help-10

[feral sentinel]

Yes

[wild swallow]

okay so

[feral sentinel]

Now ill sub 0=0

[wild swallow]

we got all these

[feral sentinel]

Get rid of some terms

[wild swallow]

yes

the ones you want to get rid of

are the xy, yz, xz terms

using the fact that they add to 0

[feral sentinel]

Hm

AH I THINK I SEE IT

HA

[wild swallow]

hmmm

if you

delete teh 48xy and the 24yz

you also gotta delete 12pxz as well

oh wait

you did

[feral sentinel]

A wat

[wild swallow]

but

badly

[feral sentinel]

Yes

Oh

[wild swallow]

because its 12pxz

not just 12xz

[feral sentinel]

I left the o out

P

[wild swallow]

yeah

[wild swallow]

good good

write it factorised tho

not as two separate terms

[feral sentinel]

Now i will yeet out xz

[wild swallow]

LOL

yes

yeet it out

[feral sentinel]

WE ARE CLOSE

The entire left side

Is just 0

36x^2 till 9z^`

2

[wild swallow]

no it

isnt

[feral sentinel]

Oh

Wait wut

Why

[wild swallow]

well

why would it be 0

[feral sentinel]

Cuz of 6x + 4y + 3z=0

Then i square

So ill be left with

3 - p =0

[wild swallow]

that process led us to

the whole LHS being 0

not just up to 9z^2

[feral sentinel]

Wait wut

Wait

[wild swallow]

you cant just delete the first 3 terms

the only things you can delete

are these

[feral sentinel]

This wouldnt work?

[wild swallow]

not the same

[feral sentinel]

Oh

[wild swallow]

this is the first line squared

you even calculated it

[feral sentinel]

Oh wait

Lmao

LMAO

NEVERMIND

[wild swallow]

[feral sentinel]

IM STOOPID

AAAAA

Anyway

[wild swallow]

hmm yes

[feral sentinel]

For some reason i thought (a+b+c)^2 = a^2 + b^2 +c^2

ANYWAY

Lets continue

[feral sentinel]

Wats next

[wild swallow]

alright

[feral sentinel]

Wait

Give me clues

I wanna tink

[wild swallow]

do you know of the famed

AM-GM

[feral sentinel]

Train my brain

[wild swallow]

inequality

[feral sentinel]

What is that

[wild swallow]

its the inequality that says

[feral sentinel]

I have never sene that

Before

[wild swallow]

the LHS is the AM - the arithmetic mean

the RHS is the GM - the geometric mean

[feral sentinel]

Damn

[wild swallow]

and the arithmetic mean is always >= the geometric mean

[feral sentinel]

Is that a olympiad thing

[wild swallow]

equality holds if and only if all the a's are equal

yeah

[feral sentinel]

O

[wild swallow]

its a pretty olympiad thing

[feral sentinel]

Ohh

Have u been to opympiads

[wild swallow]

have i

[feral sentinel]

Yes

I mean ur a grad student

Currently doing research

[wild swallow]

i dont want you stalking my name

[feral sentinel]

So ur either doing

Masters

Or phd

Rn

[wild swallow]

do you really believe @stable rain

[feral sentinel]

Yes

She said it very willingly

[wild swallow]

[feral sentinel]

Yes

LETS CONTINUE

[wild swallow]

alright

we only need AM-GM for two numbers

so lemme restate

this one i think we can even prove

[feral sentinel]

I dont think i can but yes

[wild swallow]

lets prove it

[feral sentinel]

HM

[wild swallow]

so

[feral sentinel]

Direct proof?

[wild swallow]

agreed?

[feral sentinel]

Yes

[wild swallow]

alright

so we expand

wait

thats probably not a good starting point

[feral sentinel]

Nani

[wild swallow]

agreed?

[feral sentinel]

Yes

[wild swallow]

so we expand

[wild swallow]

agreed?

[feral sentinel]

Yes

[wild swallow]

so now we simpliy

agreed?

[feral sentinel]

Wut

[wild swallow]

wait shit

[feral sentinel]

Lmao

[wild swallow]

now agreed?

[feral sentinel]

TESTING ME HMM

Yes

[wild swallow]

alright

so then

[feral sentinel]

OH

[wild swallow]

and then

[feral sentinel]

I see

It

[wild swallow]

alright so when does equality hold

[wild swallow]

equality holds when this equality holds right?

[feral sentinel]

Wait but how the heck did u know we needed sqrt(a) instead of just a

I seriously dont get how u see it

[wild swallow]

no like

i know this proof

like

its memorised lmao

[feral sentinel]

Ah

Oh

[feral sentinel]

Yes

[wild swallow]

anyway

yes

so we need

[wild swallow]

in other words

[feral sentinel]

Why do we need this again

Oh

[wild swallow]

thats when equality holds right

[feral sentinel]

Ohhhhhhhhhh

[wild swallow]

when a = b

[feral sentinel]

Yesh

[wild swallow]

good

and this only works for a, b >= 0

because square roots

[wild swallow]

over here

[feral sentinel]

Yes

[wild swallow]

good

so we've prove the whole thing

[feral sentinel]

So we have proven

Yes

[wild swallow]

AM-GM for 2 numbers

[feral sentinel]

Ues

Now for n?

[wild swallow]

i cant remember if i know the proof for n numbers

its probably jensens

but we dont need it

[feral sentinel]

Wat is that

Ok

Lets contjue

[wild swallow]

alright

so we just gotta

apply AM-GM

a couple times

in good places

[feral sentinel]

Ó

Oka

[wild swallow]

what do we know

well

we know

[feral sentinel]

Yes

[wild swallow]

lets

shove the 12xz(3-p) onto the right first

[feral sentinel]

Wat

[wild swallow]

i just moved it over

to the right

[feral sentinel]

Wait

Is there a -

Oh

OH

[wild swallow]

yeah the p and the 3

switched

[feral sentinel]

Amazing

Okay

[wild swallow]

cool

now the idea is

the LHS

is a sum of square numbers

if we can show that

its gonna be too large

then itll never be equal to the RHS

[feral sentinel]

Wait how does that help

[wild swallow]

well

if its never equal to the RHS

theres no solution

[feral sentinel]

But i thought we wanna find a solttution

[wild swallow]

yeah but

we gotta show that

there is no solution

to find when solutions start appearing

because at some point

it goes from there being no more solutions

to there being more solutions

we need to figure out when and how that happens

[feral sentinel]

I still dont quite get why but lets just conunue

Maybe ill see it soon

[wild swallow]

okay

so

first of all

the LHS is >= 0 right

[wild swallow]

here

[feral sentinel]

Yes

[wild swallow]

cuz squares

[feral sentinel]

Ue

[wild swallow]

hmmmm wait

[feral sentinel]

Wot

[wild swallow]

i think i missed a case in my solution

[feral sentinel]

Ó

[wild swallow]

lets fix that later

assume it all works lol

[feral sentinel]

Yes

Waw

[wild swallow]

actually hmmm

[feral sentinel]

Okay

Hm

[wild swallow]

considering time

[feral sentinel]

Wats happening

[wild swallow]

this might need

more algebra

[feral sentinel]

Huh

[wild swallow]

because

the idea is we want to apply AM-GM

actually we might be fine

because it just doesnt matter lol

alright

[feral sentinel]

Whats happening

[wild swallow]

lets just continue

[feral sentinel]

Oki

[wild swallow]

so

the RHS has xz

as the term

[feral sentinel]

ye

bring it over?

[wild swallow]

how do we produce an xz

no no

we need to produce an xz

using AM-GM

so that we can compare sizes

because comparing x^2 and z^2 against xz is weird

[feral sentinel]

this right

[warm shale]

HellO

[wild swallow]

so

[feral sentinel]

o oops

[wild swallow]

what do you think are good choices

of terms to pick

from the LHS

to create xz

[feral sentinel]

x and z

lmao

[wild swallow]

yes good

lmao

so

[feral sentinel]

i do tat

[wild swallow]

the 36x^2 and the 9z^2

apply AM-GM on those two

what do you get

[feral sentinel]

This?

[wild swallow]

yes

but simplify the square root a bit

and then move the 2 over to the other side

[feral sentinel]

Ah wait

[wild swallow]

LOL

[wild swallow]

[feral sentinel]

Forgot my z

But yes

[wild swallow]

you did forget your z

and also

unfortunately

we dont know that x and z are positive

[feral sentinel]

My 2 is also gone

[wild swallow]

yes its

kinda gone

[feral sentinel]

Okay

Its back now

[wild swallow]

lets fix it

[feral sentinel]

Yes

[wild swallow]

so what should we actually have

as the inequality

[feral sentinel]

Wtf

Yes

Whats next

[wild swallow]

okay cool

[feral sentinel]

Tatata

[wild swallow]

its almost right

we just need to say instead

[feral sentinel]

Almost?

[feral sentinel]

Ó

Okay

[wild swallow]

cuz we dont know x and z are positive

[feral sentinel]

Yes

[wild swallow]

but x^2 and z^2 certainly are

[feral sentinel]

Yes

[wild swallow]

actually this is still right

[wild swallow]

its just weaker than this

[feral sentinel]

Yes

[wild swallow]

cuz |xz| >= xz

[feral sentinel]

Yes

[wild swallow]

alright great

so

if we sub that into the equation

[feral sentinel]

We replace

[wild swallow]

this one

what do we get

https://cdn.discordapp.com/attachments/903430332324405288/1023159322311663676/118663865264242688.png

[wild swallow]

lol okay maybe i was a bit misleading

[feral sentinel]

wot

[wild swallow]

should not have said sub in

we actually want to create a bound on the LHS from below

so we have

[feral sentinel]

this is kinda long for just 1 mark

olympiad people are smart af

[wild swallow]

LOL

[feral sentinel]

huh

[wild swallow]

36

[wild swallow]

lol

[feral sentinel]

o

wait im so confusde

where is this

[wild swallow]

this

if we can show that the LHS is too large

to be equal to the RHS

then they wont be equal right?

[feral sentinel]

Yes

[wild swallow]

we're now doing that

[wild swallow]

so we have a lower bound for the LHS right here

the LHS must be at least as large as this

[feral sentinel]

O

Where did 16y^2 come from

[wild swallow]

this LHS has the 16y^2

[feral sentinel]

WHY IS THIS QUESTION SO HARD

[wild swallow]

im just keeping it in

LOL

thats kinda how it is

alright

[feral sentinel]

Wtffff

[wild swallow]

we need

to convert the y^2

into xs and zs now

to do that

we use the other equation

[wild swallow]

so

[feral sentinel]

Wait

[feral sentinel]

Where do i write this

[wild swallow]

its just

a separate computation

[feral sentinel]

This is everything till now right

[wild swallow]

yes

[feral sentinel]

Then now

[wild swallow]

just write it as a separate computation

where we work towards the lower bound

[feral sentinel]

Okay

Wats next

[wild swallow]

this

[feral sentinel]

So another computation?

[wild swallow]

kinda

i guess

[feral sentinel]

HEH

[wild swallow]

we're gonna sub it into the y^2 eventually tho

[feral sentinel]

Where did u eben get this

[wild swallow]

what

[feral sentinel]

this

[wild swallow]

here

wait shit

LOL

wrong equation

[feral sentinel]

im so lost

[wild swallow]

so

[wild swallow]

i divided all the constants out in my working

so i didnt have the 4s and 6s and 3s flying around

[feral sentinel]

okay

im like following and not following a the same time

i dont get why are there so many new computations

[wild swallow]

its all to create the lower bound

on the 36x^2 + 16y^2 + 9z^2

[wild swallow]

so if we square this

we get 16y^2 right

[feral sentinel]

why would we need the llower bound

;-;

[wild swallow]

you will see it

soon enough

[feral sentinel]

o

ok so now

[feral sentinel]

i wrote this

[wild swallow]

so what is it squared

[wild swallow]

hmmm yes

looks kinda familiar doesnt it

[feral sentinel]

yes

BUT HOW DID U SEE IT

I SERIOUSLY DONT UNDERSTANDD

AAAAAAAAAAAAAAAAAA

[wild swallow]

so we can

rewrite

[feral sentinel]

sub in

[wild swallow]

yes

but also use the inequality from before

[wild swallow]

oh wait nooo

these arent equalities

because these arent equalities

[feral sentinel]

OH YE

Ok

Now i simplify them

[wild swallow]

nooooo

[feral sentinel]

;-;

[wild swallow]

thats not an inequality either

it just

cant be subbed in

like theres no relation right no

between the inequalities we're deriving

and the original equation

[feral sentinel]

oh there isnt?

but the terms are the same

[wild swallow]

not yet

yeah but

we cant sub an inequality into the equality like that

well we can

but

you gotta do it differently

[feral sentinel]

so ill erase

the bototm two lines

of the main equation

[wild swallow]

bottom 3

these dont work

you only want the LHS

[feral sentinel]

pok done

[wild swallow]

but not the RHS

alright so

we have now that

[feral sentinel]

nice

[feral sentinel]

wait wat

so thats ANOTHER COMPUTATION

[wild swallow]

that was the whole inequality chain

[feral sentinel]

yes

bruh i legit forgot what we were tryna find

cus we did so much more stuff

we are trying to find the value of p

yes

[wild swallow]

oh god

okay tahts not a good idea

[feral sentinel]

how do u type so quickly wtf

ye

[wild swallow]

so the blue is AM-GM

the orange is that substitution for y we did

[feral sentinel]

yes

[wild swallow]

alright cool

[feral sentinel]

yes

[wild swallow]

so then

[feral sentinel]

kind of braindead

[wild swallow]

what if we add in

[feral sentinel]

but we will finish this

[feral sentinel]

WHAT

[wild swallow]

from here

[feral sentinel]

wait

OH

[wild swallow]

so we actually have

[feral sentinel]

wait

Is this all i need

Dude the sun is setting LMAO

[feral sentinel]

Thats how long weve spent on this

[wild swallow]

this is whats changed

[feral sentinel]

Do i have to write this out

[wild swallow]

since here

[feral sentinel]

Somewhere

[wild swallow]

maybe

because its part of the computations

[feral sentinel]

i feel so stupid

sad

ok ive written it out

[wild swallow]

okay good

now is where it gets a bit fucky

because i didnt consider this case in my original proof

lemme see what happens

[feral sentinel]

do u mind

if i go eat first

im dying

[wild swallow]

sure

LOL

[feral sentinel]

my brain legit cant process this

this is fr harder than linear algebra

[wild swallow]

possibly

[feral sentinel]

sad

ill tag u in an hour

[wild swallow]

[feral sentinel]

aaa

[obtuse pebble]

@feral sentinel Has your question been resolved?

[obtuse pebble]

@feral sentinel Has your question been resolved?

[feral sentinel]

@wild swallow HI

[devout sable]

Snow liquified

[feral sentinel]

Sad

I will avenge him

[lethal sand]

what is the question?

[feral sentinel]

hello

another god

um it would be weird to continue out of nowhere tbh

[lethal sand]

oh it was the old question?

[feral sentinel]

we are like in the middle of doing it

ye

do u mind

helping me

with another qn

[lethal sand]

just curious tho, what does it meant by more than one set of solutions?

[feral sentinel]

i think its that there exist another solution which isnt the trivial solution

like x,y,z = 0

[lethal sand]

ah right then

[feral sentinel]

yes

imma start a new channel

with a new question

[lethal sand]

uh, you might not want to do that tho

[feral sentinel]

oh why

[lethal sand]

multiple occupied channels stuff

[feral sentinel]

oh the max is 2

i think

[lethal sand]

also, i think i have a solution for this

[feral sentinel]

i kinda wanna do another qn ngl

lol

[lethal sand]

maximum channel occupied is just 1

[feral sentinel]

oh wot

[lethal sand]

oh well, you probably should just ask here

or close this one

[lethal sand]

yes

[feral sentinel]

lemme just

make a new one

lul

[lethal sand]

one channel at a time

[feral sentinel]

what if

i use a secondary account

im joking XD

lemme make a new channel

[lethal sand]

im gonna ignore that there is a message above this one

[feral sentinel]

yesyes

[lethal sand]

[feral sentinel]

.close sesimi

[obtuse pebble]

Channel closed

Closed by @feral sentinel

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

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

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

Channel closed

Closed due to the original message being deleted

[final thunder]

Hilarious

[obtuse pebble]

Available help channel!

Send your question here to claim the channel.

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

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

[quiet dome]

Can someone walk me through this pls?

[obtuse pebble]

@quiet dome Has your question been resolved?

[obtuse pebble]

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.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[pliant trail]

Can anyone help me explain how "S(n) = n/2 [2a + (n-1)d]" works?

[devout sable]

I assume you mean for AP

[pliant trail]

Yes

[final thunder]

Wouldn’t it be easier for you look for some videos on YouTube

[devout sable]

Well let S_n = (a)+(a+d)+….(a+(n-1)d)

[devout sable]

Well yeah also that

[pliant trail]

Wait, what happens after (a+d?)

[devout sable]

a+2d

[pliant trail]

(a+2d)?

[devout sable]

a+3d

a+4d

..

Well ig I’ll just link a good yt

[pliant trail]

OH I GET THE (N-1)D NOW.

[devout sable]

It could save us so much time

[pliant trail]

But why 2a?

[pliant trail]

Go ahead..

[devout sable]

https://www.youtube.com/watch?v=EkmYAGdxRBY

I mean the thumbnail itself has all the required details

[pliant trail]

Oh ok, I got it! Thank you!

[devout sable]

,tex \np

[devout sable]

Perhaps close if done

[wild swallow]

deep is all about the closing today

[devout sable]

[wild swallow]

what about openings

new beginnings

[devout sable]

I’m very active now

[timid silo]

[devout sable]

e4e5

[wild swallow]

:0

LOL

[devout sable]

No d4 best opening

[timid silo]

d4

yes

nah e5 better

[devout sable]

e5

[timid silo]

then bring out the bishop

[wild swallow]

e5

[timid silo]

and then use some bad gambit

and finish it

[wild swallow]

[devout sable]

Someone.close

[timid silo]

helpers will .close probably

.close

[obtuse pebble]

Channel closed

Closed by @faint meteor

Use .reopen if this was a mistake.

[timid silo]

wait th

.reopen

[obtuse pebble]

✅

[timid silo]

ig ppl with helpfull can close channels now

[wild swallow]

you have helpful role

yeah

helpers is also self assignable

[timid silo]

helpers can't .close anymore?

[wild swallow]

not sure

if they could anyone could do it

[timid silo]

ya

[final thunder]

.close

[obtuse pebble]

Channel closed

Closed by @final thunder

Use .reopen if this was a mistake.

[final thunder]

Oh

.reopen

[obtuse pebble]

✅

[timid silo]

they can

[wild swallow]

aight guess it's everyone then

[devout sable]

Bruh

No

[timid silo]

even deep

[devout sable]

Need to have helpers role

[wild swallow]

deep even you can do it

become helper

[devout sable]

I don’t want helpers

Ew

[timid silo]

me too

too much ping

[wild swallow]

ping simulator

[devout sable]

I am very active and thats what i want 😌

[wild swallow]

[timid silo]

very active takes alot of patience tbh

[devout sable]

No

[wild swallow]

does it

[timid silo]

will miss my blue name tho

[wild swallow]

i got it in less than a month

[timid silo]

for me ya

[devout sable]

Snow did it in less that 28 days lol

[timid silo]

[timid silo]

must be very active then

[wild swallow]

[devout sable]

Yeah that very is like VERY

[timid silo]

took me more than 2 months sure

after getting active

[devout sable]

OO

[wild swallow]

i got it like

3 weeks after getting active

[devout sable]

It took me ig 2 months yeah

[timid silo]

u have very active?

[devout sable]

[wild swallow]

deep is very active now

[devout sable]

[timid silo]

ohh now she does

nice

[devout sable]

Now i need kawaiiCat

[timid silo]

welcome to no life role

[wild swallow]

lol

[timid silo]

[wild swallow]

[devout sable]

Lol bot only shows ur name in very active @timid silo

[timid silo]

where

[devout sable]

,whohas very active

[warm shale]

Members in Very Active
======================
1.   ᲼᲼#0326```

[devout sable]

[wild swallow]

wtf lol

.whohas very active

[devout sable]

,

[wild swallow]

,whohas very active

[warm shale]

Members in Very Active
======================
1.   ᲼᲼#0326```

[tidal hawk]

,whohas helper

[warm shale]

2 roles found matching helper!

1.   Adv Helper*
2.   Helpers```
Please type the number corresponding to your selection, or type `c` now to cancel.

[devout sable]

Everything is a lie now

[tidal hawk]

,whohas active

[warm shale]

2 roles found matching active!

1.   Active
2.   Very Active```
Please type the number corresponding to your selection, or type `c` now to cancel.

[wild swallow]

is it broken?

[devout sable]

My existence is a lie

[tidal hawk]

????

,w existence

[devout sable]

[warm shale]

Results provided by WolframAlpha

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

Requested by swaggofishballs#1529

[devout sable]

Did we close this thing?

[tidal hawk]

o

[wild swallow]

nope

[devout sable]

.close

[tidal hawk]

HAHAHAH

[wild swallow]

.close

[obtuse pebble]

Channel closed

Closed by @wild swallow

Use .reopen if this was a mistake.

[devout sable]

Close it lol

[warm shale]

Role selection timed out.

Sep 24, 2022, 12:55 PM

[tidal hawk]

someone get @devout sable help role

[warm shale]

Role selection timed out.

Sep 24, 2022, 12:55 PM

[devout sable]

What if bot opens new help channel

[tidal hawk]

O:

[wild swallow]

[devout sable]

,whohas a

[warm shale]

32 roles found matching a!

Page 1/2
========
1.   Role that does nothing
2.   Algebraic Geometry
3.   Applied Math
4.   Abstract Algebra
5.   Foundations
6.   Category Theory
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8.   Analysis
9.   :banach_alg_hermitian_involution:
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18.  Graduate+
19.  Adv Helper*
20.  Mathematics Minecraft```
Please type the number corresponding to your selection, or type `c` now to cancel.

[devout sable]

Lets see

[wild swallow]

there's probably a line of code somewhere

if bot: don't open channel

[devout sable]

No don’t type any new messages

It delays then

,whohas g

[warm shale]

10 roles found matching g!

1.   Role that does nothing
2.   Algebraic Geometry
3.   Abstract Algebra
4.   Category Theory
5.   Topology
6.   :banach_alg_hermitian_involution:
7.   Undergraduate
8.   Graduate+
9.   studying!
10.  perma-studying!```
Please type the number corresponding to your selection, or type `c` now to cancel.

Role selection timed out.

Sep 24, 2022, 12:57 PM

Role selection timed out.

Sep 24, 2022, 12:58 PM

[devout sable]

,whohas o

[warm shale]

26 roles found matching o!

Page 1/2
========
1.   @​​everyone
2.   Role that does nothing
3.   Algebraic Geometry
4.   Foundations
5.   Number Theory
6.   Category Theory
7.   Combinatorics
8.   Topology
9.   :banach_alg_hermitian_involution:
10.  Talk Announcements
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Please type the number corresponding to your selection, or type `c` now to cancel.

[obtuse pebble]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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• 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.

[warm shale]

Role selection timed out.

Sep 24, 2022, 1:00 PM

[woven cloud]

can a improper fraction be a rational number

[woven cloud]

sooo?

[fallow cove]

for example?

[woven cloud]

idk

you are supposed to be the master

[high lily]

what's your definition of imporper fraction
and what's your definition of rational number

[woven cloud]

idk

i am dumb

i would guess like 1 is rational

pie is not

1.333333333 is

1.2432534123124325 is not

1.252525 is

[high lily]

why don't you think
1.2432534123124325 is not
isn't rational

also doesn't answer the questions i asked earlier

[fierce lagoon]

This is rational

[woven cloud]

whaaaaaaaaaaaa

IT DOES NOT REPEAT

[high lily]

it terminates

which makes it rational

[fierce lagoon]

It terminates

[woven cloud]

so it would be irratiol

[fierce lagoon]

^^

No

[woven cloud]

i hate math

bye

fuck this

[high lily]

terminating decimals are rational

repeating decimals are rational

[fierce lagoon]

Rational fractions have decimals that terminate;

Irrational keep on going forever

[high lily]

you could view terminating decimals as decimals with a repeating trailing 0

[obtuse pebble]

@woven cloud Has your question been resolved?

[obtuse pebble]

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• After 15 minutes, feel free to ping <@&286206848099549185>.
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• 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 nova]

Hi, I don't understand this definition of a limit.

[graceful nova]

more specifically I get lost at the part after: x element of domain(f)

[timid silo]

the upsidedown v is an and symbol, so its saying that if x is in the domain and x is delta away from a, then f(x) is epsilon away from L

[graceful nova]

wait so '0 < | x - a | < delta' means what exactly?

and where does the 'a' come from?

and why absolute value?

[timid silo]

a comes from that

Am I right?

[timid silo]

#❓how-to-get-help

[timid silo]

its basically saying the distance between x and a is less than delta

the absolute value just says it doesnt matter if x>a or x<a

[graceful nova]

so a is the like asymptote

and x is just a point you plug into the formula?

and delta is some line close to the asymptote?

if that's right, what about the part afterwards?

| f(x) - L | < e

[timid silo]

its saying that as x gets closer to a, then f(x) gets closer to L

[graceful nova]

okay so wait

this is what im imagining rn

where in this picture does 'e' go?

and why are 'L' and 'a' different things?

[timid silo]

L is on the f(x) axis, and a is on the x axis

[graceful nova]

ohhh okay

[timid silo]

lemme draw a pic

[graceful nova]

what about the difference between 'e' and 'delta'?

[graceful nova]

great !

[timid silo]

(disclaimer, its gonna be bad lol)

[graceful nova]

yes cause this is really high quality xd

[graceful nova]

ahhh, okay

[timid silo]

if we are within delta of a on the x axis, then we are within epsilon of L on the y axis

[graceful nova]

one more thing

$\forall \epsilon > 0 \exists \delta > 0$

[warm shale]

Gigabyte

[graceful nova]

why is this like

why isnt it for example

$\forall \delta > 0 \exists \epsilon > 0$

[timid silo]

that means that if I give you any positive epsilon, you can always find delta so that this image holds

[warm shale]

Gigabyte

[graceful nova]

do you like

understand my confusion

[timid silo]

we want f(x) to get close to L

[graceful nova]

yes

[timid silo]

we can always get close to a (by assumption)

so we want to prove that f(x) can be epsilon close to L, not x delta close to a

[timid silo]

yeah, but idk if im doing a good job at explaining it to you lol

[graceful nova]

you're doing great so far!

[timid silo]

thx 😀

[graceful nova]

uhm, im not following this entirely though

what would exactly be wrong about doing it the other way around or for example 1 sec

let me get all other possible permutations(?)

$\forall \delta > 0 \forall \epsilon> 0$

$\forall \delta > 0 \exists \epsilon> 0$

[warm shale]

Gigabyte

[graceful nova]

$\exists \delta > 0 \forall \epsilon> 0$

$\exists \delta > 0 \exists \epsilon> 0$

can you like say what's wrong about them

[warm shale]

Gigabyte

Gigabyte

Gigabyte

[graceful nova]

this looks ugly af btw

i hate inequalities

xd

[timid silo]

same :p

we are looking at the definition of $\lim_{x\to a}f(x)=L$, ie, we want to show that f(x) gets close to L when x gets close to a. $\forall\epsilon\exists\delta$ does this.

$\forall\delta\exists\epsilon$ would be saying we can make x close to a as long as f(x) is close to L, which I guess says something about the inverse of f(x). The following, the limit doesnt exist but this permutation of exists/forall works

[warm shale]

Toby

[timid silo]

this is because "exists" means we can make a choice of epsilon

"forall" means it needs to work for any epsilon, in particular very small ones

[graceful nova]

this clears up a lot of confusion !!

[timid silo]

so here, for any small delta (ie we want to get as close to a as possible), we can choose an epsilon (such as the one drawn) so that f(x) is close to L

[graceful nova]

so i get why that permutation is logical, but why would the other ones be incorrect?

like actually mathematically incorrect

so right now it says "for each delta we can pick any epsilon" and P(x) should hold

why would "for each epsilon we can pick any delta" and P(x) should hold be exactly wrong

[timid silo]

$\exists\delta\forall\epsilon$ would be too strong

[warm shale]

Toby

[timid silo]

ie it says that there is a single delta that works for any epsilon

(normally, delta depends on epsilon)

[graceful nova]

oh so there exists a single delta for all possible epsilon

[graceful nova]

and this there exists a single epsilon for all possible delta

[timid silo]

no, otherway. thats for all possible delta, there exists an epsilon (possibly depending on delta)

$\forall\delta\exists\epsilon$

[warm shale]

Toby

[timid silo]

yup

(logic hurts my brain :p)

[graceful nova]

same

okay wait so i also study computer science so

this is like a for loop and if statement right

for (delta) if (epsilon) P(x)

this is the correct one

[timid silo]

yes for the for, not sure about using if

[graceful nova]

i think if is right cause logic is the foundation of programming languages im pretty sure

for (epsilon) if (delta) P(x)

you said this was too strong right? but would it be incorrect?

[timid silo]

ig this is psudo code for the limit definition

  set delta=delta_epsilon
    for x in (a-delta,a+delta):
      if not(|f(x)-L|< epsilon):
        print("incorrect limit :(")
        break
print("correct limit :)")```

and your question is about the order of the top two lines

[graceful nova]

isnt set delta=delta_epsilon always the same?

set delta=delta_epsilon
for epsilon in positive reals:
    for x in (a-delta,a+delta):
      if not(|f(x)-L|< epsilon):
        print("incorrect limit :(")
        break
print("correct limit :)")```

can you do this?

[timid silo]

there exists delta for all epsilon would mean the limit exists, but it does not have to hold if the limit exists (ie converse doesnt hold)

[timid silo]

ig im emphasising that delta depends on epsilon there

[graceful nova]

oh yeah cause delta comes from x and epsilon from f(x)

thats what you mean right

and L is never reached by the graph, but a is always accesible

[timid silo]

if f(x)=x and a=0, then you cannot find a delta that works for all epsilon

[graceful nova]

f(x) = x does not have an asymptote?

[timid silo]

your definition works for continuous functions too

[graceful nova]

wut, i thought this was for limits

specifically

[timid silo]

continuous functions also have limits

[graceful nova]

like a precise definition

[graceful nova]

how does f(x) = x have a limit?

[timid silo]

$\lim_{x\to a}x=a$

[warm shale]

Toby

[graceful nova]

yeah but that doesnt say anything

why would you take a limit of a function that can reach everywhere

[timid silo]

sure, but it still satisfies the defition

[graceful nova]

fucking math

xd

this really does improve my thinking though

[timid silo]

think about $f(x)=\begin{cases}x&\text{if }x\neq0\\pi&\text{if }x=0\end{cases}$. Then $\lim_{x\to 0}f(x)=0\neq \pi=f(0)$

[warm shale]

Toby

[graceful nova]

then $\lim_{x\to 0}f(x)=0\neq \pi=f(0)$

[warm shale]

Gigabyte

[graceful nova]

why $0 \neq \pi$

[warm shale]

Gigabyte

[graceful nova]

what did mean with that

writing math formulas (in latex) is so cool btw

best part by far

[timid silo]

pi=3.14 here

could have been any number

I sure do hope pi isnt 0

:p

[graceful nova]

wait but from your definition

shouldn't it be

$f(0) = \pi \neq 0$

[warm shale]

Gigabyte

[timid silo]

yeah?

but the limit of f(x) to 0 is 0

[graceful nova]

you said $f(0) = 0 \neq \pi$

[warm shale]

Gigabyte

[graceful nova]

is this the same?

[timid silo]

I said $\lim_{x\to 0}f(x)=0\neq \pi$

[warm shale]

Toby

[graceful nova]

yeah so you plug in 0 for the x right?

[timid silo]

no

[graceful nova]

and then you get f(0)

[graceful nova]

huh

[timid silo]

thats only true for continuous functions

[timid silo]

whenever we see the limit symbol, we have to use this definition

(it just so happens that for continuous functions we can substitute and skip the definition)

[graceful nova]

wait lemme copy your message 1 sec

think about $f(x)=\begin{cases}x&\text{if }x\neq0\pi&\text{if }x=0\end{cases}$. Then $\lim_{x\to 0}f(x)=0\neq \pi=f(0)$

[warm shale]

Gigabyte

[graceful nova]

wait why does it equal 0

i dont get it

like x approaches 0, but never becomes it

so shouldnt it be f(x) = x != pi?

[timid silo]

yeah, the limit only cares about what it approaches, not what it becomes

[graceful nova]

then why does it equal 0

cause it never becomes 0 right