garnet]

45 for a 60 for b

[frozen aurora]

correct

now let's consider what's happening at some time t

how far away is A from the station at time t?

[somber garnet]

can we let x represent some distance

[frozen aurora]

(we can let t be in hours)

[somber garnet]

oh rlly?

why?

[frozen aurora]

why not

[somber garnet]

nvm

i thoght you said

we cant

mbbb

okay yea

ohhhhhhhhh wait

so we can rewrite the distance at time t

using v = d/t

?

[frozen aurora]

yes

[somber garnet]

okay

okay

whats next

[frozen aurora]

what did you get

[frozen aurora]

for this

[somber garnet]

for a: 60t for b: 45t

wait

do we care about how far they traveled after 1 hour

[frozen aurora]

not quite

A is correct, B is wrong

[somber garnet]

oh

sign conventions?

[frozen aurora]

how far away from the station is B at time t?

at t=1 hour, how far away is B from the station?

[somber garnet]

wait a sec

45-45t?

[frozen aurora]

yes precisely

[somber garnet]

ohh so we want their position relative to station

[frozen aurora]

yes

[somber garnet]

got it

okay

whats next

[frozen aurora]

it would be helpful to draw a little diagram to help visualize this

[somber garnet]

i did

1 sec

,rotate

[frozen aurora]

great

[somber garnet]

do we find the hypotenuse now lol

[frozen aurora]

yes

[somber garnet]

omg

[frozen aurora]

you find its length in terms of those two equations

[somber garnet]

okay

let me try

[frozen aurora]

hint if needed: ||distance between two points formula||

[somber garnet]

uh

why are we

using distance formula

cant we use a^2 +b^2 = c^2

[frozen aurora]

oh yes you are right that's sufficient

sorry it's 2:47 am here 😭

[somber garnet]

no worries

i genuinely appreciate it (but you should consider sleeping)

[frozen aurora]

i will

[frozen aurora]

what do you get for c?

[somber garnet]

sorry 1 sec

😭

[frozen aurora]

no worries

[somber garnet]

we need deirvaive riht

[frozen aurora]

after you get the expression yea

[somber garnet]

uhhhhhhhhhhhhhhh

[frozen aurora]

what's up

[somber garnet]

i got..

0.086..

😭

[frozen aurora]

oh damn

no that ain't right

show your work

[somber garnet]

😭

d is the hypotenuse btw

[frozen aurora]

bro........

it's (60t)^2

[somber garnet]

OMG

[frozen aurora]

and (45-45t)^2

the whole thing

[somber garnet]

okay i did that for 45

i forgot the otyher one

[frozen aurora]

I've done that mistake SO many times

[somber garnet]

mbbb

[somber garnet]

😭

omg

i hate this

[frozen aurora]

i knowwww 😭😭

[somber garnet]

okay

new anser

naswe

i got

0.93

istg 😭

[frozen aurora]

still wrong :(

show work

[somber garnet]

omfg

wait

[frozen aurora]

the answer is gonna be nice

[somber garnet]

oh what

[frozen aurora]

there's something wrong on line 3

look closely

[somber garnet]

huhhh

[frozen aurora]

wait actually on line 2 aswell

[somber garnet]

wait

did i do the binomial expansion wrong

[frozen aurora]

,w expand (45-45t^2)^2

[woven radish]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by arte_m_etra

[frozen aurora]

ye

[somber garnet]

OH

omfg

sorry

wait

[frozen aurora]

don't do the whole thing

let's take it step by step here

[somber garnet]

okay :(

[echo tangle]

could i get some help when you guys are done?

[frozen aurora]

no shame in that – simply sparing you the misery of having to redo a question 5 times from scratch :)

[frozen aurora]

take a look in #❓how-to-get-help

[echo tangle]

ok, sorry

[somber garnet]

i rlly appreciate it 😭

idk why

im not getting a perfect answer

[frozen aurora]

show

[somber garnet]

i swear

omg

😭

[frozen aurora]

wait

fuck i didn't notice that

why t^2?

it's (45-45t)^2

[somber garnet]

omfg

omfg

omfg

[frozen aurora]

no two squares

[somber garnet]

IM SO SORRY

WTF HELP

😭

sorry

[frozen aurora]

nah i missed it too 🫠

it's okay

[somber garnet]

im acc tripping

okay

1 last time.

i swear

4th time's the charm fr

[frozen aurora]

you got this broski

[somber garnet]

OKAY

OKAY

i got

0.36

[frozen aurora]

yessir

that's your final answer

4th time was the charm indeed

good job!

[somber garnet]

yayyyyyyy

sorry btw

😭

and tysm for your patience

[frozen aurora]

don't apologise

it's why I'm here lol

[somber garnet]

i have an optimization test tmr, do you have any tips cuz i feel like with optimization there's way too many different types of problems to study

[somber garnet]

🥹

[frozen aurora]

it's allll about getting the model right

cuz after that it's just derivatives

[somber garnet]

yea the model is so hard omg

like howd you even think of this

[frozen aurora]

so try to use both fornulas and common sense

plug in values to test your model

and you want everything to be in consistent units and (most likely in your case) dependent on exactly one variable

[frozen aurora]

the question asks to find the time, so i figured using t will be helpful

the only issue is that it's not always like that, for example of they asked to find the least distance between trains you would still pick time to be the variable

that's because time is consistent between both trains: their velocities and distances are completely unrelated

[somber garnet]

okayy

[frozen aurora]

if you tried picking distances as your variables, you'd either get stuck with two variables, or eventually stumble upon what we did here anyway

[somber garnet]

yea omg 😭

[frozen aurora]

i thought i reacted to this lol but my internal reaction was like "that's just not gonna work"

[somber garnet]

lol

is it bc

it says what time

so we need to use time as variable

[frozen aurora]

you have like 3 different distances

distsnce between trains and the station, distance between the trains themselves

you can't pick one x

[somber garnet]

ohhhhhhhhhhhhhhh

[frozen aurora]

but! all of those depend on time

[somber garnet]

and they have the same time?

[frozen aurora]

yes

[somber garnet]

oh okay

[frozen aurora]

they all depend on the same time variable

which is why we use time

so yeah i think that's about it i can give advice-wise

[somber garnet]

ohh i see

thank you smmmmmmm

[frozen aurora]

of course :)

any more questions?

[somber garnet]

i think imight later 😭

you should rest tho if its late there

[frozen aurora]

I'll probably go to bed now so i think you'll end up talking to someone else

are you from the US?

[somber garnet]

nope im from canada

[somber garnet]

okayy sleep well

[frozen aurora]

aha i see

[somber garnet]

🙇

[frozen aurora]

7 hour difference damn

[somber garnet]

yeaaa

are you from ukraine

[frozen aurora]

yes

i live in sweden for now tho

[somber garnet]

oh for uni?

[frozen aurora]

yes

[frozen aurora]

well if that's it for now, feel free to type ".close"

[somber garnet]

oo so nicee

gn

.close

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[royal grail]

i need help knowing which trig function to use for which right triangle scenario.

for example:
in the first one, i am given the opposite angle and hypotenuse. this means i use sin.

for triangles #2 and #6, they are the opposite angle & opposite side and adjacent angle & adjacent side. how do i know what function to use for these?

and am i getting any others wrong?

please ping when responding :)

[wispy oyster]

there are no trig functions for opp/opp

that is just 1

we are related ratios of triangle's sides to the angles

they either need to give u two sides (means u r likely finding an angle) or one side and one angle to find the other sides

[royal grail]

what function would i use for triangle #2 then? since they're both opposite, there's no function that would work

[devout snow]

@royal grail Has your question been resolved?

[royal grail]

whatever i'll try later

[devout snow]

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[tough vigil]

!help

[devout snow]

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

[tough vigil]

idk know what to even do

[supple knot]

Try partial fractions and write it as a telescoping sum

[tough vigil]

no it does not telescopes

no it does

yeah

i got 99 which is correct

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[placid trout]

i got my answers but how do i graph it?

horiz tan at (2,-1)

vert tab (1,0)

d^2 y/dx^2 = (-2t+4)/(9t^5)

[lilac moat]

on paper or with a computer/calculator?

[placid trout]

paper

[lilac moat]

make a chart, t|x|y

for t = -1, x = 0, y = 3

and so on, for enough values of t that you get a curve

[placid trout]

ohh ok

thanks

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[hybrid maple]

I’m double integrals

I don’t know how to horizontal strips for this example
I feel like I have to split it up but I’m not sure

[devout snow]

@hybrid maple Has your question been resolved?

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[unreal falcon]

This just confused the hell out of me what happened

[unreal falcon]

The y= 3+3.5lamda to z = -1 - 4 lamba also does not make sense to me

oh wait

Why did they pick x to be the lambda?

[supple knot]

You can pick any of the three. x, y, or z. x is just conventional I think

[pseudo basin]

one of the unknowns gotta become a parameter

there's no sacred meaning to picking x specifically for the role

[unreal falcon]

Oh ok and I get why the x in the directional is 1 but why is the x in the positional vector 0? is it always zero when the unknown is the lambda?

[pseudo basin]

x = 0 + λ

[unreal falcon]

ok i get it now thank you

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[barren gulch]

Can someone help me with this

[versed juniper]

,rccw

[versed juniper]

what's the question

quadratic equation for what

[barren gulch]

For the area of shaded region which is the triangle

[versed juniper]

the area is base times height over two

what is the base and height?

[barren gulch]

I tried to find the base QS using pythogaras theorem but I don’t think i did it correctly since the answer is diff

[versed juniper]

you know the height doesn't have to be inside the triangle

[barren gulch]

But i thought h is supposed to be in the middle of triangle

[versed juniper]

nop

it doesn't have be

[barren gulch]

Oh

[versed juniper]

if you are not convinced, mirror it over the base

it's a parallelogram

which is also base times height

[barren gulch]

Wdym? 😭

Im confused

[versed juniper]

not really mirror but if you add another triangle it makes a parallelogram

area if parallelogram is bh

so the triangle is 1/2 bh

and if you arent convinced that a parallelograms area is bh then i also have something to show you

[barren gulch]

Ohhh i see”

I understand noww thank youu

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[night cypress]

Could someone help me understand what did they just did

[patent token]

Which step did u not understand

[night cypress]

The one that I have marked

How top red went to bottom red

[patent token]

oh they used the half angle formula

wait a sec

They used this formula

I dont remember the derivation tho

[night cypress]

Huh

This exists

[patent token]

Yea

[night cypress]

Didn't they opposite of that?

They started with 1-cos

[patent token]

u mean 1 - cos^2 = sin^2?

[night cypress]

1-cosA = 2sin²A/2

Got it

[patent token]

yea

[night cypress]

Thank you!

[patent token]

np

[night cypress]

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[jaunty mantle]

is this a game

something is missing here

[faint zinc]

@steady marsh every 4th Y means literally if you number the Ys that occur with indices, then the ones numbered 4, 8, 12, etc are guaranteed Xs? Or does it mean that if you have a string of 3 Ys in a row the next one is guaranteed to be an X?

[jaunty mantle]

you are asking for P(X) but clearly it depends on which Y you talk about because of the every 4th Y thing

so are you really asking what is the P(X, n) where n is the number of trials?

each trial here will have a 4% to be Y and of those 25% will be X, and every 4th Y will be X as well

so the probability X occurs is a function of how many trials are done

[faint zinc]

Maybe the value of the number of X/n as n->inf?

[jaunty mantle]

i dont think that's what's stumping you because i dont think you've precisely asked your question

[faint zinc]

This sounds like a gacha system.

[jaunty mantle]

is this what you're looking for

100 in particular?

[faint zinc]

And I'm guessing you want to know the total occurrence rate of X per pull

As the number of pulls get large

[jaunty mantle]

this is not good because each pull has a different probability

you can ask of its limiting probability yes but is that what you're looking for

you need to be clear

any finite trial will not be exactly this limiting probability, but will asymptotically approach it

[faint zinc]

(especially because it will take a lot of pulls to reach the limiting probability, so it's probably only relevant to whales rather than f2players)

[jaunty mantle]

well okay in this case we can probably just use the limiting probability

i think given the numbers thousands of trial are sufficiently large in magnitude?

i would say tho

if you arent good at probability you can simulate this

it will do a great job and not be too difficult

it's a good skill to be able to write simple scripts like this!

[faint zinc]

So every time you pull an X the counter is reset, so we have

X
YX
YYX
YYYX

Each of these we can label with a probability of appearing.

X = 25%
YX = (75%)(25%)
YYX = (75%)^2 (25%)
YYYX = (75%)^3

We can then label each string with the proportion of Xs

X = 1
YX = 1/2
YYX = 1/3
YYYX = 1/4

And finally perform a weighted average

1 * 0.25 + 1/2 * 0.75 * 0.25 + 1/3 * 0.75^2 * 0.25 + 1/4 * 0.75^3

,w 1 * 0.25 + 1/2 * 0.75 * 0.25 + 1/3 * 0.75^2 * 0.25 + 1/4 * 0.75^3

[woven radish]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by omnipotententity

[faint zinc]

A little bit under half

[jaunty mantle]

this is the non-math perspective on conditional probabilities lol

but i really like it for explanatory purposes

[faint zinc]

It's only ≈ 0.5 so it's slightly less than 2% but yes

[jaunty mantle]

there's a similar mechanic in world of warcraft actually

it's a sort of badluck protection

[faint zinc]

And Fire Emblem Heroes

[jaunty mantle]

everytime X fails the occur when Y happens, the probability that X happens when Y happens increases until 100%

[faint zinc]

Actually, FEH has 3 layers of bad luck protection

[jaunty mantle]

so here it stays at a constant 25% until the 3rd fail

[faint zinc]

Which makes calculating probabilities obnoxious as hell lol

[jaunty mantle]

the mechanic im describing goes the first fail makes the next one go from 50% to 70% then to 90% then 100%

they just make the odds hard as hell to actually calculate haha

that we can calculate :p

that's what probability is for

we can quantify exactly what the variance will be

[faint zinc]

I had a streak of luck so bad I went full tilt and made an old spreadsheet where I kept track of my pulls, to verify that the odds were more or less the same as advertised. And keeping track of the probability state was maddening lol

Which, I'm F2P, so I don't even have skin in the game

[jaunty mantle]

ive just realised that your wording of variance is not what i was thinking

the % are deterministic by number of trials it's not random

[faint zinc]

Exponential distributions have a very long tail, but have probability mass concentrated near the start

So it will probably take thousands of pulls to approach the limit

Yes

[jaunty mantle]

i mean given n trials, E[the number of appearances of X / n] is a deterministic number

for n trials, the expected number of appearances of X is not random, so the % at each trial value is deterministic

it's a sequence of % that approaches the 2% value

uh let's see, this seems really complicated 😭
P(X = 0, 1) = 0.96 + 0.04 * 0.75 = 99%
P(X = 1, 1) = 0.04 * 0.25 = 1%
P(X = 0, 2) = 0.96 * 0.96 + 0.96 * 0.04 * 0.75 + 0.04 * 0.75 * 0.96 + 0.04 * 0.75 * 0.04 * 0.75= 98.01%
P(X = 1, 2) = 0.96 * 0.04 * 0.25 + 0.04 * 0.25 * 0.96 + 0.04 * 0.75 * 0.04 * 0.25 + 0.04 * 0.25 * 0.04 * 0.75= 1.98%
P(X = 2, 2) = 0.04 * 0.25 * 0.04 * 0.25 = 0.1%

so this P(X = 1, 2) means the probability that with 2 trials, X shows up exactly once

ideally we have some formula for P(X=k, n)

because then E[X]/n is just 1/n sum k=0 to n of kP(X=k, n)

$\mathbb E[X, n] = \sum_{k=0}^n kP(X=k, n)$

[woven radish]

frosst

[jaunty mantle]

that's the formula for the expected number of X occurences in n trials

but idk if P(X=k, n) has a nice closed form

no yeah i know what you mean

you're looking for |E[X, n]/n - 2%|

how far is this exact value from the limiting value for each n

but the problem is i cant think of an easy way to find E[X, n]

it's an interesting question for sure

[devout snow]

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[twilit scroll]

I need help with 31 if anyone knows how to do this

[honest meadow]

what have you tried so far?

[wind mason]

Do you know how this would be represented?

[twilit scroll]

no i dont know where to start

at all

[wind mason]

Okay.

When you have f(g(x)), you want to take the input and plug it into g first.

[twilit scroll]

so log 5v

?

[wind mason]

Then plug the resulting output into f.

[wind mason]

Huh?

[twilit scroll]

whats the input

[wind mason]

You tell me. We can start with part a.

[twilit scroll]

ok

so

plug 6 into 3x+7?

[wind mason]

Yes.

[twilit scroll]

so 6(3x+7)

[honest meadow]

what's your understanding of composite functions?

[twilit scroll]

nothing

[honest meadow]

https://www.youtube.com/watch?v=ZFPkQkURSxk&themeRefresh=1

how about you watch this

and then see if that helps with your question

[wind mason]

Do you know what g(6) represents?

[twilit scroll]

no i funny

dont

[honest meadow]

https://www.youtube.com/watch?v=1xATmTI-YY8

id recommend watching the videos so you can see how much you understand and so we can help clarify content

[wind mason]

It represents the value of x that you’re going to subsitute for g.

Hence why g(x)=3x+7.

[twilit scroll]

can you show me the steps its alot easier for me to learn that way

[wind mason]

Okay.

So we have g(x)=3x+7.

g(6)=3(6)+7.

Whatever you see inside the parentheses, you replace the x for it.

Does that make sense?

[real mason]

The function g(x) is basically saying, you give me x and I give you 3x+7

[twilit scroll]

yes

[wind mason]

Okay. Try to apply the principle, then.

[twilit scroll]

so log 5 25?

which =2

[wind mason]

Yes.

[twilit scroll]

ok so for B

its different

and idk where to plug in 125

is it

log 5 125 =3x+7

that doesnt make sense

[wind mason]

Okay.

Which one is on the outside? f or g?

[twilit scroll]

g

g(f(125))

[wind mason]

Okay.

The one on the outside is applied last.

You apply the innermost one first.

Does that make sense?

[twilit scroll]

so so i need to put 125 into log5

so log 5 125

3

?

so

x=3

3(3)+7

[wind mason]

Yes.

Which is?

[devout snow]

@twilit scroll Has your question been resolved?

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[pallid stream]

i need help with a geometry task:
there is a rhombus ABCD with an acute angle a. A belongs to a line that divides the angle a by a ratio of 1:3 and |CD| by a ratio of 2:3.
prove that cosa = 1/8

[pallid stream]

so i ended up with the last equation and earlier i decided to do something like cos(3*1/4a) = cos(3b) but it didnt really work and it made things a lot messier

[devout snow]

@pallid stream Has your question been resolved?

[devout snow]

@pallid stream Has your question been resolved?

[devout snow]

@pallid stream Has your question been resolved?

[devout snow]

@pallid stream Has your question been resolved?

[pallid stream]

<@&286206848099549185>

[devout snow]

@pallid stream Has your question been resolved?

[muted sapphire]

Have u solve it?

[devout snow]

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[sand quarry]

Hey, so this is a semi-mathematical, semi-programmatic question I have. I am trying to implement an algorithm that computes the integer square root of a number. Meaning, for a non-negative integer $x$, the largest integer $r$ that satisfies the inequality [
r \2 r \le x
]
There are several algorithms one could use, but my question is to gain a better intuition on the following iterative, numerical method:
\begin{itemize}
\item Start with a guess $r = x$
\item Repeat: [
r = \lfloor\4{r +\lfloor\4xr\rfloor}2\rfloor
]
\item Stop when $r$ stabilises.
\end{itemize}
I can't really understand the mathematical intuition behind why the above should eventually converge. Could I gain some insight on this?

[woven radish]

Aero ݁˖ ❀ ⋆｡˚

[stone stump]

if r is bigger than the sqrt, then x/r is smaller

and you are taking the average of them

so you get put somewhere in the middle

and then you repeat

so intuitively the numbers r and x/r get closer and closer together

and always one bigger than sqrt, the other smaller

(if you didnt round then this would converge to the actual sqrt)

[sand quarry]

ahh that makes sense now

If your guess r is too big, then x/r is too small
If r is too small, then x / r is too big

thanks

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[stuck vortex]

is there any formula to find the sides of a polygon
i remember studying something liek that in class but cant recall

[winter patrol]

length of sides or number of side
and with what given info

[stuck vortex]

calculate number of sides @stuck vortex

[autumn girder]

given?

[stuck vortex]

angle

[winter patrol]

what angle

[autumn girder]

of?

[stuck vortex]

interior angle is given

of a polygon

156 degree

[drifting mauve]

strange

[winter patrol]

as in all interior angles are 156 degrees?

[drifting mauve]

no polygon has 156 degrees sum of internal angles 💀

[winter patrol]

as in this is a regular polygon?

[winter patrol]

supposedly not what they mean

[drifting mauve]

oh

[stuck vortex]

The diagram shows part of a regular polyugon
The interior angle of the polygon is 132 degree larger than the exterioor angle

Calculate the number of sides of this polygon

[drifting mauve]

hmmm so all them are 156 degrees

[winter patrol]

the sum of the exterior angles is the constant 360°

[stuck vortex]

so x + 132 + x = 180

so x = 24

[winter patrol]

and getting the number of those will get you the number of sides

[stuck vortex]

24 + 132
we get 156 degree as intereio angle

[drifting mauve]

i think n * 156 = (n-2) * 180 something like this maybe

[winter patrol]

i.e.
(number of side/interior/exterior angles)*(your exterior angle x) = 360

[drifting mauve]

i get 15 from this hmmm

since all the sides are 156, the sum should be n * 156 as there are obviously n vertices then just apply the angle formula...

hmmm

[stuck vortex]

i swear tehre was a formula to find the no of side

[winter patrol]

yes there is

and i've mentionedit

[stuck vortex]

what

$(number of side/interior/exterior angles)*(your exterior angle x) = 360$

[winter patrol]

yes

[woven radish]

BrutalCandor

[winter patrol]

try applying that

[stuck vortex]

found the formula

thanks guys

.close

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[winter patrol]

There's another that uses the exterior angle which is more efficient

which saves you a few lines of work

[devout snow]

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[violet vault]

Can anybody help me with z tables?

[violet vault]

I don't understand why the solution in this case is Z=+/- 1.96

Cause our critical z value is -2.12, and our alpha/2 is 0.025

So to me, our Z value should be 0.1357 to like 0.0951

[supple knot]

you need a wider z table

[supple knot]

-1.96 is not shown here

[violet vault]

Yeah, it's not shown anywhere, but given my critical z value and alpha, it shouldn't be greater than one, right?

This is what I'm given

I found the Z-test and the alpha values

Z_test is -2.12 and alpha/2 is 0.025

So I should get a value that's in the region of 0.0132 and 0.017, right?

I genuinely don't understand where they get their 1.96 from

[supple knot]

the critical z value and alpha values you found are from the data. it has to be measured against the significance level alpha = 0.05 to see if you reject or do not reject the null hypothesis.

[violet vault]

I think I understand that part, I just don't understand where they got their 1.96 value from at all cause it's in no place in the Z table except for the values go that high for the critical z values

[supple knot]

.

[violet vault]

I've looked everywhere online and I can't reverse engineer this.

Ok, could you show me an example?

Cause here's the thing

GIVEN that critical z value (-2.12), that should be my first column, furthest to the left, right?

[supple knot]

You're working backwards

You're supposed to find the z value from alpha=0.05

[violet vault]

And GIVEN that alpha is 0.05, that would put me at like ROUGHLY some value between 0.012 or something

[violet vault]

But that's a whole column??

I'm fonfused

[supple knot]

What's a whole column

[violet vault]

Like z could be anything in that column

[supple knot]

Nope. Should be 1.96

[violet vault]

You're saying all I have is 0.05

Cause to me, my critical z values was this column...

[supple knot]

Yea you're not reading z tables correctly

[violet vault]

So how do I read it correctly? I get that like this is an area under the curve

And the area under the curve can't be more than one

[supple knot]

https://www.statology.org/how-to-use-z-table/

[violet vault]

So that's why I'm so confused about it being 1.96

And I looked online and everything's about how to find the area

Like I'm not given a percentage to find the critical z value, I thought that was this column

Is the z-score my alpha?

Could you just walk through this problem with me? Cause the link you sent me is the same thing as every other tutorial

I would need the alpha and the percentage to find the z value, but I thought that was my -2.12

@supple knot sorry to bother you, I just seriously don't find any relevance in the tutorial you sent me. Is there any way you could walk me through one problem?

[supple knot]

You have to read the z table correctly first

On the column under z, -3.4 is just the first part

Each column afterwards corresponds to the next decimal place

-3.40,-3.41,..., -3.49

Are the z values for the first row

Then the entries in the table are the area under the normal distribution to the left of that z value

[violet vault]

The next decimal place of what? My z value?

[supple knot]

Yes

[supple knot]

Hence

Stare at that until it makes sense

[violet vault]

Ok, so my z value is -2.12

My alpha is the first row, right?

So if I go to -2.1

The ROW of -2.1

And I go to the COLUMN of my alpha, which is 0.05

It unsurprisingly does not give me -1.96

Or positive 1.96

[supple knot]

No

There is no column of alpha

alpha is the value in the table

[supple knot]

.

[violet vault]

You're saying that0.5987 is my area?

[supple knot]

Yes

[violet vault]

God damn that took way too long to get an answer for. I looked literally everywhere for that information

Ok

[supple knot]

This z table tells you the value (area under normal distribution) left of z=0.25 is 0.5987

[violet vault]

So in my case we're look for the ROW -2.1, and the alpha 0.05

[supple knot]

No

alpha = 0.05 is a value in the table, not the first column or row header

Find z=-1.9

Then move to the right until 0.06

Tell me that alpha value

[violet vault]

So I don't have the table going far enough to the right

Cause there is no 0.05 in this row

[supple knot]

.

[violet vault]

I thought my critical z value was -2.12? Is the first row the critical z values?

Somebody should have put axes on this bitch

[supple knot]

#help-27 message

[supple knot]

.

[violet vault]

So somehow I was magically supposed to know that my answer would be -1.96

So ALL I'm given is my alha, that makes sense

[supple knot]

Since this is a two sided test, alpha = 0.05 is split into area to the right of 1.96 + area left of -1.96

[violet vault]

But then what are the columns? What are those headers?

[supple knot]

0.05/2 = ?

[violet vault]

0.025

[supple knot]

What's the value in the table at -1.96

[supple knot]

I explained that here
#help-27 message

[supple knot]

.

[supple knot]

.

[supple knot]

.

[violet vault]

This is explaining the first column, I'm talking about the header of each column, meaning the first row

With like the 0.00 and 0.01 and so on

[supple knot]

Yes keep reading

[supple knot]

.

[violet vault]

You're a saint, riemann thank you so much for helping me out. I'm instantly much more relieved cause I couldn't find this information ANYWHERE

[violet vault]

So it's just the decimal place of the alpha value?

[supple knot]

No

[supple knot]

Do you see this list

[supple knot]

And this table as an example

[violet vault]

Yes but it didn't make any sense

[supple knot]

0.05 is the 0.05 in 0.25

[supple knot]

You need to understand these things before trying to jump ahead

[violet vault]

Oh, so that's my alpha values and my row x column entries are my alpha/2?

[supple knot]

If you don't say you don't understand and you jump ahead, that signals you're either not paying attention or you understand what I wrote

[violet vault]

I paid attention and didnt' understand

If I could have understood immediately, I would have

[supple knot]

Then don't jump ahead

[supple knot]

Did you understand this