serious-discussion

[leaden torrent]

then in high school you pick your courses but theres still minimum requirements

e.g. 1 math course and 2 science courses a year

french generally becomes optional

that kinda stuff

[rose dock]

I think the man is clearly just trying to attack systems he has very little familiarity with

[lime sorrel]

I can critique my own system very well too, but nobody would be interested in hearing it

But I definetively feel like there's something different and dangerous from there that it's somehow trying to be inherited

Also how come many people online get shocked when I tell them in high school I keep 11 subjects for 5 years?

[azure nymph]

Here's the course selection sheet for a sophomore at my old high school it was nicer than average but some of this stuff is standard like English, Math, Social Studies, and Science https://drive.google.com/file/d/1kOoRIJSocnh0nt2bCSY2DhZFEucI5r3M/view

[frozen merlin]

at least you could produce examples

[lime sorrel]

For sure. The critique I can make to my own country is being unable to make student understand how to merge human sciences and sciences together.

[chilly hull]

https://tenor.com/view/cute-cat-funny-tongue-out-lick-lips-gif-16787843

[lime sorrel]

What's the purpose of inventing the best technology if we are not sure what we are looking for in the future and we are not aware of its dangers and where it'll lead us

Also another critique I can make to my own country would be very political, so I may avoid it

[bronze pelican]

David Mumford might be my mathematical hero

[velvet dagger]

Mumford's pretty dank

[hybrid quest]

To Wikipedia!

I did not understand a single word of Mumford's wiki page

[bronze pelican]

Well the reason I like him is because of his writing

Check out his math books

Like Mumford Curves and their Jacobians

[vivid halo]

Mumford's red book is goated

[bronze pelican]

I just learned this book is an appendix in Mumford's red book

[hybrid quest]

For sure, then I might understand some of the words

[worthy breach]

Hey does anyone here have any cats or dogs?

[neat lintel]

two cats

[summer sandal]

I wish I did but my annoying brother is allergic.

[bright hill]

Bruh

Where was I??

[lunar spear]

my ankoyimg brother isn't allergic

I am tho

we have 2

[gritty gazelle]

for those who study in university rn, how do ur uni approach in -person exams

[charred mortar]

Wdym by approach

Like how do they give those exams?

[gritty gazelle]

yea

like open book

or just hard core in persone xams

like 2019

[lunar spear]

I mean

[gritty gazelle]

cause i havent done a exam since december 2019

[lunar spear]

idk where we are but we get maol which is some sort of a bool

[gritty gazelle]

and im due for one in december 2022

[lunar spear]

book

[neat lintel]

exams here are just as they were before covid. closed book, 2 or 3 hours

[gritty gazelle]

which country?

or uni?

[neat lintel]

canada

[gritty gazelle]

idk i feel so uncomfortable doing in persone xams

because i got so used to it

like when was ur last actual exam

[neat lintel]

last april

[lunar spear]

we had like 2 months of online school on 8th grade

[gritty gazelle]

wait why

[neat lintel]

the end of my previous semester

[gritty gazelle]

wait when did u start university?

[neat lintel]

september 2018

[gritty gazelle]

i started in feb 2018

just feels weird

are canadian unis like residential or communters

our unis are commuters

[neat lintel]

depends on the university

i would call uoft (mine) a commuter school

most people commute (either using local toronto transportation or by trains from out of city)

[gritty gazelle]

lol im from au

[neat lintel]

something in a smaller place like waterloo would probably be called a residential, but i'm speaking out of my ass here

i'm trying not to use the phrase "residential school"

[gritty gazelle]

for ours is all communter

we have 2 major in 2 biggest cities

and one in every capital

either usyd or unsw in my city

[fervent pebble]

just not here ig he does it a lot

[neat frost]

Good morning valley!

[fervent pebble]

gm slurp!

[ancient flame]

SLORPY

HI

ILY

@neat frost

!!!!

[neat frost]

MODDY

ILY

[ancient flame]

YAYYYY

[surreal sapphire]

great content

[neat frost]

Indeed

[ancient flame]

yes

[neat lintel]

Like and subscribe for more

[arctic grove]

be sure to share this with friends and family

and subscribe to my patreon

[neat lintel]

And today's video was sponsored by nordVPN

[arctic grove]

and Raid shadows legend

[turbid basin]

guess ill post it here aswell

fill it with study music

or whatever

https://open.spotify.com/playlist/6Dbrmd9PQmD6Bm8e58bK9t?si=ff3c2b81165d4119&pt=2b12ccfc4e993050c2260d3d353895a7

[unique reef]

One dog

That is sleeping on my bed rn

[worthy breach]

What breed?

[turbid basin]

i have a updog

[unique reef]

Shi Tzu

[worthy breach]

Cute.

[unique reef]

I will send some pics

Literally sleeping on my bed lmao

[worthy breach]

I want it😍

[severe swallow]

Thinking about applying for some optimization algorithm job rn. I'll probably get rejected anyways, but I still want to send the applications anyways. Problems:
1 - I have 0 work experience
2 - They want proficiency in C, C++, or C#, I've only touched C++ in an introductory semester course 3 years ago, which went well but I mainly just use python.
3 - My grades aren't stellar in the related courses, but I'm really interested in the area.

[bright hill]

you gotta show some proof of competence tho

you got a portfolio or smth?

[severe swallow]

the only thing I can show off is the courses I passed unfortunately.

so yeah💀

[hybrid quest]

For sure it's good to apply

[severe swallow]

I'll get a thicker skin at least

[hybrid quest]

I suggest, sending in your application, but also reach out directly if you can. Like go on LinkedIn and message someone who works there

Cause they probably get 10 millions apps

[severe swallow]

I don't really know who works there apart from that one contact person. it's some collaboration they do with our university. I doubt they'll get that many apps

[hybrid quest]

Oh you k ow a contact person that's good

Best of luck

Don't be afraid to beef up your skills on paper

[severe swallow]

well "know a contact person". It's just a person looking at the applications

but thanks a lot. I'll uh.. first have to even write a CV

[viscid fable]

@severe swallow mr stockfish

[severe swallow]

?

[unreal brook]

i just thought of a cool idea

what if the number line is actually a circle with one infinity on the other side of 0

[neat lintel]

this is a good idea

look in to "stereographic projection"

it has to be an s shape tho

right

[unreal brook]

why s

[neat lintel]

half circle is minus infinity

the other half is positive infinity

[deep mango]

One point compactification

[unreal brook]

eh

i like circle

then 1/0 is just infinity on the other side of the number circle

[neat lintel]

which one

[ancient flame]

I taught a student domain and range of a function today!

[neat lintel]

there are 2 infinities

[ancient flame]

kinda exciting since it's a topic that a lot of people don't understand well

and im able to give the proper intuition

so ya

[unreal brook]

nice

[neat lintel]

what's a function

[ancient flame]

a mapping from one set to another

there's more rigor but idk it lol

[neat lintel]

checkmate

[dense belfry]

A function from X to Y is a way of viewing X as a bundle over Y

[ancient flame]

oh no

[azure nymph]

A bundle?

[ancient flame]

internet + tv

[dense belfry]

Lol think of the fibers of the elements

[azure nymph]

LOL

[unreal brook]

each element in set a is mapped to one element of set b

[dense belfry]

The preimages of individual elements in Y

[ancient flame]

yeah what's a fiber bundle

[dense belfry]

So a function is the same as a partition of the elements of X among the elements of Y

[ancient flame]

alright

[dense belfry]

Do you see the picture?

Or nah

[ancient flame]

uh

could u draw it pls

sry

I think I may but idk

[dense belfry]

So let's call your function f

[ancient flame]

mhm

[azure nymph]

So you partioned the elements of set X and the elements of set Y together and that's a bundle?

[dense belfry]

For each element y of Y you can think of f^{-1}(y) as the fiber over y

[ancient flame]

oh ok

[dense belfry]

And so you can think of a function from X to Y as a way of partitioning a X up into fibers

[ancient flame]

I see

[dense belfry]

And the fibers are indexed by elements of Y

[ancient flame]

indexed?

[dense belfry]

The function being surjective means the fibers are all non-empty

[ancient flame]

like fuckin

ugh what's it xalled

representatives

?

[dense belfry]

Anyway this is just kind of a nice pov

[ancient flame]

I see

[neat lintel]

might as well call it ction now cause that explanation took the fun out of function!

[ancient flame]

that's cook

LOL

[azure nymph]

I think I kind of got what were you saying

[dense belfry]

It's developed in Goldblatt's Topoi a categorial introduction to logic

And used as a launching off point to talk about bundles in general

Lol did it Llama calculus

It's a nice visual interpretation I think

There are good pictures for it

A lot of the stuff I've been thinking about is just recontextualizations of basic things

[ancient flame]

oh ok I get it now
you take all the wifi & hotspot packages and set them on the ground

[bronze pelican]

Yesterday I got 7 books from the library

[dense belfry]

Yes

[azure nymph]

Also what's a preimage

[bronze pelican]

Now I'm just reading bits and pieces from each book

Like cheese tasting

[ancient flame]

lol

[dense belfry]

The preimage of a subset A of Y under a function f is the set of points that f maps into A in X

[neat lintel]

antecedents

[ancient flame]

nono basically do you know if you look at something and close your eyes but you can still picture that thing in your head? that's after image, but instead do it in the reverse order

[neat lintel]

how do you do it in reverse order

[ancient flame]

that's the part you have to figure out

[azure nymph]

Okay so let me see if I get this now, we take the elements of X partition them in to new non-empty subsets then we index these sets through elements of Y? Like X's elements are partitioned in sets $P_{i\in{Y}}$

[fathom swallow]

Kenshin

[dense belfry]

Yes

And even more than that you can think of the preimage of a subset as being the union of the collection of fibers indexed by that subset

[azure nymph]

Huh that's pretty neat, I just learned about index sets and stuff last night so I'm a bit slow still

[dense belfry]

No worries there are no expectations from me

[pearl moth]

i have many expectations but i don't expect anyone to actually meet them

[ripe wasp]

why does algebra have so much fucking annoying computation

multiplying cycles is my new least favorite thing

[fair mural]

we’re at the same point in algebra

i just got there

[ripe wasp]

what book u using?

[fair mural]

pinter

[ripe wasp]

ah

im using

[fair mural]

the long one

[ripe wasp]

im on this chapter

[fair mural]

i’m on chapter 8 of like 32

basically like 1/4 through the book

this book doesn’t go very deep into anything though, literally just intro AA

[arctic grove]

[wild lantern]

The exercises tend to cover a bunch more stuff in Pinter.

Like, stuff you would see covered in Fraleigh, Gallian or Judson is often pushed into the exercises in Pinter.

Which works out pretty well since it kinda forces you to prove stuff.

Sylow's thm is an example of this.

Finite fields way later too.

Also Cauchy's thm (I think)

[errant merlin]

yes

[wild lantern]

He also does a pretty good job of holding your hand and splitting up hard proofs into pieces with hints. At least to where if you get stuck you can ask for help or find hints online.

[errant merlin]

yes, pinter's book is excellent

[azure nymph]

Why is D&F almost 1000 pages?

That seems a bit much lol

[neat lintel]

my guess is that it's trying to be a reference book (contain a ton of results with proofs) while also trying to be a textbook (lots of exposition, exercises, etc.)

[brave hollow]

covers a lot in terms of topics, covers a lot in terms of contents in those topics compared to any other "intro" AA text, I don't think D&F is supposed to be used like a text which you read A to Z but more like a do these parts or reference text :/

[neat lintel]

what does 形態 mean

i dont feel like pulling up a translator

[brave hollow]

form apparently

the sentence is supposed to mean this is my final form

[neat lintel]

right

i could understand the rest

私
🤨

[brave hollow]

idk about it nami was also talking about that part

[charred mortar]

Yeah I think it’s just D&F trying to be rather comprehensive

[brave hollow]

they said they were unable to understand because they don't know the context

[charred mortar]

Lang’s Algebra is also 900 pages

[brave hollow]

which is DBZ

[charred mortar]

Jacobson’s Basic Algebra series is over a thousand in total too

[ripe wasp]

D&F also has a lot of exercises like at least 15 to sometimes more than 30 per section

[charred mortar]

Tbh I’ve also found the shorter the book the harder it tends to be

[brave hollow]

having read some bits from the 3 of them they feel so different in terms of style of writing

[charred mortar]

Since it usually feels denser

[neat lintel]

isn't the quote "this isn't even my final form" or something

i probably misremember

[brave hollow]

yes that is the original famous quote

this is a different quote tho

[neat lintel]

[charred mortar]

HS books also feel like they have quite a bit of fluff tbh

[neat lintel]

in my experience high school level textbooks tend to have a ridiculous amount of unnecessary content

[charred mortar]

Yeah that too

[azure nymph]

Yeah Stewart Algebra and Trig is 1200 apparently

[neat lintel]

stewart lol

[azure nymph]

I was trying to think of a typical textbook you could probably find lol

[charred mortar]

Also it feels like font size is usually a bit bigger in HS (which is a good thing)

[arctic grove]

HS books arent that terse

[azure nymph]

But yeah I mean if it's used as a reference, comprehensive, and has a lot of problems then 1k seems reasonable I was thinking about Pinter which I think is around 400 pages or so

[neat lintel]

god bless my highschool teacher who said "the book for class isn't very good, here's a copy of spivak instead" to me

[arctic grove]

which automatically leads to bigger page numbers

[charred mortar]

They’re usually the opposite of terse

[charred mortar]

Honestly based

[neat lintel]

or maybe it was stewart they gave me, i don't remember. either way, they were awesome

[brave hollow]

I am kind of reading a section out of a spivak book rn but its called physics for mathematicians

[azure nymph]

Stewart is fine for calc 1 I think

[arctic grove]

which book?

[brave hollow]

I wrote the name in that msg

[charred mortar]

Yeah for calc i-iii I think having a plethora of exercises is good

[arctic grove]

Oh the books name

icic

[charred mortar]

Like it’s still important to learn how to calculate some things

[azure nymph]

I wish it explored some its harder exercises in the lessons though

Cause sometimes those are a bit short then there's like 80 problems to do lol

[neat lintel]

doing all the problems, no matter the book, is usually not a good use of your time

[brave hollow]

its supposed to be equivalent to something like taylor or landau (although these books are probably better if you're interested in physics)

[neat lintel]

man spivak really liked writing books huh

[charred mortar]

Yeah not all for sure, but I think having a nice variety to choose from is pretty good

[neat lintel]

there is only one textbook in existence for which i'd advocate doing every single one of the exercises

[charred mortar]

In a way knowing how to pick good problems is also a way to test your knowledge

[brave hollow]

which one....

[arctic grove]

[arctic grove]

better not be rudin

[neat lintel]

lol i don't like rudin

[arctic grove]

based

[neat lintel]

i am referring to spivak's calculus on manifolds

[arctic grove]

Oo

[charred mortar]

Ah that

[brave hollow]

its a very thin book innit

[charred mortar]

Yeah very short

[azure nymph]

This reminds me a lot of more a calculus book than an analysis book though this is in Italian so I can't read it

[charred mortar]

Many important results are in the exercises

[neat lintel]

its shortness is part of its charm

[brave hollow]

I think Atiyah and Macdonald and CoM are the same size in terms of thickness

[arctic grove]

[charred mortar]

Folland’s analysis book seems similar tbh

[neat lintel]

i have copies of both of these books

let me compare

they are nearly exactly the same

a&m is up just a tiny bit

[charred mortar]

In Folland’s words: “when in doubt, leave it as an exercise”

Fwiw I still don’t think you need to do nearly all of them

[neat lintel]

maybe i'm just projecting, cause i certainly did all of them

[arctic grove]

isnt CoM like

150 pages

[neat lintel]

a little less than that, i think

i'm going to guess 120

146 if you include every single page in the book after the start of chapter 1

i cannot words

[arctic grove]

lol

[charred mortar]

It seems feasible enough to do all of the exercises in CoM at least

[neat lintel]

make sure you do the ones with errors too!

[brave hollow]

I mean if the exercises are major results

no way to skip em :)

[neat lintel]

everyone needs to do the bump function existence problem at least once in their lives (and only once)

[arctic grove]

yeah its like 146

[arctic grove]

what

[neat lintel]

some of the problems in spivak have errors

there's one about linear approximations in the second chapter, and one about integration in the fifth chapter, that have errors, off the top of my head

[arctic grove]

[neat lintel]

finding out what's wrong with the problems is part of the exercise

version of spivak CoM without any errors when

[arctic grove]

[brave hollow]

math server latex project

[neat lintel]

i'd do it alone

i have way too much free time

[arctic grove]

tterra, are u a 4th year student

[neat lintel]

i finished my 4th year

i am doing 5

[arctic grove]

5? as in grad school?

[neat lintel]

nope

5th undergraduate year

[devout nacelle]

Completing his physics major (formally)

[neat lintel]

lmao

[arctic grove]

[neat lintel]

you're not wrong, i plan on taking another symplectic geometry course

[devout nacelle]

Geometry-pilled

Does symplectic geometry fall under "mathematical physics"?

[arctic grove]

Tterra is a known troll geometer

[arctic grove]

symp geo is diff geo

[neat lintel]

other way around

[devout nacelle]

Based

[neat lintel]

why did discord block this message

[devout nacelle]

Weird, I cannot see any word containing one of the blocked ones

[arctic grove]

discord cant take this much maths

[azure nymph]

What do you guys do in the geometry world?

[arctic grove]

they do topology

[neat lintel]

study manifolds

usually equipped with extra structures

[devout nacelle]

Push symbols and lament doing geometry

[neat lintel]

e.g. symplectic form, connection in a vector bundle, riemannian metric

[arctic grove]

did someone say
riemannian?

[azure nymph]

Is Euclidean geometry even relevant then?

[arctic grove]

no

its useless

and should be burnt away to the depths of hell

[neat lintel]

lol

[azure nymph]

Til

[hollow ginkgo]

Yes and anyone who says otherwise is a heathen

[neat lintel]

euclidean geometry is just the riemannian geometry of a simply-connected, complete, and flat riemannian manifold

[arctic grove]

no u are a heathen

[arctic grove]

[azure nymph]

Okay Tera another question if you don't mind, are solids of revolution important and why do we cover them in calculus

[neat lintel]

euclidean geometry isn't really relevant to differential geometry

[arctic grove]

fuck yeah

i can do diff geo in peace

[neat lintel]

i say that, but it is good to keep in mind examples from euclidean geometry every once in a while

[brave hollow]

pretty sure solids of revolution is only relevant to engineering/design

[neat lintel]

i remember seeing some stuff like the sine and cosine rules on general riemannian manifolds while doing curvature comparison theorems

[azure nymph]

Pythagorean theorem helps for arc length

[wooden turret]

[arctic grove]

[wooden turret]

true

[devout nacelle]

[arctic grove]

well im fairly certain i know enough euclidean stuff

[neat lintel]

you cover them in calculus because they're easy to make difficult problems about. they are important in general as they provide a large class of differentiable manifolds which are easy to construct and use as toy examples

[azure nymph]

I had just had basically copy down 2 "proofs" for my homework on arc length and surface area so Pythagorean theorem is fresh on the mind

[azure nymph]

Oh I see so like creating a torus by rotating about y-axis but then you guy don't really care about volume or SA? Probably something about them I'm not aware of I assume?

[arctic grove]

thats just BPT

basic proportionality theorem

Lol

[brave hollow]

I thought it was the diameter inscribed angle one

[arctic grove]

thats twhat the name is here

[arctic grove]

is it?

i remember it as BPT from my oly arc

[azure nymph]

Yeah I thought it was AC is diameter then it makes a right angle

Iirc here Carnot is for thermodynamics

[arctic grove]

carnot engine moment

[brave hollow]

yeah carnot has his name on the eiffel tower

[azure nymph]

And Euler is like nt lol

[brave hollow]

euler is a lot of things

euler is a giant

[azure nymph]

India studied it first anyways

[wooden turret]

[bright hill]

ah yes

let's just change the conventional names of every theorem in existence

you know

just to fuck with everyone

[charred mortar]

I’m not really that big of a fan of naming objects/theorems after ppl tbh

I generally prefer a descriptive name

[neat lintel]

What would you call galois theory then

[charred mortar]

I dunno

I haven’t studied it enough to know what would be a better name

But for example instead of Banach or Hilbert space I’d rather just say what they are

Namely a complete normed vector space/inner product space

[neat lintel]

@charred mortar guess what they call inner product spaces in french

un espace préhilbertien

As in prehilbertarian space

[charred mortar]

Oh pre-Hilbert?

[neat lintel]

Yeah

[charred mortar]

Man

Just use inner product space

I kept trying to remember the mappings from these names to the type of space

For a while

Fwiw I do think there are cases where it’s fine

[neat lintel]

Euler's theorem

[charred mortar]

In that any descriptive name would be too long

[neat lintel]

Which one

[charred mortar]

Huh I can’t think of any rn

Maybe there are no good cases lol

[neat lintel]

Also tbh some mathematicians have cool names

[charred mortar]

(There probably are, my mind’s just blanking rn)

[neat lintel]

Zorn

[charred mortar]

yeah some names are coolios

[neat lintel]

Zermelo

Dedekind

Eudoxus

[charred mortar]

Ngl any names starting with Z sound kinda edgy

[charred mortar]

Damn that one is nice

[neat lintel]

Zarisky

Hyperreals

As a way to construct the real numbers

[torn willow]

https://onepiece.fandom.com/wiki/Z

[bright hill]

that's suuuuper long tho

[charred mortar]

Is it?

It’s just three words

Or four

[neat lintel]

Just use an acronym

[charred mortar]

Yeah that

[neat lintel]

That's what we did in class

[charred mortar]

Acronyms are everywhere anyways

[neat lintel]

K sub vector space would be written as k.s.e.v

[charred mortar]

Locally compact hausdorff as LCH

[neat lintel]

Also

"topological vector space" is very commonly abbreviated as "TVS"

[neat lintel]

Petition to name the theorems in the form "Theorem of <field of math> <number>"

[surreal sapphire]

why

[azure nymph]

Man I started reading a combinatorics books today, it's got hands for sure

[neat lintel]

Some people are sensitive to the name of theorems named after people
So this might be the solution, but not really an effective one to settle disputes

[surreal sapphire]

field of mathematics isnt as clear as you might think, nobody wants to remember numbers

naming things after people is fine tbh, often there just isnt a better alternative

[neat lintel]

[charred mortar]

Yeah now that I think about it it’s hard to describe a Hausdorff space in brief

[surreal sapphire]

i mean you can call it T2 or wtv it is

[charred mortar]

Yeah but that feels like it’s just the same problem in disguise

Maybe slightly better since at least you can follow the general gist for the separation axioms

[surreal sapphire]

its not a person 🤷

[charred mortar]

ig I should clarify a bit; I don't really have anything against naming things after ppl in particular, I just prefer descriptive names in general

[surreal sapphire]

ok sure, thats just a bit hard with math

you would have to invent a lot of words or call everything normal

[charred mortar]

yeah fair enough

[odd narwhal]

You..... Finished combinatorics?

Wow we should tell all the world's combinatorists!

[sleek wing]

yeah

all 3 of them

[brave hollow]

CMarco has finished geometry and combinatorics (idk if I am missing any other subjects) I wonder what they will finish next

[toxic gale]

What's with people having color roles that are discriminatory against us light mode users.

[severe swallow]

why the hell would you use light mode

[toxic gale]

Light mode's nice!

[severe swallow]

https://tenor.com/view/morning-sunshine-blind-bright-gif-5235310

[proud olive]

Light mode is the source of all evil

[sleek wing]

https://c.tenor.com/8nwg1J0vOL4AAAAM/soyjak-spinning-spinning-soyjak.gif

[toxic gale]

I think the default dark mode looks bad. I use either the AMOLED dark mode or light mode.

But my laptop doesn't have that option so I usually use light mode.

[fervent pebble]

betterdiscord

[olive kayak]

the problem with dark mode is

its difficult to real characters with dark colors

like dark blue

[surreal sapphire]

which? we changed recently for this reason

[fervent pebble]

active

[brave hollow]

not very ppl (the role I have)

[fervent pebble]

the only problem role is active

[olive kayak]

light mode if ine

[toxic gale]

I think it's the not very ppl roll.

[olive kayak]

only under bright sun

other than that I prefer dark high contrast mode

[fervent pebble]

oh wait for light mode users

ew

[surreal sapphire]

hm

whats default color in light mode?

[olive kayak]

also I don't make friends with people who use light mode for programming

[olive kayak]

white background with black font

at least thats what most people think of

[surreal sapphire]

ye but what color do no role ppl have

black?

[brave hollow]

white

[olive kayak]

oh

you meant that

[surreal sapphire]

its white in dark mode

[olive kayak]

sorry

[brave hollow]

#chill message

[toxic gale]

I think it's whatever's the opposite of the mode colour. So it appears black in light mode.

[surreal sapphire]

[fervent pebble]

is it worth taking a gap year for college apps if my junior year gpa wasn't very good and i want to double major in math/cs?

[olive kayak]

yes

[maiden bear]

Do applications for math/cs require a good GPA?

[rose dock]

cs yes, math less so

[bronze pelican]

Alexander Grothendeick = AG = Algebraic Geometry

[azure nymph]

Man I just could not care about the lecture today

I think I actually absorbed 0 information lmao

[vivid halo]

class field theory = CFT = conformal field theory

[dim needle]

This is sad to watch. I felt pretty similar to ComplexVaraible in this server, I would talk about the Collatz Conjecture and people would try to talk me out of it or call me crazy. My first visit here I was told I should ask engineers about the Collatz Conjecture because only those weirdos would like it (?). Now that the Collatz Conjecture is more popular, more people want to try the problem and more people are going around shunning them.

I understand the argument that chasing after big problems can be problematic for your career (which is a whole other tragedy I could go on and on about), but the idea you're crazy for going after these problems seems like it's taking it to a whole other level. Does getting excited about a puzzle make someone crazy? Isn't that the point of being a mathematician in the first place? What's wrong with letting novices play with problems that are easily accessible?

I think this attitude towards problem solving is pretty toxic. I can understand if your career is on the line, but as a now computer science major who has no stakes to worry about, working on the Collatz Conjecture has brought me not just joy but passion, inspiration, and productivity. I love working on the problem, I feel like I understand the Collatz Conjecture better than I understand other math at this point. I was inspired by this problem to write code but also to give Calculus a second look, to be more open to higher level math. And since then, I have learned about higher level math in little bits and pieces thanks to working on this problem. I will only continue to learn more, and I'm excited about the future. All of this is thrown aside with this mindset only crazy people work on the problem.

I don't understand this attitude, I think it's doing more harm than good and not just for the Collatz Conjecture but for anything in math. I don't understand, what is achieved in tearing people down for this?

[charred mortar]

Thing is, many people who do try to crack Collatz are often quite misguided or think they’ve found a solution when they haven’t

If you understand that it’s an extremely difficult problem and are just curious about it, that’s completely reasonable

But unfortunately not everyone understands that

And there have been many users in the past who just refuse to learn about the math and progress in Collatz

While stubbornly insisting their flawed way is correct

[sleek wing]

computational fluid tynamics

[charred mortar]

Fwiw you seem reasonable enough based on my first impression

[dim needle]

Even though I would like to solve it, I don't think I'll solve it anytime soon. I feel good about some recent progress I made, but I know for sure I can't call anything I come up with a "proof" until I well, for starters actually write a draft of a proof. And then after that, I'm pretty sure if I go to the tutor center at my college campus they'll find a flaw. And I also find it hard to believe people haven't found what I discovered (aside from 3x+5 stuff, nobody knows much about it).

I get being frustrated with people who don't want to let go about being wrong. I'm personally guilty of this, when I'm not trashing my self-confidence I get super arrogant and cocky and then after several minutes of someone explaining I'm wrong it (hopefully) clicks. My IQ is bellow normal, and I need to work on when I am confident how to not be over confident.

I feel intimidated by looking into the mathematics already out there. I recently looked at what might be a number theory textbook online for free and after learning what a lemma was (finally!) I became overwhelmed by the idea of reading more. I remember in the past looking at p-adic theory, asking what it is, cracking open a book on ergodic theory, and then dying after the first couple of paragraphs. I even pulled an article on the Collatz Conjecture, asked for help here, and barely made it past the first page. There's a huge learning curve that I want to do but I find it personally terrifying to climb.

[charred mortar]

There’s a truly enormous amount of mathematics behind many famous conjectures

But a lot of it is also very interesting in it’s own right

It may very well take many years to catch up to current results

This isn’t me trying to dissuade you or anything, just being realistic

But even if you don’t solve it I think you can learn a nice chunk of cool math

[dim needle]

One thing that gets me though- I remember reading in the book "The Ultimate Challenge: The 3x+1 Problem" and somewhere seeing that 3x+1 and 5x+1 are virtually the same in p-adic theory. And then I thought, well, maybe that's causing people problems!

I have the book, I need to double check if that's real or if I confabulated that though

[sleek wing]

I wish you luck on your journey

[sick burrow]

isn't the consensus that the mathematics we've currently developed is incapable of solving 3x+1

[sleek wing]

yeah so lets develop some more

[sick burrow]

that is, it would take a groundbreaking new field of math to even have a chance at solving it

I wonder what the proof will eventually look like

[dim needle]

gasp I found the NOTA generalization! (pg 67)

Wait did this dude straight up plagiarize Marc Chamberland?

[sick burrow]

but anyways I think the reason people will make fun of you if you try 3x+1 is that at best you will get nowhere

and at worst you will convince yourself you've done something you haven't and annoy people in the process

[charred mortar]

Yeah Collatz is insanely hard

[sick burrow]

and I think there are just better puzzles to play with

Collatz is such a weird problem

like it looks so much easier than it is

[charred mortar]

Yeah same with say Goldbach

NT is wonky

[sick burrow]

Goldbach looks a good deal harder than Collatz imo

[charred mortar]

I haven’t studied enough to know anything about both besides “mega super ultra hard” tbh

[sick burrow]

I'm just talking about a "at first glance" thing

[charred mortar]

Ah ic

Yeah that seems fair

Goldbach intuitively feels harder yeah

[sick burrow]

Goldbach at least has something to do with the distribution of the primes which is a known hard problem

would it be accurate to call the distribution of the primes "the hardest problem in math"?

I feel like a numberphile viewer saying that

yeah the more I think about that sentence the dumber it sounds

what would solving RH get us again?

it's like some bound on the error of pi(n) right

[charred mortar]

Yeah something like that

[dim needle]

No, He does credit Marc Chamberland in the description. I should probably re-watch this video
https://www.youtube.com/watch?v=t1I9uHF9X5Y

[sick burrow]

hi Emma

[neat lintel]

many results in analytic number theory are phrased as "if RH is true, then... (bound)"

at least that's the impression i got of the field

[dim needle]

The Collatz Conjecture is apparently also a million dollars + more, but I heard there's a lot of doubt if the Japanese company offering the 120 million Yen can actually pay that out

[sick burrow]

Hi Emma
Emma vanishes

[charred mortar]

Tbh the prize money for these big conjectures does not seem that worth considering the amount of effort you’d most likely need to sink into them

[sick burrow]

the real reward would be the clout

[charred mortar]

Yeah

[bronze pelican]

one day we will prove Collatz by assuming RH and assuming Not(RH). That's how we get an unconditional proof

[sick burrow]

[dim needle]

I don't really care for the money, I have been studying the Collatz Conjecture before that cash prize was even a thing
I'm having way too much fun trying to figure out "pinch pairs" and getting angry about twin loops

[bronze pelican]

i was a collatz reseracher in highschool

for a brief semester

[azure nymph]

I never got the collatz hype it never seemed very interesting to me

[toxic gale]

Honestly same.

I think it's just my personal lack of interest in number theory.

[sick burrow]

Griffon I hope you understand that the chance you'll actually get anything out of this is 0%

[dim needle]

What did you find? When I first started my instinct was to connect the Collatz Conjecture to the primes (and then failed miserably)

[sick burrow]

not a very small number, but literally 0%

[bronze pelican]

I just reformulated the problem in terms of trees or something i forget

[toxic gale]

I've thought of what I would do for research if I did ever go to grad school. Maybe PDEs or something.

[dim needle]

I got inspiration, something that makes me happy, and something to write code problems on. I think chasing the Collatz Conjecture has been nothing but a great side hobby

[bronze pelican]

that's great

[sick burrow]

I feel like there are probably a lot of cool things to be found in chasing Collatz as an ameteur mathematician

[dim needle]

Like computer science trees or is there an official math tree?

[bronze pelican]

collatz tre

[dim needle]

Did you know that the 19 loop is the most popular loop in the positive numbers for 3x+5? And if I ever bother to actually verify it up to 10 million or whatever, I think the 8-4-2-1 loop is the most popular because it eats a lot of if not all of the negative numbers that are not multiples of 5

[bronze pelican]

what did i do? lemme go check the pdf

[toxic gale]

I've heard a good amount of math relies on the Riemann hypothesis being true. Is the same true for Collatz?

[dim needle]

I would laugh out loud hard if the RH had anything to do with Collatz. Prime numbers... modulo 3 and 2... I don't think so

[bronze pelican]

holy shit

i wrote this in word

[toxic gale]

[bronze pelican]

and i cited a youtube video ◉◡◉

[toxic gale]

Actually math in Word can look somewhat OK I guess. My high school did their lecture notes using MathType on Word. But the built-in equation editor is actually garbage if I remember right.

[charred mortar]

Yeah the equation editor is dodgy

[dim needle]

I hate the equation editor, I just straight up give up and draw it in ms paint

[charred mortar]

It’s ok for simple math

But beyond that it just completely collapses on its face

[neat lintel]

I remember doing polynomial long division in Word. It was a lot of adjusting spaces to make everything line up right.

[toxic gale]

I do hate elitist behaviour like "hurhur only LaTeX, Word is for plebs", but I'd shamefully have to somewhat agree. Word can look OK if you use like MathType or whatever but at that point I think I'd rather just use LaTeX, since it's free.

I still haven't figured out how to do long division in LaTeX.

[charred mortar]

There should be some packages out there

[toxic gale]

At some point I'm gonna need to more properly learn LaTeX. I just know bits and pieces that can make a decent document but I haven't gone to look at more complex stuff.

[azure nymph]

The more I use latex the more I like it

[charred mortar]

Word is nice enough for simple stuff, but anything complicated I can’t work with anymore

Pictures will just go flying about the place

[dim needle]

I asked this on SE awhile ago but I feel like people don't know this.

Did you know you can completely ignore any number that's 5+6n or 13+18n assuming you are counting from 1 to n in the positive space? You don't need those numbers, you already did them. I think technically you may only need to do all of the multiples of 3, but you may have large gaps and not have everything in a pretty line from 1-n.

[toxic gale]

I do think importing pictures on LaTeX is a little more of a pain unless I'm doing it wrong.

[charred mortar]

And it can be annoying to force an equation into what I want it to be

[toxic gale]

I tried using Word to type some math notes but using the equation editor is so damn slow that I'd rather just use LaTeX.

Also LaTeX is so much better after I learnt to define \m as a command for a matrix.

[azure nymph]

What do you mean?

[toxic gale]

No more \begin{bmatrix} \end{bmatrix}.

[sleek wing]

b

[toxic gale]

Damn is there a new country edition of Wew every day?

[azure nymph]

Wew was just azerbaijan earlier

[sleek wing]

I believe this is the new meme yes

[toxic gale]

What's next? Uzbekistan?

[sleek wing]

My name is the most recent country I’ve been banned from entering

[fathom swallow]

PhenomPlasma

[sleek wing]

Light mode detected

Soylet freak out engaged

[dim needle]

I posted this in another server, sorry if it feels out of context:

[toxic gale]

Light mode is nice!

[dim needle]

Fun fact: If you want to check numbers 1-N for the Collatz Conjecture, You can skip all of the 5+6n and 13+18n numbers. Basically, you already do these numbers when calculating the other numbers, so there's no need to do these (If you calculated 3 you already know 5 goes to 1, if you calculated 7 you know 11 and 13 go to 1, etc.). There are other 'redundant numbers' in the Collatz Conjecture, but their formulas suck (every 1,000 to 10,000 numbers or something ridiculous) and I wonder if the modulo check would be more expensive than just calculating the extra redundant number's trajectories again. (Everything 1-100 is covered except for 61 and 91, and after that some exceptions are slightly more common but still pretty rare)

Boring 'proof': || 5+6n and 13+18n numbers are the result of an increase in the trajectory, so a smaller number must have come before and you would have calculated that number before calculating the 5+6n/13+18n number. For example, 5 and 11 are the end result of 3 and 7 increasing respectively. (3 -> 10 -> 5 -> [...]) , (7 -> 22 -> 11 -> [...]). 13+18n is the same idea, but you have a step where the smaller number increases before it decreases. For example, consider 13 and 31: 11 goes to 17 before going to 13, and 27 goes to 41 before going to 31. By doing 11 (or really 7 because you should skip 11) and 27, you would have already found that 13 and 27 also go to one.

I'm mostly confident this is safe to assume for all 5+6n numbers and 13+18n numbers because the rate of how much these numbers increase can be determined. 3 increases by 2 to reach 5, 7 increases by 4 to reach 11, 11 increases by 6 to reach 17, and so on. If you go backwards instead of forwards, -1 increases by 0 to reach -1, -5 decreases by 2 to reach -7, and so on. A similar thing applies to 13+18n (and yes, I suspect the -5 -> -7 loop is related to this pattern as well). As long as you're working with positive numbers, any given 5+6n or 13+18n number will continue to be the end result of a previous smaller number. I say "mostly confident" because I don't think this proof is rigorous, I am assuming that this pattern won't break for some weird reason.||

[sleek wing]

Collatz on Z localised at (3) :troll: problem, number theorists?

[dim needle]

The picture is brand new, I'm assuming that a fraction of n minus something is going to be smaller than n

But I'm starting to doubt my logic there

I think this is a logical leap, I think I should try to more rigorously prove it

"more rigorously"

[azure nymph]

Yeah it's my understanding from hearing people talk about collatz that it requires very high level math

[dim needle]

I think I fixed it by adding this part:

Am I allowed to say that 2 minus extra is going to be smaller than 3?

Is that legal?

[sleek wing]

If n is positive then yes obviously

[dim needle]

Yaaaay I kind of sort of proved something!

Impress your friends! Beat them at the coding challenge for who can do it faster!

Oh I thought it was calculate the most Collatz sequences for project Euler. If it's a matter of what's the longest sequence, I would stick to the powers of 3

Wait but it's infinite, that doesn't make sense... is there a limit to the size of the numbers? Otherwise you can find the longest sequence just by saying you started with zero

Oh... which starting number under a million
https://projecteuler.net/problem=14

837,799. I did it guys

Wait... I skipped all of the even numbers

Crap

Apparently being an even number does not help your step count when counting from 1 to n:

And it's the most non-even even number- only divisible by 2 once

[glad sail]

what is exp?

[neat lintel]

exponent $e^x$

[fathom swallow]

Pencil/Idris

[neat lintel]

exp(x) = $\sum_{i=0}^{\infty} \frac{x^i}{i!}=e^x$

[fathom swallow]

∆y/∆x=πy+π^2x

[neat lintel]

so ya just e^x

[leaden torrent]

sometimes i wonder whether people actually think they're being helpful or if they just want to show off that they know a definition.

[neat lintel]

eell

many a times

they write it as a series

ig it's like that in most programming languages

I am referring to the last part

one of my favorite parts about being on this server is having namington proclaim publicly something i think in the back of my head, like this

[leaden torrent]

i am the voice of the people.

and the people proclaimeth thus:

PLAY UMINEKO

[neat lintel]

there is a tiny

difference

between exp and e^x

they give almost same value

computed differnetly

[leaden torrent]

...

okay, so you don't know the definition

[dense belfry]

Oof

[leaden torrent]

that answers that

[neat lintel]

lmao

ig that's I thought

[dense belfry]

I am considering playing umineko nami

Does it cost money?

[neat lintel]

everything is free if you know where to look, emma

[leaden torrent]

legally the answer to that is "yes"

but not even the original developer cares enough to enforce it

[dense belfry]

I see

[leaden torrent]

take that as you may

[neat lintel]

[leaden torrent]

the best version of Umineko is the Umineko Project offering, available here https://umineko-project.org/en/

technically you're supposed to get a code from the physical game release and use that to unzip the download

[neat lintel]

[leaden torrent]

this is meant to be an anti-piracy measure

but the code is the same for everyone so uh

[dense belfry]

Lmao

[leaden torrent]

i'll let you fill in the blanks

[neat lintel]

e^x is e[a approximated constant] raised to x

exp(x) is the machlaurian expansion of e^x computed at x

but ya they yield same value

ok

[dense belfry]

I don't think that's standard

[leaden torrent]

how do you define "e raised to the x" for real x without some sort of series definition

i guess you can define it as the limit of a sequence of rational powers

[neat lintel]

exp(x) is e^x, because the series is calculated at a=0, so it makes no difference.
The definitions are equivalent

[leaden torrent]

but how do you show this is well-defined

[neat lintel]

u compute e using

$e=\sum_{i=0}^{N} \frac{1}{i!}$

the bigger N the better ur approx

[fathom swallow]

∆y/∆x=πy+π^2x

[neat lintel]

then save that constant somewhere in memory

and raise it to some x

[leaden torrent]

why the programming considerations

this is math

[neat lintel]

Lol

[dense belfry]

Huh

[neat lintel]

"e = \sum_{i=0}^N..."

[dense belfry]

This is a take I wasn't expecting

[neat lintel]

they are two different functions yileding same results vasically

[dense belfry]

If you are approximating they will yield different results

[neat lintel]

[leaden torrent]

so you're positing defining $e^{x} = \lim_{M\to\infty}(\lim_{N\to\infty}\sum_{i=1}^{N}\frac{1}{i!})^{x_{M}}$ where $x_M \to x$?

[fathom swallow]

Namington

[leaden torrent]

this seems both incredibly clunky and obviously equivalent to the usual definition

[dense belfry]

I think you're reading too much into it nami

But I guess you know that

Lol

[neat lintel]

no

[leaden torrent]

then please explain to me how to make sense of, say, $e^{\pi}$

[fathom swallow]

Namington

[leaden torrent]

even if your version of the base e is an "approximation"

[neat lintel]

exp(x) := $\sum_{i=0}^{\infty} \frac{x^i}{i!}$

then u can can prove exp(x)= $e^x$

[fathom swallow]

∆y/∆x=πy+π^2x

[leaden torrent]

how are you proving something based on an approximation?

exp(x) will not equal your version of e^x (given by the sum from 0 to N) for any N

you can prove that they're equal in the limit, sure

which is what i was getting at above

[neat lintel]

I realize approximation is thr wrong term here coz we are into math

[leaden torrent]

but just picking an N doesnt let you prove anything

[leaden torrent]

anyway just a warning here: its really slow to start and most of the characters are kind of assholes

the pacing is pretty bad in general

at least up to chapter 3

but its worth

[neat lintel]

https://math.stackexchange.com/a/1226107

i am just saying what's said here

and what I was told

[leaden torrent]

that link contradicts what you're saying and agrees with me

[neat lintel]

how about you say what you understand instead of saying what other people say

[leaden torrent]

at least the answer you linked to

[neat lintel]

I am saying what's in the answer

[neat lintel]

this

in a nutshell

[leaden torrent]

no, you dont "prove" exp(x) = e^x

you prove it makes sense to call exp(x) e^x

by showing it satisfies properties of the exponential

such as the rule that exp(x+y) = exp(x)exp(y)

and that the base of this exponential is, indeed, e (since exp(1) = e)

[neat lintel]

ig u can prove why

$e^x=\sum_{i=0}^{\infty} \frac{x^i}{i!}$

[leaden torrent]

but this is fundamentally a definition of e^x

[fathom swallow]

∆y/∆x=πy+π^2x

[leaden torrent]

e^x does not "exist" a priori as a mathematical object

you can define e^x as i mentioned previously

$e^{x} = \lim_{M\to\infty}(\lim_{N\to\infty}\sum_{i=1}^{N}\frac{1}{i!})^{x_{M}}$ where $x_M \to x$ is a rational sequence

[fathom swallow]

Namington

[leaden torrent]

this is a totally valid, if incredibly weird, way to go about it.

but usually texts just say:

[neat lintel]

e^x is not defined by the expansion ig

[leaden torrent]

• define exp(x)
• show that exp(x) satisfies various all of the properties of the exponential and that exp(1) = e
• reason that it makes sense to call exp(x) an exponential function with base e, i.e. e^x

and this is how e^x is defined

(you can show that this definition agrees with the definition i gave, i suppose)

(i'm not sure if that's immediate from some theorem or if it'd need to have its own proof)

(it doesnt seem nice in any case)

[leaden torrent]

actually i guess its the same thing as showing exp is continuous?

¯_(ツ)_/¯

it's worth noting that there are multiple ways to define exp(x), though

[neat lintel]

I guess

[leaden torrent]

the power series definition is just one of them

you could, for example, define it as the unique solution f to the differential equation f' = f with initial condition f(0) = 1

[neat lintel]

I don't think the power series version is a definition coz it can be proved

[leaden torrent]

"prove" what?

[velvet dagger]

Any time you have equivalent definitions

[leaden torrent]

prove convergence?

[velvet dagger]

You can start with one and derive the other

[neat lintel]

prove the series of exp concerges to e^x

[leaden torrent]

how are you defining e^x

if you already have a definition of e^x, then yes

as i said multiple times by now

[neat lintel]

ye like this

[leaden torrent]

you can prove that the power series coincides with it

alright, then as i said

[velvet dagger]

You can also prove that one starting with any other definition

[leaden torrent]

that's just proving that exp is continuous

you can start from either definition and prove the other

[neat lintel]

ig that's also what I said

[velvet dagger]

You haven't yet given a reason why your definition is the best definition

And the others should be considered theorems

[neat lintel]

idk if its best Or not

but I generally write it as a series

and then show that that's equal e^x

coincides basically

[velvet dagger]

No my point is this message

Why not flip it?

[leaden torrent]

yeah you make weird ass claims and then, when i argue against them, you try to say "but that's what i was saying all along"

no it isnt

you said the power series definition is not a definition

[neat lintel]

like starting from e^x?

[neat lintel]

power series def of e^x

[leaden torrent]

it's fair to say that alternate definitions exist

[velvet dagger]

Okay look let's say we're in outer space

[leaden torrent]

but calling it not a definition is simply wrong.

[velvet dagger]

We haven't defined e^x or exp or anything

[leaden torrent]

please open an analysis textbook and flip to the chapter that defines e^x

[neat lintel]

I am saying exp is not defined as e^x but u can prove they coincide

[leaden torrent]

i guarantee you they do it in one of two ways:

[velvet dagger]

When I think of e I think Markiplier

[leaden torrent]

• define e^x using either the power series formula, or a differential equation
  or
• define exp using either the power series formula, or a differential equation
• show exp satisfies various algebraic properties of an exponential function and that exp(1) = e
• justify that it makes sense to define e^x = exp(x)

(these two approaches are, in practice, of course the same)

(the latter just uses the notation exp as an "intermediate")

[velvet dagger]

I could say hmm, I'm gonna define exp as a power series. And then show that for any rational x, we have exp(1)^x = exp(x)

[leaden torrent]

(it makes no mathematical difference because we're mathematicians, not programmers, and accept that equality means "the same")

there are, of course, alternate ways to define it

there always are in mathematics

but i have never seen an analysis text use one of these alternate ways

[neat lintel]

haven't I said that here

[velvet dagger]

I could also start by defining e some like way

[leaden torrent]

it's always either "here's a random power series" or "we want to solve the differential equation f' = f given f(0) = 1, here's a power series that does it"

[velvet dagger]

But then I need to make sense of irrational powers, by taking limits

[leaden torrent]

in my experience

[velvet dagger]

There's also starting with log Namington

[velvet dagger]

Also some people like (1+x/n)^n

But those 4 exhaust what I usually see

[neat lintel]

I agree that my programming connection was wrong tho

[leaden torrent]

okay fair

thats meaningfully the same thing tho

but admittedly log properties are easier to prove.

[velvet dagger]

In any event I think this guy is trying to make some sense that like, "the fundamental nature" of e^x is the existence of e and then taking powers of it?

And the other characterizations should be seen as emergent phenomena?

But he hasn't given a reason why, and tbh I vibe more with saying that the fundamental idea is either power series or differential equation

[leaden torrent]

i suppose that's a more charitable interpretation than i was reading into it

i do think that, from a "philosophical" perspective, it probably makes sense to reason that e^x (and real numbers to arbitrary real numbers in general) "ought to exist"

and therefore the power series definition is just formalizing (and verifying) a platonistic prior

perhaps my phrasing has been too formalist

but that perspective doesnt seem helpful to actually doing mathematics, at least at an early level like this

[neat lintel]

is e^x even defined as exp(x) is or e^x=exp(x)

[leaden torrent]

i think you're putting too much weight on notation here

you define either e^x or exp(x)

it doesnt matter which one you define

or which definition you use

or what

and you say the other one is alternate notation

[neat lintel]

what are the definitions of e^x according to u?

[leaden torrent]

it's just that the e^x notation is motivated by it agreeing with how exponentials "ought to behave"

uh i wouldnt be able to list them all lmao\

[neat lintel]

well there are many

like e^(something )

where u actually

have a part

or text

conputing e

separately

or plugging into the series

without computing e

[leaden torrent]

most analysis textbooks i've seen just define e = exp(1) or e = e^1 [whichever notation they prefer], but some explicitly mention "this means that e = sum 1/n!"

or they could use the natural logarithm to define e as well

[neat lintel]

base of ln that also works

[leaden torrent]

usually by saying it's "the" antiderivative of 1/x

and then defining e as the unique real s.t. ln(e) = 1

[neat lintel]

so e^x := (sum 1/n!)^x

and exp(x) := sum(x^n/n!)

but e^x := exp(x) is not true ig
but e^x=exp(x) [which u can prove using Taylor series]

[leaden torrent]

??

[neat lintel]

if I am confusing my notation := means defined as?

[leaden torrent]

i do not understand what is meant by:

(sum 1/n!)^x
how do you raise a real number to the power of another real number?

how would i compute e^π using your definition? or e^(-1.5)?

actually i guess e^-1.5 isnt a problem

but e^π still seems problematic

[ashen pike]

Bruh

You asked me the same thing

[leaden torrent]

yes, because these are the standard queries that intro analysis exists to address

i am not some trailblazing philosopher challenging our foundations of mathematics

i am summarizing a section of a chapter of a textbook

[ashen pike]

Yeah I know. I'm just surprised how often it comes up

[neat lintel]

can we actually we actually compute the exact decimal for irrational exponents with non zero bases

[leaden torrent]

i don't care about computing a decimal representation

i care about asserting that it exists

what does e^π mean to you? if you're defining e^x as that

the usual construction is as such:

we know, at the very least, how to compute rational powers of real numbers

natural powers are "multiply it by itself n times"

which, as long as you have real multiplication, is fine

(and i wont make you justify real multiplication)

and then x^a/b is just the b'th root of x^a

for integer a, b

again, totally fine

so we take some rational sequence that converges to our exponent

for example, the sequence 3, 3.1, 3.14, 3.141, 3.1415... converges to π

[neat lintel]

$e^\pi$ in a definiton would mean to me

$(\sum_{i=0}^{\infty} \frac{1}{i!})^{\pi}$

no

[leaden torrent]

and we take the limit of e^x_n as n → infinity

[neat lintel]

not that

[ashen pike]

Then what's the problem?

[leaden torrent]

there isn't one

it's just that we have to make a specific consideration of this

and verify that it actually "works"

i.e. that it agrees with how we expect exponentials to behave

[leaden torrent]

you are missing the problem

i can accept your definition of the base e

[fathom swallow]

∆y/∆x=πy+π^2x

[leaden torrent]

i do not understand what e^π means

or, for that matter, what 2^π means

[velvet dagger]

^

[neat lintel]

[leaden torrent]

you've taken it as a prior that i can define real numbers to real numbers

but how? how do i know this exists and wont run into contradictions?

this can be proven, of course

but you need to, you know, prove it

you cant just define e^x as that

without considering whether it makes sense

[neat lintel]

actually what is 2^π either?

[ashen pike]

But you do? We just discussed it. Hes not doing real analysis here, just talking definitions loosely

[velvet dagger]

Exactly lol

[leaden torrent]

in mathematics jargon, we call this "verifying a construction is well-defined" or perhaps "verifying a construction correctly generalizes a concept" or similar

[leaden torrent]

that was rhetorical

[velvet dagger]

Worm the conversation is literally about definitions or not

[ashen pike]

I get that, but maybe its too pedantic

[leaden torrent]

again, if you define $e^x$ as the limit of $e^{3}, e^{3.1}, e^{3.14}, e^{3.141}, e^{3.1415}\dots$, this is totally fine

[fathom swallow]

Namington

[leaden torrent]

and in fact, this is what it means for e^x to be "continuous"

(well, more precisely, e^x needs to agree with all such sequence limits)

(not just the decimal one)

(but close enough)

but you need to verify that this actually makes sense

and agrees with how we expect exponentials to behave

this verification is not particularly hard

but it needs to be done

[neat lintel]

e

$\lim_{m \to \pi}(\sum_{i=0}^{\infty} \frac{1}{i!})^{m}$

[leaden torrent]

or else you have no clue that your notion is well-defined

[fathom swallow]

∆y/∆x=πy+π^2x

[neat lintel]

something like that?

[leaden torrent]

that is close to what i was referring to

[neat lintel]

ignore the e on the top

[leaden torrent]

but dodging the question

when you compute a limit of a function, "most" inputs into the limit will be irrational

that's why i phrased it as computing the limit of a sequence

e^3, e^3.1, e^3.14, e^3.141, ...

we know how rational exponents work

[neat lintel]

interesting

[leaden torrent]

the problem is that we don't know (a priori) that irrational exponents work as well

of course, in practice, we "know" that they do

but we have to verify that it works

[ashen pike]

Let m be a rational and the then take the limit - is that a well defined shorthand for taking the limit of sequence approaching a value?

[leaden torrent]

(at the very least, to show the way we think about mathematics makes sense lmao)

[neat lintel]

I can't see how this relates to the orginak topic

[leaden torrent]

this seems... dangerous

you generally cant like, "quantify" over the domain of a function limit like you can over, say, the index of a sum

at the very least it's nonconstructive

i would just say "let x_m be a rational sequence converging to x"

(where here x is π)

and if you do that, then it's just completeness of the reals

[ashen pike]

Cool

[neat lintel]

I see

[leaden torrent]

you can compute limits of the restriction of a function to a domain, i suppose

but if your domain consists of isolated points (like the rationals) then your limits wont be very interesting lmao

so that doesnt really work for what you want

[stray kite]

Anyone else get burnt out by maths and then have to take a short break to regain the interest?

[neat lintel]

Happened to me once

Took like uh

Ig 3 to 4 months break

[severe swallow]

I get burnt out by certain areas or certain kinds of maths regularly, but personally, it is too large of an area to get burnt out by every aspect of it simultaneously

If I get burnt one by one thing, there is usually one other math related thing I want to look at instead

[rose dock]

often

[viscid pecan]

One thing that will help you is that you won’t be able to learn all maths. So just enjoy studying maths

Learn what is important for you*

[proud olive]

That took me a while

Approximately 1 hour and 58 minutes

[arctic grove]

what does this mean

[proud olive]

What's the first thing @arctic grove ?

[arctic grove]

e

[proud olive]

You only need to figure out the first one really to crack the rest

Anything peculiar about that e?

[fervent pebble]

OH

that makes sense

[proud olive]

Shudufu up valley

[fervent pebble]