help-23

[opaque fern]

i forgor about this

.close

[safe radish]

Channel closed

Closed by @opaque fern

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[strong pike]

Can I get some hint. I initially had intuition that triangle BOA is equilateral which is wrong according the given answer.

[strong pike]

How should I proceed to find angle BOA?

[zealous ingot]

join OE

assume angle EOA = a, angle EOB = b

you can prove certain triangles congruent

obtain angle COD = 2(a+b)

your required angle is a+b

[strong pike]

Ok, I'll try

[zealous ingot]

ping me if you got any doubts

[strong pike]

Thank you it worked out as angle COD comes out to be 2(a + b). Then I have to just equate all angles of quadrilateral PDOC = 360 degree to find a + b as 70 degree.

.close

[safe radish]

Channel closed

Closed by @strong pike

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[flint thicket]

I need help

[flint thicket]

I’ll send the question

I need help with number 15

How can I set it up?

[lucid eagle]

what have you tried

[flint thicket]

I’ll show you

I’m getting the wrong answer

[mighty mango]

,rccw

oh nice

[lucid eagle]

right now what you have is if you looked down 36 degrees, not the angle of depression

[flint thicket]

Wdym

Please show me

How do you draw it?

[lucid eagle]

so the angle of depression is like if you were looking straight forward and at what angle you would like see the object or point
so basically this angle would be an angle of depression

[mighty mango]

what a weird term for it 😭

[flint thicket]

I’m a bit confused

So did I draw the angle of elevation?

I thought depression was like looking down

[mighty mango]

angle of elevation is looking up.

[flint thicket]

Yeah but how does that differ from my picture

I though I drew the angle looking down

[lucid eagle]

angle of depression is from a horizontal line looking down

[flint thicket]

Ooooh

[flint thicket]

So it’s always from a horizontal line?

Thanks!

[lucid eagle]

these are the elevation and depression basically

[flint thicket]

Didn’t realize that

.closed

[mighty mango]

no d 🤩 .close

[flint thicket]

.close

[safe radish]

Channel closed

Closed by @flint thicket

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[oak river]

So, I'm a little confused.. When im trying to evalute if 31.4 is opposite, I think it's opposite.. but AI tells me it's adjacent? even when I watch videos it feels like it's opposite and not adjacent, am I wrong? if I'm wrong whats wrong with my thinking

[prisma wren]

it's opposite (to the 57 degree angle). AI is bad at interpreting images- especially math-ey images.

More generally, terms like "opposite" or "adjacent" aren't universal in a triangle- it depends on what angle you're looking at. If you were considering the small acute angle on the bottom right, d would be opposite to it while 31.4 would be adjacent to it.

This is in contrast to "hypotenuse" which does have a universal definition: the longest side in the triangle

[oak river]

Gotcha, thank you appreciate it.. I was going crazy trying to figure out why 3 different AI models saying it was adjacent

.close

[safe radish]

Channel closed

Closed by @oak river

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[real gulch]

What's an example of an affine variety with a nontrivial Picard group? The only affine schemes with nontrivial Picard groups that I can think of , are rings of integers in number fields. Someone suggested elliptic curves? But elliptic curves are projective.

[real gulch]

Variety = integral separated scheme of finite type over an algebraically closed field.

[safe radish]

@real gulch Has your question been resolved?

[safe radish]

Channel closed

Closed due to timeout

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[modern spade]

I want my answer checked for this question: Determine the interval of convergence for the series defined in the image.

[modern spade]

I use the ratio test and determine it’s convergent for all x

[marsh walrus]

yea, i believe its correct

i was trying to see if theres something more clever to be done here by comparison

but this seems the most straightforward

[modern spade]

alr thx

.close

[safe radish]

Channel closed

Closed by @modern spade

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[oak river]

Just wondering if this is correct, and if wrong where did I go wrong! ty

[oak river]

Find (\theta ) to the nearest degree.

[rain basin]

what is the question

[flat frigate]

Orchy

[rain basin]

u need to do arc tan

[oak river]

Ah, yeah I did in the calculation

[rain basin]

yeah so ur right

just make sure its tan^-1

[oak river]

yeah, it is mb on the sketch 😓

ty!

.close

[safe radish]

Channel closed

Closed by @oak river

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[rocky pond]

Am I allowed to row swap as such

[rocky pond]

Also for this question I don’t get why there’s only 2 columns for the linearly indepent set because technically isn’t there 3 pivot columns on the right

[safe radish]

@rocky pond Has your question been resolved?

[rocky pond]

<@&286206848099549185>

[safe radish]

@rocky pond Has your question been resolved?

[safe radish]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[rocky pond]

@tardy mango hi I fixed it

[rocky pond]

but ghe theta is so weird

[safe radish]

@rocky pond Has your question been resolved?

[elfin cobalt]

hi do you still need help with this integral

@rocky pond ?

[rocky pond]

Yes please

[elfin cobalt]

did you need help with rewriting the answer back in terms of x?

[safe radish]

Channel closed

Closed due to timeout

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[unkempt saffron]

Im so confused on how to find the height

Wait

Nvm im still confused

[worldly summit]

do you mean the max height?

[unkempt saffron]

Yeah

Im also confused what the x is

[worldly summit]

the x is the horizontal distance

[unkempt saffron]

Woild I use like the -b/2a?

[worldly summit]

I believe that should work yes

[unkempt saffron]

Also do I need to solve this?

[worldly summit]

,rccw

[worldly summit]

you should be able to just get a number from this yeah

[unkempt saffron]

I went into the negatives

Thatd mean my ball went into the grouns?5

[worldly summit]

yeah that would mean the ball went into the ground so you must've made a mistake

[unkempt saffron]

yeah

1.19x15 for b

And thats 17.85

[worldly summit]

what did you get for your turning point?

15?

[unkempt saffron]

15 was the horizontal distance from the hoop

[unkempt saffron]

How would I find that

[worldly summit]

Oh

[worldly summit]

your turning point is the -b/2a

[unkempt saffron]

And I got a weird number for that

[unkempt saffron]

This

[worldly summit]

Let's go through the questions one by one

What did you get for the max height?

[unkempt saffron]

Ok

Max height

Wouldn't it be just 10ft?

Because thats where the hoop would be right?

[worldly summit]

No

[unkempt saffron]

Oh

[worldly summit]

Your max height is found using the x = -b/2a

and you plug that into your height formula

[unkempt saffron]

So the -0.06223

But im pretty sure thats wrong tho

[worldly summit]

-0.06223 is just your a coefficient

you've got to take your b coefficient and divide it by a

[unkempt saffron]

2a?

[worldly summit]

yes

[unkempt saffron]

My b coefficient

Would be 1.19 multiplied by 15?

[worldly summit]

No

$H(x) = ax^2 + bx + c$

[flat frigate]

Katrro

[unkempt saffron]

Oh

[worldly summit]

the coefficients a and b are the numbers attached to x

[unkempt saffron]

Just 1.19 then

[worldly summit]

Yea

[unkempt saffron]

Then divided bt the first part

And I'd have to plug a number in for v

Which I got told to pick 2p

20

[worldly summit]

yes

that's what your -0.06223 value is

[unkempt saffron]

But how would this be my max height

[worldly summit]

you gotta half that first

And this value represents where your max height is

[unkempt saffron]

Bht wouldn't thst be 9ft underground

[worldly summit]

no

[unkempt saffron]

If i didvide that in half?

[worldly summit]

$x = \frac{-b}{2a} = \frac{-1.19}{2*-0.06223} = 9.56$

[flat frigate]

Katrro

[worldly summit]

So this is where our turning point happens horizontally

So we put this number into the function H

and that's what your max height will be

[unkempt saffron]

Alrigjt

So now its js simple algebra?

[worldly summit]

kind of yeah

[unkempt saffron]

This would be 0.37?

[worldly summit]

What would?

[unkempt saffron]

-B/2A

[worldly summit]

no it's 9.56

[unkempt saffron]

What am I typing in my calculator to get that

[worldly summit]

you just need to type the numbers in the second fraction and it should give you 9.56

[unkempt saffron]

And that would mean I'd miss the basketball shot

[worldly summit]

no we're not up to that yet

Do you understand what the 9.56 value is?

[unkempt saffron]

Oh wait

Yeah

Wait katrro

That would mean it would hit the backboard then rifht?

[worldly summit]

No

not necessarily

This 11.44 value is the max height the ball reaches when you throw it

[worldly summit]

So for part 3, we now have the max height and where it occurs

and now we can move onto part 4

[unkempt saffron]

Which would be determined if if makes it in

Hit backboard

Or just airballed

[worldly summit]

Yeah we need to figure out which happens

this is where the 15ft comes into play

[unkempt saffron]

Because od the horizontal distance

[worldly summit]

Yes

[unkempt saffron]

Would we plug thst into the x?

[worldly summit]

yep

x = horizontal distance

[unkempt saffron]

Which is

9.59

So nvm it would just touch the net

[worldly summit]

no

[unkempt saffron]

Why not?

[worldly summit]

9.59 is the horizontal distance where your ball reaches its max height

But the net is a horizontal distance of 15ft

And we need to find the vertical distance at 15ft using H(x)

[unkempt saffron]

So I have to plug 15 in the hx?

[worldly summit]

Yes

[unkempt saffron]

I got 11.43 again

Which is basically what we got for the max height,[

no?

[worldly summit]

Sorry this was right

9.59 was the vertical distance at 15

I thought you were repeating the wrong thing

[unkempt saffron]

So yeah it would touch the net?

[worldly summit]

Well the hoop is at 10 ft exactly

and we're 0.4 ft away from it

so we'll just say it's an airball since we're underneath it

[unkempt saffron]

Yeah

That makes sdnese

[worldly summit]

Now you just have to figure out if you should increase the speed or decrease it for the ball to land in the hoop

[unkempt saffron]

It would js be increased speed right?

[worldly summit]

yes that's right

[unkempt saffron]

It dosent say I need to find out the exact speed

I just need reasoning

You think

"Since the ball went under and was an airball it would just be logical to increase the speed"

[worldly summit]

you might want to explain a little as to why increasing the speed would make this happen

[unkempt saffron]

Alright

Anyways

Thank you so much for the help tonight

I haven't been understanding much since I switched to online

You helped alot thank you

[worldly summit]

you're welcome

goodluck with the rest of your studies

[unkempt saffron]

Close

!close

close

?close

,close

.close

[safe radish]

Channel closed

Closed by @unkempt saffron

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[high perch]

yo i really need somebody to teach me related rates

i don't understand it and its confusing

i got 2 assignments due in a hour

[timid escarp]

I suppose you are allowed/supposed to use calculus?

[safe radish]

@high perch Has your question been resolved?

Channel closed

Closed by @high perch

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[chrome relic]

how does the denominator

[chrome relic]

simplify like that

[prisma scarab]

Rewrite a = √(a^2)

Then you have √(a² * 1 - a² * (x²/a²))

a² cancels out and you end up with √(a²-x²)

[chrome relic]

ohh okk

rt

ty

.close

[safe radish]

Channel closed

Closed by @chrome relic

Use .reopen if this was a mistake.

[prisma scarab]

Np

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[frozen quarry]

Bro I got something like this on my math test and I literally didn’t do the equation properly

[frozen quarry]

Can’t I just bring 5 to 9 and then multiplying 1,8(2x+4y) ?

[quasi bison]

what do you mean by "bring 5 to 9"?

[molten acorn]

I think they meant substracting 1.8 times row 1 from row 2

[molten acorn]

Have you learnt matrices?

[quasi bison]

ok well let OP explain what she meant.

[quasi bison]

definitely not appropriate for this context

i mean you could but also this is a 2×2 system and you learn these way earlier, before the word "matrix" so much as crosses your lips

[molten acorn]

Alright, I was just thinking that maybe they wanted to solve it in that matrix way. Couldn't it be possible that this question was given as an example for basic matrix systems? Also, I have absolutely no intent to be disrespectful

[quasi bison]

@frozen quarry please explain to us what you meant when you said "bring 5 to 9"

that, or show what that one step will look like when you do it on paper (so we can see if it is valid or not)

[half perch]

use gaussian elimination?

[molten acorn]

That's why I asked whether OP has learnt matrices or not, but I think they dissapeared

[quasi bison]

let nenni come back and answer me

but then again like... if op is just up and gone then there is nothing we can do

[dawn saffron]

i feel like they mean multiply 5 by 1.8 so its up to 9 and we can get direct relation between x and y

or something like that

[quasi bison]

why is everyone suddenly a telepath now

[dawn saffron]

idk i used to do that ig

[molten acorn]

I think it's just the most reasonable possibility

[dawn saffron]

are we overthinking this and the intended solution is just aligning one of the two coefficients?

[quasi bison]

i mean like if i were solving this system i would do it by elimination

[dawn saffron]

same here

[quasi bison]

whether i would eliminate x or y depends on the phase of the moon and whether my left leg itches or not

(which is to say it's about the same effort)

[dawn saffron]

i mean i usually just pick the one which is the most convenient to do

or pick x if idk what to choose

[quasi bison]

i will reiterate that OP i.e. @frozen quarry is online but still gone.

[safe radish]

@frozen quarry Has your question been resolved?

[dawn saffron]

Nenni is in fact probably not gone

[frozen quarry]

.close

[safe radish]

Channel closed

Closed by @frozen quarry

Use .reopen if this was a mistake.

[ember socket]

<@&268886789983436800>

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[kindred venture]

@safe radish

[kindred venture]

oh shit mb

.close

[safe radish]

Channel closed

Closed by @kindred venture

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[tired minnow]

Imma keep it a buck i just need recommendations for where to learn the entirety of calculus 1 including the function part like ik it but most of my sources are scattered around the internet which makes learning quite difficult as each of these sources have a different method of teaching, so i just need somewhere proven, thank you

[zealous ingot]

well ig you'd want to check #book-recommendations

but james stewart's book is pretty good

[tired minnow]

thxs ill check it out

.close

[safe radish]

Channel closed

Closed by @tired minnow

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[river ledge]

I was solving the question of regular expressions to finite automata in FLA. The issue in my question is that I am not able to figure out about splitting the a + b in the later part of the expression

[river ledge]

This is the question I was solving and it's written on top too

<@&286206848099549185>

[plucky elk]

can also try asking in #theoretical-cs

[river ledge]

okay

thanks

.close

[safe radish]

Channel closed

Closed by @river ledge

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[astral glacier]

Well what's the formula for cos 2A

[split fulcrum]

Hello, I'm not sure what to do with this question, my initial thought was to use pithagerous therum to get the adj but that won't work as I would end up with a negative number..

^ this is why I was confused at start too

[astral glacier]

Have you not done angle sum formulae

[split fulcrum]

Oh yea, forgot about that lol

2 secs, let me find my formula sheet

[astral glacier]

Lmao

,tex .double angle

[flat frigate]

Xavier 🌺

[split fulcrum]

3 options:

Cos²A-sin²a
2cos²A-1
1-2sin²A

[astral glacier]

Well which one applies here

[split fulcrum]

Well, I don't have Cos²A so would it be 1-2sin²A?

[astral glacier]

Yup

Also hold on

How the fuck is sin A = 4/√7

What the fuck

,calc 4/sqrt(7)

[split fulcrum]

Exactly, makes no sense...

[flat frigate]

Result:

1.5118578920369

[astral glacier]

Wtf

[split fulcrum]

Exactly

[astral glacier]

,w sin x = 4/sqrt(7) find cos 2x

[flat frigate]

Results provided by WolframAlpha

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

Requested by xavier_dragon

[astral glacier]

I wanna know what wolfram alpha says

Amazing thanks wa

[astral glacier]

Ask your teacher

[split fulcrum]

Ok lol

So do I do the angle formula?

[sharp sentinel]

lmao

[split fulcrum]

Or would it not work

[astral glacier]

I mean I guess?

It will give you an answer

,w arcsin(4/sqrt(7))

[flat frigate]

Results provided by WolframAlpha

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

Requested by xavier_dragon

[split fulcrum]

I'm so confused...

[astral glacier]

Right I guess the complex plane does exist

Honestly just ask your teacher

[split fulcrum]

I'm only doing higher maths lol. I don't understand

[astral glacier]

For now just substitute into the formula

,w cos(2 arcsin(4/sqrt(7)))

Yup that's the expected solution ig

[split fulcrum]

How do I get 2sin²a?

[astral glacier]

Well you have sin A

[split fulcrum]

Yea

I don't understand that bit... sorry

How do I convert it

Do I square sin a?

[sharp sentinel]

sin²a means (sin a)²

[split fulcrum]

And times by 2?

[sharp sentinel]

yeah exactly

[split fulcrum]

Oh!

Now i kinda understand

This correct?

Should I leave as mixed or top heavy?

.close

[safe radish]

Channel closed

Closed by @split fulcrum

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[echo pelican]

I don’t understand shiz😭

[zealous ingot]

you know trigonometry?

[echo pelican]

Uh

Not really

But im supposed to be able to solve this with my knowledge

[edgy breach]

sin cos tan?

[vast trail]

30 60 90 triangles?

[zealous ingot]

yeah like basic definitions only

[echo pelican]

Nope

[edgy breach]

alr whats ur knowledge on 30 60 90 triangles

[echo pelican]

Uh

[vast trail]

e.g.

[echo pelican]

yeah never seen this before

[edgy breach]

yes so imagine 48 is x sqrt3

[echo pelican]

Okay

[edgy breach]

you are trying to find the square of the side 'x'

so basically if x sqrt3 = 48 find x^2

[lavish silo]

Whoops mb

[echo pelican]

why would i need to find x^2

[edgy breach]

because thats what the question is asking

find the square of the side BC

[echo pelican]

OHH

wait so

3x+(xsqrt3)= circumfrence

[edgy breach]

where does circumference come from

[echo pelican]

Idk dont i need to know that

To know what x is

So i can get x^2

[edgy breach]

no, its algebra

solve for x: x*sqrt3 = 48

[vast trail]

then when you isolate x it will be equal to the length of BC

then you square that value

[echo pelican]

Yeah and how am i supposed to solve that withour a calculator

[edgy breach]

thats why it asks for the square

x is 48/sqrt3

[vast trail]

divide both sides by sqrt 3, then square it

[echo pelican]

Yeah i figurwd that out

So

wqit

^3

Idk

[echo pelican]

How do i square it

[vast trail]

multiply (48/sqrt(3)) by itself

[echo pelican]

oH YEAHH

dont i also need to have 48^3

[vast trail]

you do not need to cube 48 here

[echo pelican]

48^3/3

[edgy breach]

48^2

[echo pelican]

Wouldnt that chabge the value

[edgy breach]

cube is completely irrelevant youre just multiplying 48 by 48

[vast trail]

and sqrt 3 by sqrt 3

[echo pelican]

But if im cubing sqrt3

Dont i need to cube 48

[vast trail]

you’re squaring the value here

not cubing

[echo pelican]

OHHHH

shi

Okay but tjat would leave me with 48^2

[vast trail]

“what is the square of the distance (in m)…”

[echo pelican]

So 48x48

[vast trail]

divided by our sqrt3 term squared

and sqrt 3 squared is simply 3

[echo pelican]

48x48/3

[vast trail]

yes! exactly

[echo pelican]

48x48=2 304

2304/3=

Uhh

768

[vast trail]

yep!

[echo pelican]

I HAVE AN ANSWER

WOHOO

[vast trail]

nice job!

[echo pelican]

i got one more

[vast trail]

b + 3w + n = 55
b + w + 2n = 42
b + 2w + 3n = 56

are you familiar with systems of equations?

[echo pelican]

Isnt this the same as those thibgs youd do as a kid

When its like

2 flowers +4 candys=20

😞

And then its like

2flowers=10

Reminds me of tjat

Atleast

[vast trail]

btw
bolts = b
washers = w
nuts = n

[echo pelican]

Yes

I just dont know how to solve it without testing

[vast trail]

the best approach would be subtracting the second equation from the first and third so you can isolate the variables

[echo pelican]

Uhh

Could you elaborate

[vast trail]

yeah so

b + 3w + n = 55
-(b + w + 2n = 42)
—————————
0 + 2w - n = 13

and then we can isolate n here by adding n to both sides and subtracting 13 from both sides to get
n = 2w - 13

then we can do this with our other equation

b + 2w + 3n = 56
-(b + w + 2n = 42)
0 + w + n = 14

then we can plug in n = 2w -13 into w + n = 14 and solve for n and w

afterwards you can plug in the values of n and w into b + w + 2n = 42 to solve for b!

[echo pelican]

I see

Ohhhh

Okayy

I got it

Thanks!!

.close

[safe radish]

Channel closed

Closed by @echo pelican

Use .reopen if this was a mistake.

[vast trail]

gotchu

[neon vapor]

30-60-90 triangle

there is a ratio

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[compact imp]

Please help
1+2=?

[dull sequoia]

3

Asking troll questions can get you punished as well

[safe radish]

@compact imp Has your question been resolved?

[carmine tulip]

@dull sequoia He doesn't speak much English. Excuse him.

[compact imp]

Mr Fort craft can you help me?

[carmine tulip]

sure what's your problem

?

[compact imp]

2x^3 - 5x^2 + 4x - 7 = 0

[carmine tulip]

let me grab a pen and paper

[compact imp]

OK

[meager owl]

you could use the cubic formula

[carmine tulip]

Is this your problem?
$2x^{3}-5x^{2}+4x-7=0\$
or\
$(2x)^{3}-(5x)^{2}+4x-7=0$

[flat frigate]

fort craft 🇵🇸

[carmine tulip]

@compact imp

[compact imp]

2x³ - 5x² + 4x - 7 = 0

[carmine tulip]

up or down?

[compact imp]

Up

[carmine tulip]

ok

[compact imp]

I went to sleep. You'll send it tomorrow.

[carmine tulip]

sure

[safe radish]

@compact imp Has your question been resolved?

[safe radish]

Channel closed

Closed due to timeout

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[molten prism]

Anna wanted to buy 10 identical cans of dog food, but she was short of 11 zloty. She bought 6 such cans of dog food and was left with 3.40 zloty. How much does one can of dog food cost?
Save calculations.

[molten prism]

how do you even do this

[brisk shard]

let the cost of 10 cans be x

cost of one can is x/10

now, since she was short of 11 zloty--> that means she had x-11 zloty

she bought 6 cans of price 6x/10

and had 3.40 zloty left = x-11-6x/10

[molten prism]

i dont understand

[brisk shard]

try yourself, assume the cost of cans to be something as x or y

[safe radish]

@molten prism Has your question been resolved?

Channel closed

Closed by @molten prism

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[round shore]

Real numbers x and y satisfy the following system $$x^3 - 8x^2 + px - q = 0 $$$$ y^3 - 8y^2 + py - q = 0 $$$$ (x+y)^3 - 8(x+y)^2 + p(x+y) - q = 0$$ There exists a unique $k$ such that $pk - q$ is constant for all x and y. What is $pk + q$?

[round shore]

the options are -64,-58,-32,-10, and 26

[flat frigate]

dork9399

[round shore]

what i found so far was if you subtract the first two from the third, you get

$3x^2y + 3xy^2 - 16xy + q = 0$

[flat frigate]

dork9399

[round shore]

I tried solving for q and plugging in to solve for p but that didn't seem to lead anywhere

im kinda stuck for ideas rn

[keen tulip]

Have you tried using Vietas?

[round shore]

how does vietas help

[keen tulip]

\rb*{$x,y,x+y$ are all roots of $t^3-8t^2+pt-q$.}

[flat frigate]

Cooly

[round shore]

oh yeah

but still

how does that help

[keen tulip]

Try it first. This should be quite straight forward.

[round shore]

ah i got it ty

.close

[safe radish]

Channel closed

Closed by @round shore

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[tulip mortar]

I tried thinking
u_5 + 2d = u_7
u_5 - 2d = u_3
u_7 = u_3 + 4d
then we need to get a common term of the sequence then we'll do algebra to find out the dfference but i got stuck cus there were more than 2 variables and we cant work with multiplication from the squared cus its an arithmetic sequence

[knotty carbon]

If u_7 is u_3^2

Then you can substitute u_3^2 into u_7

u_5 is 1

So you have that variable

And then it should be solvable

[tulip mortar]

alr thx

hold on its still 2 variables, u_3 and d?

[severe pond]

no

$a_7 = a_3^2 = a_5 + 2d = 1 + 2d$

[flat frigate]

knief

[tulip mortar]

u_3 + 4d = u_3^2
divide by u_3 on both sides
1 + 1 = u_3

?

[severe pond]

what

where did the d go

you have

[tulip mortar]

oh nvm

cus its adding

ye

[severe pond]

$a_3^2 = a_3 + 4d$ and $a_3 = a_5 - 2d = 1 - 2d$

[flat frigate]

knief

[severe pond]

or $a_3^2 = 1 + 2d = (1 - 2d)^2$

[flat frigate]

knief

[tulip mortar]

$1 + 2d = (1 - 2d)(1 - 2d)$

[flat frigate]

clippy

[tulip mortar]

$that equals 1+2d/1-2d = (1 - 2d)$

[flat frigate]

clippy

[severe pond]

🤔

wut

[tulip mortar]

its squared so then divide both sides by 1 - 2d

[severe pond]

$1 + 2d = (1 - 2d)^2 \iff 1 + 2d = 1 - 4d + 4d^2 \iff 4d^2 - 6d = 0$

[flat frigate]

knief

[severe pond]

why are you dividing

just expand and collect like terms

remember that we know d \neq 0

so we can ignore that root

[tulip mortar]

ok but theres no c term in 4d^2 - 6d = 0 how can i solve it

[severe pond]

wat

$2d(2d - 3) = 0$

[flat frigate]

knief

[severe pond]

d = 0 or 2d - 3 = 0

but d \neq 0

so 2d - 3 = 0 which implies that d = 3/2

[tulip mortar]

d = 3/2

[severe pond]

🔥

[tulip mortar]

thx

.close

[safe radish]

Channel closed

Closed by @tulip mortar

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[severe pond]

yep

[steep magnet]

I understand up until the end of the first line

[severe pond]

you wrote it?

[steep magnet]

No its my lecturers solution

I dont understand how they separated the sin fraction and cos fraction

[severe pond]

what is confusing

[steep magnet]

This bit

[severe pond]

$\frac{\frac{a}{b}}{c} = \frac{a}{bc}$

[flat frigate]

knief

[steep magnet]

Omg 😭

[severe pond]

they just split it into two limits?

[steep magnet]

Yep im just being a dumbass haha

Thanks!

❤️

.close

[safe radish]

Channel closed

Closed by @steep magnet

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[small falcon]

can someone let me know if the work is correct AP physics

[safe radish]

@small falcon Has your question been resolved?

[iron estuary]

i think you used 196 N for the normal force when you should subtract the upward pull

normal force is Fn = mg − Fy = 196 − 90.3 = 105.7 N

then friction = 0.5 × 105.7 = 52.85 N

[tardy mango]

discord.gg/physics

[safe radish]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[exotic stump]

pls i need help with all

[meager owl]

for the first question, the unit is squared bc it's a change in area

so area is going to be two lengths multiplied together always

[exotic stump]

ohhhh okay okay

so everytime i see the area i square it?

[meager owl]

no I'm just talking about the unit

specifically

[exotic stump]

okay okay

[meager owl]

the formulas will be different

[exotic stump]

okay so for the second question

why 9?

[magic junco]

,rccw

[flat frigate]

Couldn't find an attached image in the last 10 messages.

Nov 4, 2025, 2:16 AM

[meager owl]

Ok so

[meager owl]

we know the question gives us the derivative of radius with respect to time

[exotic stump]

yeah

[meager owl]

and we have to find the area when the circumference of 18pi,

so as done there, we first write down the formula for the area of a circle

[exotic stump]

pi r ^ 2?

[meager owl]

yeah

and take the derivative with respect to t on both sides

[exotic stump]

uhhh

[meager owl]

this is an implicit derivative

[exotic stump]

see that’s where i get lost

[meager owl]

implicit differentiation

[exotic stump]

yeah i could do it at first

[meager owl]

pretty much we treat area and radius as if they were functions of time

[exotic stump]

but now i don’t get it

i just don’t understand how to lay it out

[meager owl]

ok so you know the chain rule

[exotic stump]

yes

[meager owl]

the chain rule states

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

[flat frigate]

A MESSAGE

[exotic stump]

hmm

[meager owl]

or in languages notation

$h'(x) = f'(g(x)) \cdot g'(x)$

[flat frigate]

A MESSAGE

[exotic stump]

yes

it’s the same thing?

[meager owl]

yes those two are both chain rule

just written in different notation

now apply the chain rule

but instead of writing A

write A(t)

and r(t)^2

and it may become more clear

[cosmic shoal]

Suppp

[meager owl]

hi

[cosmic shoal]

Oh calc type shi

What question is it?

[meager owl]

explaining implicit, tho I wish I knew how to use TeXit effectively

I was explaining 2

[exotic stump]

sorry

i really don’t understand

😭

[meager owl]

ok let me give you an example not related to this question

[exotic stump]

oki

[meager owl]

actually imma just post my notes and then explain

[exotic stump]

yeah that’s the one i semi understand

[meager owl]

semi?

[exotic stump]

like

kinda

[meager owl]

ill just explain both sides

[exotic stump]

like i can do it

i got 90 on that topic

but now that we got this newer unit i don’t get how they do it

i feel like i need to learn from the scratch

[meager owl]

ok so related rates is just the same

A=pir^2

pretend a and r are functions of time

[exotic stump]

so the new x and y?

[meager owl]

kinda

[meager owl]

here y is a function of x

in the related ratez

[exotic stump]

okay okay

[meager owl]

so try to take the derivative with respect to t on both sides

$A(t) = \pi (r(t))^2$

[flat frigate]

A MESSAGE

[meager owl]

Remember, $h'(t) = f'(g(t)) \cdot g'(t)$

[flat frigate]

A MESSAGE

[exotic stump]

wait so it would be

2pi(r(t) times r’(t)

i think i’m doing something wrong

[meager owl]

no that's correct

[exotic stump]

oh okay okay

[meager owl]

and what happens to the left

taking the derivative of area with respect to time

[exotic stump]

plug in 9?

[meager owl]

not yet

ok so if there is no outside function, what is the derivative of f(t) with respect to t

[exotic stump]

outside function? uhhh f’(t)..?

[meager owl]

correct

[exotic stump]

okioki

[meager owl]

so what is the derivative of A with respect to t

[exotic stump]

1

[meager owl]

no

[exotic stump]

oh

[meager owl]

a is a function of t

[exotic stump]

5

i’m sorry i promise im trying

[meager owl]

no numbers yet

[exotic stump]

okay

[meager owl]

no no your fine

[meager owl]

f is a function of t here

and A is not being composed

[exotic stump]

so we have to compose our own a?

[meager owl]

no

ok I'm using too complex language sorry

[exotic stump]

😅

[meager owl]

just plain and simple

how do you write the derivative of a function with respect to a variable

[exotic stump]

da/dt?

[meager owl]

yes

correct

so what's the full equation

when you take the derivative with respect to t on both sides

you got this

your doing great

[exotic stump]

ummmmmmmmmmmm da/dt = f’(x)?

wait

take da/dt from both sides?

[meager owl]

no

take d/dt on both sides

[exotic stump]

why just d alone?

[meager owl]

$[
\frac{d}{dt}(A) = \frac{d}{dt}(\pi r^2)
]$

wrong

wait

[exotic stump]

there’s a formula for that?!!

[flat frigate]

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

[exotic stump]

oh oh

[meager owl]

sorry I wrote it wrong originally

[exotic stump]

i didn’t even know

[meager owl]

ok so d/dt is the derivative operator

[exotic stump]

what’s that

[meager owl]

it just means

take the derivative of what's in the perenthesis with respect to x

it's kind of like you know in algebra

if you have $x^2 = 1$, you square root both sides: $\sqrt{x^2} = \sqrt{1}$

[exotic stump]

uh

maybe

[flat frigate]

A MESSAGE

[exotic stump]

ohhh to cancel the ^2?

[meager owl]

yeah

and then you simplify

you can see it as applying the square root operator to both sides of the functioon

[exotic stump]

i’m really lost now though

[meager owl]

same way, we apply the derivative operator to both sides first to just show we are going to

$[
\frac{d}{dt}(A) = \frac{d}{dt}(\pi r^2)
]$

[flat frigate]

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

[meager owl]

and then we just take the derivative

[exotic stump]

okay hold on

question 9 is the one we are doing right?

i’m gonna write down what i know so far based on what we have gone over together right now

[meager owl]

yeah question 9

write on the back

btw all work on 9 looks correct

[exotic stump]

ya i got answers

but i dont get it

is that right?

it looks wrong

[meager owl]

not really but your doing fine tho

[exotic stump]

oh oh d/dt only on one side

right?

[meager owl]

take your area formula

[exotic stump]

uh huh

[meager owl]

write d/dt on both sides of it

and then take the derivative with respect to time

using area and radius as functions of time

[meager owl]

ok not quite

[meager owl]

I meant this

[exotic stump]

but you said derivative 🥹

okay i’m gonna go shower to try and refresh my brain then i’ll come back is that okay?

[meager owl]

I don't have to say derivative operator on both sides

[meager owl]

np

[exotic stump]

okioki

[meager owl]

ping me when you are back

[safe radish]

@exotic stump Has your question been resolved?

[exotic stump]

no im back

[exotic stump]

i don’t know what that means 🥲

[meager owl]

@meager owl

ping

[exotic stump]

ohhhh

oki so

okay i think i have a better way for me to understand

maybe you can explain what’s happening here?

i understand how to get da/dt now

but where does d/dr and dr/dt come out of?

[safe radish]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[exotic stump]

i understand 2pir is derivative

huh?

.reopen

[safe radish]

✅ Original question: #help-23 message

[meager owl]

idk where d/dr comes from that's incorrect

[meager owl]

ok but what values do the question tell you

[exotic stump]

5 and 18 right?

[safe radish]

@exotic stump Has your question been resolved?

[meager owl]

yesh but do those values mean

[meager owl]

like

what are they

[exotic stump]

oh

circumference and rate of radius

[meager owl]

rate of radius does not have enough information

well I guess it's implied

but I'll just ask it

what is the radius changing with respect to

[exotic stump]

5 in respect to time?

i did some research and found out d/dr just means find the derivative

[meager owl]

it means the derivative with respect to radius

well derivative with respect to r

but r in this context means radisu

[safe radish]

@exotic stump Has your question been resolved?

[safe radish]

Channel closed

Closed due to timeout

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[graceful lichen]

!status

[safe radish]

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 was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[zealous ingot]

its like a center of mass thing

[graceful lichen]

yes

[zealous ingot]

just write the two equations then

assume forces to be a,b

$\frac{bd}{b+a} = k$ and $\frac{ad}{a+b} = k + \frac{d}{4}$

[flat frigate]

Donkey

[safe radish]

@graceful lichen Has your question been resolved?

[graceful lichen]

How do you get it?

[safe radish]

@graceful lichen Has your question been resolved?

Channel closed

Closed by @graceful lichen

Use .reopen if this was a mistake.

[safe radish]

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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[sly aurora]

I am struggling to understand this proof.

[quasi bison]

there are 2 proofs here

also hi name buddies lmao

[silk cove]

Ann with Ann

[sly aurora]

Hiii, so I am referring to the first one

I am absent from the lecture

[quasi bison]

i thought somehow a channel got opened in my name lol

[sly aurora]

Hahaha

[quasi bison]

lemme think about how best to express the basic idea of the proof

[sly aurora]

Alright, I will patiently wait :)

[quasi bison]

so basically you split up the interval (0,1] into ranges like

(1/2, 1]
(1/4, 1/2]
(1/8, 1/4]
etc going down by powers of 2 like this

and then you "turn each range upside down"

so (1/2, 1] gets mapped to [1/2, 1) and so on for all the others

does that make sense

[sly aurora]

Hmm, kind of, what I do not understand is why we are splitting the interval into these ranges or rather how do we know into which ranges to split?

[safe radish]

@sly aurora Has your question been resolved?

[safe radish]

@sly aurora Has your question been resolved?

[smoky nacelle]

ann and ann

[sly aurora]

Everyone is saying Ann and Ann...

yet still I have not understood the proof..

[smoky nacelle]

you must wait for ann ig

[sly aurora]

I am waiting, do not worry... for enlightenment? for something? for the end of the world?

[smoky nacelle]

I do not think Ann is gonna cause the end of the world.

Yet.

Maybe by the time she's over 40 she'll be done with humanity and Doofenshmirtz us all into the parallel (euclidean) universe

[sly aurora]

That is a very specific way to cause a calamity.

I approve of this.

[smoky nacelle]

uh, hit no or else itll timeout ig

[sly aurora]

Oh, right, forgot.

Thanks

[hoary seal]

ann 😭

[sly aurora]

here we go again.

Hahah

[astral glacier]

Oh god cardinality stuff

[sly aurora]

Yeah, indeed

[astral glacier]

Well the main idea is to make smaller intervals and slowly move them

Firstly let's start with the easiest thing

Actually no

What does it mean for two sets to have the same cardinality

Are you clear about that

[sly aurora]

There is a bijection from one set to another.

[astral glacier]

Well for a bijection direction doesn't matter

But yes

They're in bijection

Alr let's do the easiest pair first

You understand how (0,1] and [0,1) have the same cardinality right

[sly aurora]

Not really?

[astral glacier]

Well this one's the easiest

Both of them have (0,1)

One has 1 and the other has 0

Can you construct a bijection now

Do note that this is a set map. So it doesn't need to be "continuous"

[sly aurora]

That is good since I am a first semester undergraduate student and have not even had Analysis 1 yet...

Hmm, so f(x) = 0, if x=1 should be one part.

[astral glacier]

Good

And what about the rest

[sly aurora]

f(x) = x, if 0<x<1?

[astral glacier]

Yup

Now is this a bijection

[sly aurora]

Well, it is injective since he only exceptional value at x=1 which goes to 0. We have no other input that goes to zero. For surjectivity any y is hit by f(x) = x.

[astral glacier]

Yup

Now another quick thing

If A and B are in bijection, and B and C are in bijection, what can we say about A and C

[sly aurora]

Since |A| = |B| is an equivalence relation, by transitivity, there also exists a bijection between A and C.

[astral glacier]

Yup

Now we show that there's a bijection between (0,1) and (0,1]

This is the hard part

Well "hard" in quotes cuz we don't ask for continuity

[sly aurora]

May I ask a general question for a moment?

[astral glacier]

Dure

Sure

[sly aurora]

How does one usually "construct" a bijection? How do I know to find a bijection in general if I want to prove two sets have the same cardinality, not referring to any specific example? Or does the method always vary depending on what sets we are considering?

[astral glacier]

I'm not aware of any general way

Thankfully usually we just end up proving that all intervals are bijective

And never look back

[sly aurora]

Hahah, okay.

[astral glacier]

So this is just smth you prove once

Anyway

All we need to do is construct a bijection from (0,1) to (0,1]

That will give us everything else

I'll explain why later

[sly aurora]

Alright, so far so good.

[astral glacier]

So here's an easy way

Take every element of the form 1/2^n

What now

[sly aurora]

We will have some set {1, 1/2, 1,/4,...} which is in (0,1]?

[astral glacier]

Well we are starting in (0,1) so we don't have 1

[sly aurora]

Oh, so only {1/2,....}?

[astral glacier]

Yes

Now what can we do to this set to get 1

[sly aurora]

Well, currently the sequence is approaching 0.

[astral glacier]

Yes

[sly aurora]

If we want to have one we need to somehow alter the sequence.

[astral glacier]

We don't want the sequence to approach 1

We just want it to contain 1

We want to map it in a way so that the image contains 1

[sly aurora]

Okay, okay, so we need to change 1/2^n to something else.

[astral glacier]

There's an answer that's staring you in the face

[sly aurora]

if we want every term to get slightly larger, perhaps 1/2^n-1?

[astral glacier]

Double every term yes

And leave everything else as it is

Now is this a bijection

[sly aurora]

The cases do not collide for injectivity, I guess? and every y is also hit once.

[astral glacier]

Yup

So this is a bijection

So far we have

(0,1) ≅ (0,1]
(0,1] ≅ [0,1)

So we automatically have (0,1) ≅ [0,1)

Now we want to throw [0,1] into the mix

This is easy using the map we just made

Tell me how

[sly aurora]

Perhaps by composition of all three bijections?

[astral glacier]

There's a much easier way