question about point at infinity

[bold jay]

It's constant in that it would be a single value

[hollow drift]

but is it constant in that 1/0 + k doesnt equal 1/0

a property of all vaules that are constant is that adding/subtracting another constant must not equal the orignial value

like inf is not constnat because inf + 1 = inf

[bold jay]

Where are you getting this?

[hollow drift]

I saw it one time ill try to find it again

oh way idk maybe its just something I came up with

nvm then

but I think that should also be trivial right?

if a is constant and k is another constnat than a + k cant equal a

but if a is not constant and k is a constant than a + k can equal a

or perhaps you cant maintuplate a value that is constnat to be nonconstnat

since 0 is a constant the reciprical must also be constant

maybe

[jaunty magnet]

look up field axioms

we don't consider infinity a real number or complex number since they're both fields, including infinity violates one of the axioms

[hollow drift]

right

but what about when 1/0 is deifned as a point at infinity

is it a constant?

[jaunty magnet]

what set are we working in here

[hollow drift]

I think

projective number line : R U {1/0}

[jaunty magnet]

okay what do you mean by constant?

also you might wanna check out the Riemann sphere since that seems relevant

[hollow drift]

yes

[hollow drift]

it can only ever be its self

or like

1/0 + another constant doesnt equal 1/0

[hollow drift]

actually this is only true from the perspective of 1/0

or "locally" although Im probably using that math term wrong

[jaunty magnet]

It's hard to make sense of (1/0) unless you define it rigorously

[hollow drift]

I know

Im saying in the context of the remainn sphere

[jaunty magnet]

ohh

[hollow drift]

or projective number line

1/0 is used the same in both

reiamm sphere is just the projective number line but complex

[hollow drift]

ohhh

nice

I didnt see that part

[jaunty magnet]

I still don't exactly know what you mean by constant in this context

[hollow drift]

only exists at one point

[jaunty magnet]

as opposed to?

[hollow drift]

infintity which doesnt exist at only one point

[jaunty magnet]

like?

[hollow drift]

like on the projective number line

any value thats not at the point at 0 or 1/0 has a value of inf or - inf right?

becuase every location is infinitly far away from 0 on the projective real number line

not include the location of 0 or 1/0

actually this is my assumption I could be wrong

[jaunty magnet]

not sure sorry lol

♾️=♾️ on the Riemann sphere idk if that means anything here

[hollow drift]

what

wait

send the pic again

[jaunty magnet]

This is for Riemann sphere

[hollow drift]

where does it say inf = inf

maybe inf = - inf?

[jaunty magnet]

it's implied by the definition

[hollow drift]

oh okay

[jaunty magnet]

z+inf=inf

[hollow drift]

ah

[jaunty magnet]

substitute z+inf for inf and you get inf=inf lol

[hollow drift]

right right

but also

like

okay imagine if you zoom into 1/0

from this perspecitve 0 is infinitly far away

so from that perspective using the same logic that inf + z = inf

0 + z = 0 since all finite number are equally infinitly far away

but we know 0 to be constant so perhaps 1/0 is also

[jaunty magnet]

I don't know if this answers your question but on the Riemann sphere z/0 is always infinity regardless of z
(unless z is 0)

[hollow drift]

this is my understanding not rigoursly defined

[jaunty magnet]

meaning 1/0=2/0

[hollow drift]

yes I agree

[hollow drift]

thats what I thought

hmm

ya idk

wait do you understand

that like

what I was trying to ask like does 1/0 exist at only one point compared to that inf can exist at any point on an infinitly long line projected into a finite distance

becuase all points are infinitly far away

[bold jay]

What else would it be? When is a value ever something besides itself?

[hollow drift]

0/0

can be anything

[jaunty magnet]

does 3 only exist at one point? I'm still confused lol

[hollow drift]

so I was just making sure that 1/0 is always its self

[hollow drift]

yes

[hollow drift]

sorry its becuase im bad at communicating rigoursly and often assume what I say intutivily makes sense haha

my bad

I havent been taught/learned proofs or logic yet

so

[jaunty magnet]

oh wait are you confused because some infinities are different sizes (in reference to cardinality)

[hollow drift]

um

well

not in the context of this

[jaunty magnet]

ok cool nvm lol

[hollow drift]

ya ya

[hollow drift]

in the context of cardinality could "different" 1/0 have different sizes?

[jaunty magnet]

not really

[hollow drift]

unless 1/0 doesnt mean anything in that context

[jaunty magnet]

that doesn't really make sense lol

[bold jay]

No you're reaching the same problem here. If you define 0/0 to be something, then it's going to be just that.

[jaunty magnet]

ignore my comment on cardinality

[hollow drift]

oh you know I also that that 1/0 is a numerical represntation for uncountable infinity?

[hollow drift]

i think this might be wrong though idk

[bold jay]

0/0 is undefined because division by zero with finite numbers does not work if you want to keep all the field axioms

[hollow drift]

wait really

[hollow drift]

right

[bold jay]

If you don't care about the field axioms, then you can define 0/0 to be any value you want without issue

[jaunty magnet]

we're throwing out field axioms here this is about the projective real line I think

[hollow drift]

wait does any model define 0/0 to be meaningful?

[bold jay]

Same with 1/0

[bold jay]

Not that i am aware of, no

[hollow drift]

before trying to think of 1/0 as something that could exist people said 0/0 and 1/0 equally dont exist but I always felt 1/0 to have more signifciance and be more consistient

[jaunty magnet]

are you trying to say like "are all 1/0s equal to all other 1/0s?"

[hollow drift]

does this make sense or nah lmaoo

[hollow drift]

im pretty sure this should be the cast

case*

since all 0s should also be equal to all other zeros righ

[jaunty magnet]

In the riemann sphere that's the case

[hollow drift]

you cant have a smaller magnitude than zero thats like saying how do you get closer to something your touching or how do you move slower than standing still

[hollow drift]

right

[hollow drift]

does this make sense?

[jaunty magnet]

I don't think the question makes sense by itself, you need to state what set/system you're working in. There could be some system where 1/0≠1/0

[hollow drift]

hmmm

[jaunty magnet]

if we're working in real numbers it doesn't make sense either way so we don't even consider it

[hollow drift]

do you know of any?

[jaunty magnet]

same with complex numbers

[jaunty magnet]

Riemann sphere

[hollow drift]

no but 1/0 does = 1/0

[jaunty magnet]

ohh

[hollow drift]

cause its only a single point

[jaunty magnet]

I mean I could come up with one right now lmao

[hollow drift]

just like 1=1 or 2=2

all things that equal 1 or 2 must exist at the exact same point

[hollow drift]

do it

[jaunty magnet]

actually nvm lmao

that'd contradict deeper axioms I don't know enough about to talk about

but I could define a set where 1/0≠2/0

[hollow drift]

hmm

are you trying to say 1/0 < 2/0?

[bold jay]

I feel like we're getting off track

[hollow drift]

yes

but

[jaunty magnet]

sure, in this system

I don't think we are

[hollow drift]

i mean idk Im okay with it

[jaunty magnet]

as opposed to the Riemann sphere which defines anything/0 as infinity

[hollow drift]

it does relate I think

[hollow drift]

right

oh also did you know the infinity it uses is a "nuetral" infinity

neither postive nor negative

which is pretty cool

[jaunty magnet]

sure

[hollow drift]

I think its becuase the sign of 1/a is the same sign as a and since 0 is neither postive or negative 1/0 must also be

[jaunty magnet]

not necessarily

[hollow drift]

how

[jaunty magnet]

you're thinking with limits rn

[hollow drift]

no

im not

there is no general limit of 1/0

but 1/0 its self is neutral

I think

[jaunty magnet]

well you're applying field algebra rules to things that aren't fields

[hollow drift]

but can it still be an explation?

i think it can

[jaunty magnet]

we don't like to rely on intuition in pure math

[hollow drift]

I know

oh ya

hm

but I think its still rigours like that

here

on a line

the closer you get to zero

the reciprical of that value would be inversly farther from zero

as the denominator gets closer to zero right?

[jaunty magnet]

my point is I could say 1/0 is -infinity and it'd still work the exact same way

[hollow drift]

you could but it would be redundant

+1/0 = -1/0 the same way +0 = -0

becuaes there is no positve or negative

[jaunty magnet]

well I could assert that "infinity" isn't a member of the set

only "-infinity"

[hollow drift]

hm

okay

[jaunty magnet]

which works exactly like what used to be "infinity"

this is part of why you can't rely on intuition here lol

[hollow drift]

okay okay

[hollow drift]

what?

[hollow drift]

then its not equal to 1/0

so shouuldnt it not be a valid comparrison?

[jaunty magnet]

1/0=-infinity

[hollow drift]

um

[jaunty magnet]

z/0=-infinity for all z

[hollow drift]

sure

but this would be a differnet defintion and a diffenretion set

so my intution could still be correct becuase im strictly using an explation to descirbe the behvior in the remiann sphere

[jaunty magnet]

wait you're talking specifically about the Riemann sphere?

[hollow drift]

do you wanna hear the other intutive way of thinking I had

[hollow drift]

yes

or projective number line

[jaunty magnet]

oh that makes things easier lol

[hollow drift]

its defined in the same way

[hollow drift]

I literlly already said that 😭

right?

or maybe idk

but ya

[jaunty magnet]

on the Riemann sphere infinity is as much of a constant as any other number

[hollow drift]

like I was thinking as a gets closer to 0 on a number line 1/a would get further but when a is on the same point as 0 "a = 0" its the closest possible distance to 0

so the recipricle of 1/0 would inversely be the maxium distance away from zero

and on the projective real number line the number line is a circle and the maxium distance away from a point on a circle is the point directially above on the other side of the circle 😎

1/0

since 0 isnt on either the postive or negative side of the number line a point directally above would also not be on either side and exactly in the middle

niether postive or negative

[jaunty magnet]

when you say projective real number line do you mean the projectively extended real line

[hollow drift]

yes

OH SHOOT

its also defined that 1/(1/0) equals 0

yoo this is what I thought at the very beggining as well

althought 1/0 * 0 is still undefined

[jaunty magnet]

1/0 is as constant as any other number on the projectively extended real line

[hollow drift]

nice

thats what I was thinking

[jaunty magnet]

anything else lol

[hollow drift]

does this make sense?

[hollow drift]

and this

[jaunty magnet]

you're thinking of it intuitively which is good for brainstorming a proof

[hollow drift]

I know

I cant do it rigoursly

[jaunty magnet]

but you could draw the set in a different way and your intuition would break

[hollow drift]

but would it be the same set?

[hollow drift]

do it

[jaunty magnet]

two sets are equal if all of the members are the same yeah

okay

[hollow drift]

but projecting 2 sets diffentially can sometimes lead to diffenret outcomes right?

[jaunty magnet]

as long as you trust me when I say that the set containing every element in this representation is equal to the set of the projectively extended real line, then this is the exact same set but represented differently

[jaunty magnet]

I'm not sure what you mean and I haven't taken differential geometry yet

[jaunty magnet]

this just kills any intuition you have

obviously the other representation is better since it actually gives you an intuitive understanding of what's going on

[hollow drift]

this is not the same set

[jaunty magnet]

but you can't rely on your intuition to answer a question

intuition can lead you in the right direction

[hollow drift]

is that 2 diffenret points of infiniy?

[jaunty magnet]

they're equal

because I said so lol

[hollow drift]

?

i know

but im saying

[jaunty magnet]

two sets are equal if they contain the same elements

[hollow drift]

but it doesnt mean the model is equal

your values arent following the same defined rules

its not compactified by the point at infinity

so its not the same

right>

[jaunty magnet]

it's still compactified but "compact" has a different geometric meaning than what you're used to

[hollow drift]

I know

also

are you sure

how is it compact

you put points that arent connected to the real number line

so how could they compact it

there doesnt exist a finite subcover

right?

[jaunty magnet]

compact doesn't mean points need to be touching in the geometric sense

that's a property of how we usually represent it

[hollow drift]

i think it doesnt

does*

in the one point compactificaiton

1/0 acts as the finite subcover

[jaunty magnet]

negative numbers U positive numbers U 0 U infinity

[hollow drift]

but in your diagrram 1/0 doesnt touch +inf and -inf

so how can it be the same

surely this doesnt count as one point compactifcaiton

[jaunty magnet]

lets simplify to real numbers for a sec

actually integers

[hollow drift]

also the line you drew is finite with the ending value being 69 lmao thats not even the real number line

[jaunty magnet]

are these the integers?

assume there's nothing between 0 and 1 or 1 and 2 or -1 and -2 etc

[hollow drift]

what

okay so there is no direction to this line

this isnt a number line

number lines must increase or decrease in a specfic direction correct?

[jaunty magnet]

I'm not sure what the definition of "number line" is but if that's the case then don't call it a number line lol

[hollow drift]

hm

[jaunty magnet]

the point is that this is still the set of integers just represented differently

and represented in a way that kind of deletes your intuition of how integers work

[hollow drift]

also this isnt equvilent to the remainn sphere cause its not only a set right its also the geomtry

[jaunty magnet]

-2 is still smaller than 2, it's just not intuitively implied by this representation

[hollow drift]

like 2 sets projected diffenretially can be defined as seperate things

[hollow drift]

yes

becuase its not on a number line

so take away the line

{0,1,-1,2,-2,3,-3,4,-4,5,-5...}

[jaunty magnet]

"riemann sphere" refers to the model whereas "extended complex numbers" refers to the set the model is describing

[hollow drift]

right

but also one more then

thing*

try again with a differnet example

because

[hollow drift]

this isnt a diffenret projection of the reimmann sphere and since my intution was in the context of the reimann sphere it is not a valid counter argument

its not homo something

homotogical

wait

homological

perhaps

[jaunty magnet]

what do you mean by "projection" here

also I thought we were talking about the projectively extended real number line not the riemann sphere

[hollow drift]

as in

you cant just tear the 2 ends apart of the remainn spehere and still claim it is equvilvant

becuase the reiamman sphere is only unqiue becuase it has connected ends

[jaunty magnet]

well the concept of a limit takes a different intuition when you represent it in this way

it's still the exact same set just represented with a different model

infinity still compactifies the real line it's just that "compact" no longer has anything to do with numbers "approaching" other numbers and only really makes sense in the formal definition of a limit

[hollow drift]

then there would be differnet intution depending on the model right?

[jaunty magnet]

yeah exactly

which is why you can't rely on intuiton to prove things (or verify a solution to something)
proofs have nothing to do with models they have to do with what the models actually describe

intuition varies depending on the model you conceptualize

which is why topology is OP lol you can find your way around difficult math questions with basic geometric intuition

[hollow drift]

oh ya well also LMAO im not trying to "prove" it i know you cant with intution but I already know its proved and am using my own thinking

did it make sense to you?

[jaunty magnet]

well but you were trying to find an answer to something

[hollow drift]

no

its already known

that 1/0 is neither postive or negative

and I think I gave valid explations

although there most likely exists rigours ones

[jaunty magnet]

oh

the model most people think of when they think of the projectively extended real line is a good intuitive way to remember that infinity is neither positive nor negative yeah

I thought you were trying to use that model to prove that that's the case

[hollow drift]

no

I was trying to say 1/0 is neither postive or negative

did you read my intution?

I said 1/0 would exist as a point dirrectionally above 0 and since 0 is neither on the postive or negative side 1/0 would also not be either postive or negative

[jaunty magnet]

yeah

I'm saying that's a good way to remember that lol

it's just not a sufficient proof which I now assume isn't your goal here

👍

[hollow drift]

yaaa

[hollow drift]

i know

but

does it make sense

[jaunty magnet]

yeah that makes sense

[hollow drift]

nice becuase

did you know this was actually my thinking before I even knew these math concpets existed LMAO

in math class I thought this property existed and I got really lucky that it actually does

[jaunty magnet]

that's cool

[hollow drift]

yes

anyways im tired

[jaunty magnet]

you could've come up with the projectively extended real line lol

there are other models where that isn't true though

[hollow drift]

perhaps but only the intuion not the proofs

[hollow drift]

i know

[jaunty magnet]

okay cool lol

[hollow drift]

for my career path I wanna go into therotical fields and I think i can think abstractly

[jaunty magnet]

keep coming up with random shit in math class that's how you get both creative and good at math

[hollow drift]

im always fasinated by things that people say "dont exist" but sometimes they do

[hollow drift]

yaaa

but lowkey im behind compared to other math people we werent taught intergrals 😭

and Im defintly belove average in this math discord LMAO

[jaunty magnet]

you can literally invent a new field of math by figuring out a way to make something that conventionally "doesn't exist" to make sense

what year/grade are you?

[hollow drift]

I think if I can learn proofs and logic it will help me a lot becuase i can only make ideas with intution which wont be taken seriouslly or formal

[hollow drift]

just graduated grade 12

[hollow drift]

yaaa

[jaunty magnet]

what was the last math class you took?

[hollow drift]

calc

intro

but no intergrals

[jaunty magnet]

weird

[hollow drift]

yes

its candain

canadain*

so

I think they had it before but it was too hard for the classmaybe

[jaunty magnet]

linear algebra is where that starts for most people

[hollow drift]

yaaa

[jaunty magnet]

might wanna take a proof writing course first though

[hollow drift]

ohhh ya also I thought of a physics thing as well like if time was plotted on a projective number line

[jaunty magnet]

in fact here want a basic proof exercise from my linear algebra textbook lol

[hollow drift]

then after an infinite amount of time it then loops

so a new universe would begin

and I found out this property also already exist in a theroy 😎

penrose diagrahms

[jaunty magnet]

cool

[hollow drift]

yes

each vertical length of the "universe square" represnet an infinite length of time

[jaunty magnet]

idk shit about this lol I need to take differential geometry

this is einstein's field equations right

[hollow drift]

i dont either I just regonize the looping property lmao

[hollow drift]

um

i think its derived from it

like its called penrose diagrams

[jaunty magnet]

oh

[hollow drift]

and the curved line in the middle is a therotical path you could take to travel to differnet universes

although it requires a type of negative mass which most likely doesnt exist according to the majority of scientists

[hollow drift]

does this make sense?

[jaunty magnet]

kind of yeah

[hollow drift]

niceee

[jaunty magnet]

it'll make more sense after I learn more math I think lol

[hollow drift]

rightttt well I mean i probably only "think" it maekse sense to me becuase I dont know what I dont know

and dont understand the proofs really

so like

the way I compherend things is kinda of cheating lma

lmao

[jaunty magnet]

no that's okay

[hollow drift]

like a fake compheresion haha

[jaunty magnet]

it's not like intuition is useless

[hollow drift]

ya I know its like not my fault

[jaunty magnet]

no it's good to have though

if you can think of problems intuitively it'll lead you in the right direction to proving stuff (most of the time)

[hollow drift]

its jus the only way for me to do it so its not as good espially when I can only communicate that way becuase others then need to take extra time to decripher my informal methods and stuff

[hollow drift]

ya thats what I was wonderin

but someone in the server said this

[jaunty magnet]

wanna try this proof exercise

[hollow drift]

oh wiat nvm

[hollow drift]

oh ya bet

[jaunty magnet]

ok I have to explain what a field is

[hollow drift]

wait I gotta help you? oh bruh im cooked LMAO

[jaunty magnet]

no you're not helping me I'm giving you a proof problem

[hollow drift]

LMAO oh okay

haha

[jaunty magnet]

so you can kind of see what very basic proofs are like

I can explain it too

[hollow drift]

okay okay

also I may not know the methods

or like proof notaion

but ill try

[jaunty magnet]

you don't need notation

[hollow drift]

oh okay

[jaunty magnet]

you can use english

[hollow drift]

cool

bet

[jaunty magnet]

a field is just a set with two binary operations (addition and multiplication) such that these properties all hold:

real numbers are a field for example

rational numbers are a field

complex numbers are a field

[hollow drift]

oh ya

wait I thought these were rings

[jaunty magnet]

they're similar

[hollow drift]

isnt additon and multipllication rings?

oh

diffenret types of rings are fields?

[jaunty magnet]

in a ring multiplication doesn't need to be commutative

in a field multiplication is commutative

[hollow drift]

oh

i dont understand

also what does ring mean

in this contet

like a circular number line?

[jaunty magnet]

let's focus on fields for now lol

[hollow drift]

aight bet

[jaunty magnet]

basically if you have a set, and multiplication and addition follow each of these rules in that set, it's a field

[hollow drift]

okay okay

give the question

[jaunty magnet]

an example of a non-field would be the set {1,0} for example, with the traditional definitions of addition and multiplication, since the additive inverse property isn't guaranteed (1 doesn't have an additive inverse in this set)

[jaunty magnet]

prove that, if x is a member of a field, -(-x))=x
that's the question

actually here use these axioms instead

they actually explain it a little bit better

{1,0} isn't a set since it violates the additive inverse axiom: there's no additive inverse of 1 in {1,0}; there's no value "-1" such that 1+(-1)=0

[jaunty magnet]

if you're stuck I can give you the answer

take a few minutes though

[hollow drift]

okay im back

i was doing something

[hollow drift]

hmm okay lets see

[jaunty magnet]

don't use the first axioms I posted use the second image

[hollow drift]

k

[hollow drift]

i think I got it

wait

[jaunty magnet]

that was fast lol

what'd you get

[hollow drift]

no hold on

lmao

i got it

-(-x)) = x
-1 * (-x) = x
-x = x/-1
-x =-1 * x/1
-x = -1 * x
-x = -x

right?

[jaunty magnet]

close

[hollow drift]

wait

also

[jaunty magnet]

we don't know that -1*x=-x yet

[hollow drift]

hmmm

wait

[jaunty magnet]

so you'll have to prove that for it to be sufficient

[hollow drift]

thats what I was thinking

[jaunty magnet]

and then also division isn't defined yet, but you can multiply by multiplicative inverses

[hollow drift]

got it

[hollow drift]

can i just say that then

[jaunty magnet]

yeah so in step 3 you'd have to write -x=x*(-1)^-1 instead

[hollow drift]

-1 *x = -x
0 = -x + 1 * x
0 = -x +x

this is true as the additive inverseinverse therofore -1 * x = -x

[hollow drift]

kk

[jaunty magnet]

your proof is backwards but yeah that works

[hollow drift]

ya I thought that lmao

[jaunty magnet]

actually wait

[hollow drift]

but nice did I do it? 😎

[jaunty magnet]

this is ALMOST right

[hollow drift]

BRUH

RELEASE ME

[jaunty magnet]

if you add the additive inverse of 1x to both sides you get that -(1x)=-x

between step 2 and step 3 (reading it backwards)

[hollow drift]

oh okay

right

[jaunty magnet]

so you need to prove that this is equivalent to (-1)*x=-x

[hollow drift]

got it

1 * x = x

a* (b * c) = (a * b ) * (c)

-1 * (x * 1) = (-1 * x) * 1 = (-1 * x) = -1x

YUHHH 🔥

[jaunty magnet]

good job

[hollow drift]

i am ensiten

[jaunty magnet]

ok can I embarass you now lol

[hollow drift]

RAHHHHH

fine

[hollow drift]

time for topology

LMAO

[jaunty magnet]

yeah this is what proofs are like

[hollow drift]

oh okay

cool

[jaunty magnet]

come up with an elaborate solution just for someone else to prove it in like 3 steps lmao

[hollow drift]

thats more simple ha

[hollow drift]

well ya though right often in new theroies there are first eleborate theroys then simiplifed theroys in the future

just like black hole singularitys

[jaunty magnet]

yeah

[hollow drift]

only recnetly were we able to simplify them

[jaunty magnet]

that's called abstraction

[hollow drift]

hmmm okay

[jaunty magnet]

which is kind of controversial for beginning undergraduates lol (like me)

[hollow drift]

i think my brain thinks longer proof = better/more rigours proof LMFAO

[jaunty magnet]

since often it gets rid of intuition

[hollow drift]

which is wrong

[hollow drift]

how so?

[jaunty magnet]

what the fuck is a field

that's how lol

[hollow drift]

i was thinking about doing this but I didnt think it was defined in the axioms that you can add something to both sides and it still being equal

[jaunty magnet]

oh that's a basic property in all of arithmetic

[hollow drift]

i know

[jaunty magnet]

x=x therefore f(x)=f(x) or f(x,y)=f(x,y)
meaning if we let f(x,y)=x+y, then
x=x
f(x,y)=f(x,y)
x+y=x+y

[hollow drift]

but so is -1*x = x but I was still asked to prove it

[hollow drift]

i think this is cheating

[jaunty magnet]

nope

[hollow drift]

LMAO

sorry

[jaunty magnet]

there can be a set where -1 isn't even defined

[hollow drift]

brh

[hollow drift]

isnt this cheating though cause like this proof doesnt explain that adding something to both sides maintains equvilenace

and you cant say its trivial

[jaunty magnet]

that actually got me at first when I started linear algebra

[hollow drift]

what did

[jaunty magnet]

this

[hollow drift]

well

isnt it?

like sure its simplfied but i mean

it missed a step no?

ill ask chat gpt and ill see what it says

[jaunty magnet]

usually in algebra classes that's automatically assumed since unless you're working in some extremely nonstandard area that's always true in math

at that point you'd have to define what it even means for two things to be equal

[hollow drift]

in the context you gave me the only things you are allowed to assume are the axioms

and there isnt an axiom in the ones you gave me that claim this

[jaunty magnet]

sorry lol

[hollow drift]

HA so its not my fault

[jaunty magnet]

these axioms add to the ZFC axioms

[hollow drift]

yes

[jaunty magnet]

whenever someone asks you a math problem assume you're working in zfc as well lmao

[hollow drift]

it was just so basic that like i thought you could only assume the axioms were true and nothin else

[jaunty magnet]

you might like set theory then lol

[jaunty magnet]

you didn't seem to have a problem with dividing both sides by -1 though lol

[hollow drift]

oh ya

idk

maybe im giving exuces so make me self not feel dumb HA

[jaunty magnet]

I don't mean this in a condescending way but get used to feeling dumb in math lol

[hollow drift]

oh ya

no its okay that makes sense

I thought of this quote

"the more you discover the less you know"

i think somone already said that though

[jaunty magnet]

exactly

[hollow drift]

people always stealing my ideas 😭

lmao just playing

[jaunty magnet]

also there will always be moments where you're thinking "what the fuck is happening what does that mean why is that true huh??? etc" when you're learning math

the higher you go the more that happens

[hollow drift]

"what the fuck is happening what does that mean why is that true huh??? etc" when you're learning math" i had a moment like this were I tried to understand why a harmonic series was divergent

[jaunty magnet]

I had to learn to get not hate myself when that happens lol

[hollow drift]

and I tried with intuion

but i only came to the wrong conclusion becuase someone gave me the wrong awsner for something

otherwise I would have been correct

[hollow drift]

ya

[jaunty magnet]

damn

[hollow drift]

it was like half a year ago i wasnt as good with math but bassically someone asked does 1 + 1/2 + 1/3 + 1/4 + ...

and I said it converges (incorrrect) but I later made my own intutive proof

this series can be represnt with an infinite number of finite sseries

however the sum of an infinite amount of finite sums is infinite

1 + 1/2 + 1/3 + 1/4 + 1/5 + ...

can be expressed as (1 + 1/2 + 1/4 + 1/16 + ...) + (1/3 + 1/9 + 1/81 + ... ) + (1/4 + 1/16 + 1/256 + ... ) + ...

these are finite sums however it would take an infinite amount of fininite sums to fully represent the intial sum therfore the intial sum is also infinite and divergent

[jaunty magnet]

that's an interesting proof lol

[hollow drift]

is it valid?

[jaunty magnet]

I'm not sure

[hollow drift]

im sure not rigours

[jaunty magnet]

I don't know enough about infinite series

[hollow drift]

does the proof make sense

[jaunty magnet]

it does

[hollow drift]

okay bet

[jaunty magnet]

it looks rigorous but I'm not sure if rearranging elements in an infinite series yields the same result

[hollow drift]

it does

[jaunty magnet]

well it doesn't if it's conditionally convergent

https://en.wikipedia.org/wiki/Riemann_series_theorem#:~:text=In mathematics%2C the Riemann series,new series converges to an

[hollow drift]

oh

[jaunty magnet]

which is extremely weird LOL

I haven't really thought about infinite series in a while I'll probably learn more about them in analysis

[hollow drift]

WHAT

THATS IMPOSSIBLE

HOW DOES IT DO THAT

wait my brain is on to something

but I dont know what its thinking

LMFAO

I THINK

IT

um

i think it stretched out larger values further

which makes it differnt

wait

RAH

my brain is working

okay

hmmm

it doesnt have enough time to converge back maybe to zer

wait no idk

[jaunty magnet]

also there's the famous 1+2+3+4+5...=-1/12

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_⋯

[hollow drift]

fuck that bull shit

i know that

[jaunty magnet]

it's not equal to -1/12
but it's a way to give an answer to a divergent series

[hollow drift]

it shouldnt be aloud to say that

its not in the same way

i have heard

if we can do this 1+2+3+4+5...=-1/12 then we cant ever have a sum that equals anything

righht?

[jaunty magnet]

it's not "correct"

it's a way to assign a value to a divergent series

since it's obviously divergent lmao

it just shows that you have to be really careful manipulating infinite series since they act really weird sometimes

[hollow drift]

okay good

[jaunty magnet]

I'm guessing the reason it's "wrong" is because the concept of a limit isn't being used here or theres some rule of limits being broken

[hollow drift]

oh here btw

[jaunty magnet]

I don't know enough about this subject though

[hollow drift]

this is chat gpt

but i think its accurate

Notice how the partial sums begin to approach a positive value and diverge from zero. This arrangement manipulates the balance between positive and negative terms, causing the series to converge to
ln2 instead of 0

i think in the series the rearrange meant makes it so thats there "more" postive terms

[jaunty magnet]

yeah I think that's what's happening too

again I'm not qualified to speak though

[hollow drift]

like same thing with a series thats 1-1 +2-2 + 3-3 + 4-4 + 5-5 + 6-6 + 7-7 + ...

but does this series 1 +2 +3 -2 +4 +5 +6 - 3 + 7 +8 +9 +10 -4 also not converge to 0 because I rearranged it

i think i understand

[jaunty magnet]

it looks like unconditionally convergent series can be rearranged to get the same answer though

which means I think your proof is correct

[hollow drift]

wait no it diverges to infinity

[hollow drift]

yaya

[jaunty magnet]

ok I gotta go byebye

[hollow drift]

same LMFAO

[jaunty magnet]

fun math talk lol

[hollow drift]

good convo though

do you wanna talk again sometime?

[jaunty magnet]

sure lol dm me shit

[hollow drift]

sometimes I be doing comp math questions if you wanna try to figure them out with me sometime

[hollow drift]

bet bet

[hollow drift]

solved