serious-discussion

[proud olive]

Next year of uni is going to be even harder

For me that is

First semester will go smooth sailing

Second semester will not

[neat lintel]

How do I remember lectures

[proud olive]

Second semester will very much not go smooth sailing

[neat lintel]

I am going insane

[proud olive]

Notes

High quality notes

[surreal sapphire]

from my experience students struggle more in their first semester

[neat lintel]

I literally wrote by pen all material

[surreal sapphire]

and then it gets "easier" by third semester

[neat lintel]

But I have exam in 3 days and I feel paranoid

[proud olive]

Forth semester here i supposedly one of the worst ones

[neat lintel]

I am on third semester and first one was bad because of mine skill issue.

Aka I got lots of debts

Second was the easiest and now I am struggling with current exam period.

Maybe if I don’t sleep before exam and just spend entire night learning

I will remember stuff better

During exam

Free?

I use gnuplot and matplotlib

Although latter is not very good

Because of python(I am biased)

Gnuplot is nice for making latex pdf files

Theoretically hypothetically if you buy Wolfram it should give good tools

But I can’t say much about it because I don’t own it

So I am stuck to Gnuplot and Matplotlib

[grave wagon]

be me
prints out Hatcher's point set notes to read on vacation
bag gets stolen

[stark charm]

thats what you get for not using munkres

[open aspen]

where does one ask a hyperbolic geometry question

[velvet dagger]

Munkres is prob a bit much for most people anyway tbh

@open aspen #diff-geo-diff-top probably

[mossy oriole]

good night all. Does anyone recommend a (free) program to do math exercises? I must put images (graphics) and many symbols and equations... which one is the most comfortable?

(note: LaTex is very difficult for me, I have no way to learn it because I don't speak English and most tutorials are in English)

[open aspen]

you mean a program to typeset handwritten math?

[mossy oriole]

I am looking for something similar to the equation editor that the Word program has (the desktop one), the web version is not complete. the original version has equation input via mouse And also a lot of variety of symbols. I didn't like Libreoffice either.

[next schooner]

You might like mathcha.io

[mossy oriole]

perfect, I think I'm going to have to learn latex haha

[open aspen]

lmaoo

[next schooner]

Good idea

I don't know what language you speak MathInfinite but surely there are resources for learning LaTeX?

[little vine]

What language do you speak@mossy oriole

[mossy oriole]

Spanish

[mossy oriole]

most are in English, as I have searched. I'll look for more tomorrow.

[solemn laurel]

Anyone have the server profile picture link I’d like to have it

Like this one

But link

[compact tartan]

there's a couple versions

there's a 1:2 image that loops around that we use for the icon background and the banner

and then there's a 9:16 image that we use for the invite splash with the torus in the foreground

oops it's actually 1:1

[fair mural]

why do i see a pair of headphones in the top left corner

[ancient flame]

LOL

[compact tartan]

you may also like

[neat lintel]

I thought that was some Identity morphism or smth

[compact tartan]

no it's horseshoe integration

[neat lintel]

Ooh

[compact tartan]

cc @solemn laurel

[ancient flame]

who designed these

[compact tartan]

me

[wild lantern]

Looks noice

[compact tartan]

if you didn't figure out yet I do everything here

[ancient flame]

oh damn

thats insane tho

[tribal terrace]

xdd1 converted me to math enjoyer

[solemn laurel]

Thanks

[steep mountain]

wow i just noticed there is writings behind the figure

[valid compass]

Finally figured out long division after all these years

Skated by without it in high school. Decided I’m going to just learn it and not feel weird anymore

Honestly if I had spent like 2 hours actually focusing on it it would have saved me so much hassle and stress

[charred mortar]

Ngl that sentiments has held quite true for me in general

Something seems hard/annoying, and then I sit down and go over it in a few hours/days, and then it clicks and I wonder what the fuss was all about

[arctic grove]

wtf

that sucks so much

[brave hollow]

whoever stole your bag is gonna get into math because of you now :3

[arctic grove]

trueeee

an intro to topology will leave them starving for more

[toxic badge]

hi

[atomic wraith]

There's a crank in the Manim (mathematical animation software) server who thinks Pi is not 3.14159... and talking to him is making me lose my sanity

He relates it to the "inverse square law" yet he clearly does not know what it means

[torn willow]

Ok?

[neat lintel]

Nice pfp drake

[lunar spear]

$\pi=5$

[fathom swallow]

.,..

[neat lintel]

What the

[lunar spear]

Im an engineer

[neat lintel]

Not

Hot*

[lunar spear]

not really

[neat lintel]

What type?

Like what type of engineering

[lunar spear]

Im not an engineer

but I might go to electrical

[neat lintel]

Oooo nice

[lunar spear]

after I finish this one

in 2-3 years lol

[neat lintel]

This one?

[lunar spear]

Im in trade school or whatever its in english

electrician and automation technician

I think

[neat lintel]

That’s what you’re studying?

[lunar spear]

I think thats what its in english

with dual qualification I think

my english is bad

[neat lintel]

That’s AMAZING

what

How you so smart

[lunar spear]

Im a dumbass

wym smart

[neat lintel]

Smarter than me bruh

[lunar spear]

lol

[neat lintel]

Fr smart af

[lunar spear]

Tf

I get horrible grades from swedish and finnish

and it was the first year

so I dont have to take swedish anymore which is good

[neat lintel]

Nah bro

Trusttt yoselfffff I get trash grades too, and academic performance doesn't necessarily show how "intelligent" you are

[restive bough]

what is your native language?

[lunar spear]

finnish

[neat lintel]

It just shows your academic ability 🤓🤝

[restive bough]

ah, that would make germanics annoying, yeah

[lunar spear]

physics chemistry and math have gone well ig

and english but that shouldnt have happened

[neat lintel]

i think my help question might be to hard

[echo fox]

230 - 220 * 0.5 = 230 - (200 * 0.5) = 230 - 110 = 120
that's what i chose :|

[neat lintel]

I'm pretty sure that's correct and it's 220, not 200

[echo fox]

yes tiny mistake forgive me dear god

[neat lintel]

Unless the subtraction operation there has a higher precedence than multiplication operation, which is nonsense if true

[echo fox]

imagine answering 5 lol that would be your age

[neat lintel]

[sour fox]

Hello I've got beautiful drawings here ya go

[cyan goblet]

when taking notes over a textbook, do you write down every theorem? if not, how do you choose which theorems to take note of (or in general, what to take note of)?

[proud olive]

The one that's the least obvious

Idk tho. I haven't seen the hardest theorems yet

[next schooner]

I generally note stuff that is of utility but that I am also likely to forget

Stuff with niggly details, usually

Then again I am extremely sparse with note taking

[modest rune]

I personally do not write down anything that is written down in the book

[next schooner]

I see it primarily as a means to record things I'd otherwise not remember, and even then it's mostly just to help it go into memory through the whole weird pen-paper brain memory magic. I don't think I've ever actually gone back and read any notes I've taken when the original text is available

[modest rune]

And only write down things that are not in the book

i.e. details left out of a proof or intuitions or connections i find

[cyan goblet]

interesting

[next schooner]

Yeah I also do that max

[cyan goblet]

i wrote down like

[modest rune]

I feel like there is no reason to be a scribe and duplicate material that i know how to find

[cyan goblet]

every theorem of hoffman kunze

but in its respective note file

and i found it a bit tedious and want to try something new for analysis

[modest rune]

My honest take is that almost every second spent on taking notes out of a textbook

could be better spent doing exercises

[cyan goblet]

amen

i like that idea

[next schooner]

I do, however, transcribe any and every commutative diagram I find just because I find them satisfying lmao

[modest rune]

lmfao

i could send you some that might change ur mind

i wonder if the cursed one i sent here the other day is findable

[next schooner]

My notes on hatcher are just a big unannotated list of commutative diagrams and arcane pictures of blobs and arrows

[cyan goblet]

like this is one of my notes pages on F[x]

idk it just

isnt notes anymore

it's just my textbook reformatted lmao

[next schooner]

Yeah I feel like people either transcribe everything or nothing

By and large

I found my undergraduate folder a couple weeks ago. Four years worth of work and it's just 100 or so pages of scribbled calculations that are completely indecipherable now

[cyan goblet]

lmfao

[modest rune]

this is not as bad as the final version

but its still awful

[next schooner]

I hate that

[cyan goblet]

oh also side question, if i've already read goldrei set theory is it necessary to read chapters 1-5

[next schooner]

Why does zero make three appearances

[cyan goblet]

goldrei already talks about all of this stuff i think

[next schooner]

Meh I skip basically everything I already have a decent grasp on. I'll refer back when required but context generally makes that unnecessary

[cyan goblet]

kk

i'll probably just skim it for review then

[modest rune]

@next schooner

i found the final version

[next schooner]

Gross

What even is that for

[modest rune]

It was a project I was working on

[next schooner]

None of that notation is familiar to me

[modest rune]

its my own notation

[next schooner]

Oh based

[modest rune]

its horrible lol

[next schooner]

Yeah I had to make my own notation for one of my recent papers

[modest rune]

but i did not have a better idea

i still dont really

[next schooner]

Coauthors and I spent like two weeks trying to figure out a better, more canonical way of writing things

All the sub and superscripts contain bespoke permutation operators

Most hideous calculation I've ever done lmao

[modest rune]

yeah

sometimes things are just awful

[next schooner]

Indeed

Numerics is particularly bad for it to be fair so I kinda brought it upon myself

[surreal sapphire]

have you seen the cursed commutative diagrams i posted recently

[modest rune]

lmfao

[maiden bear]

good lord

[next schooner]

Loch that is absolutely heinous

[proud olive]

I refuse to study that

[surreal sapphire]

this isnt the worst one i think

[velvet dagger]

Wow

[crimson whale]

Hi everyone, hope I don't interrupt anything

I have two question and I hope I get some detailed answer on it, not necessarily write it by you even if there an external link for good information about my question I'll be more then thankful !

1 - it's has been years since I graduated, I only " for real " remember how to ( + , - , / , * ), and know I'm studying CS & CE on my own kinda, and planing to dig deeper and be a real CS and have a solid understanding for the Fundamentals of this world.

and for that I realize how mathematics are great and how essential it's for everything we had today, for that I need to go over all stuff again and more, but I don't know the pathway should I take and the order, I would really appreciate someone Help for that, I would love to be able to have the material that I'll learn from available on my Personal computer like books or videos I can download, so Resource can Help me that I can learn from without a lot for search or asking every bit for help " something will help to learn and gather everything no missing pieces that what I mean "

at the same time an exercise that I can test what learn and my understanding with .

2 - if I have an equation or an expression and want to know what the symbols in it means so that I can search and learn more about it and make sense of it is there any tool could help me with this ? " for example I'm reading book about something and it's has a lot of a expression forms " I want to know what it's mean to try to keep up with it

[modest rune]

You can post a screenshot or picture here

otherwise a website called de-texify can help you at least put a name to certain symbols

As for a path for learning, the best thing would be to go through something like khan academy until like calculus and then ask for specific recs

[crimson whale]

I'll check that, thank you!
also more then one person recommend khan and I'll go with it

[empty stratus]

¡my eyes!

ignites; rolls on the floor, trying to snuff out the flames

[vast surge]

These are the best commutative diagrams: https://math.ucr.edu/home/baez/trimble/tetracategories.html

At least, the best ones I could find.

[vestal silo]

Just a bit confused on the definition of a field; would a set containing all primes, for example, be a field? Arithmetic operations are possible but, for example, 5 + 7 would produce a value not in the set, so it's not a valid calculation?

[dense belfry]

So you don't have division@vestal silo

That's one thing that goes wrong here

You also don't have closure under multiplication or addition

Do you want some examples of fields?

[vestal silo]

Please, yeah

[dense belfry]

Okay so the first one is the rational numbers

[empty stratus]

The real numbers are another example.

[sleek wing]

wtf new emma pfp

[dense belfry]

You have the ability to add two rationals to get a third, similarly for multiplication and division by nonzero rationals

And the rationals satisfy commutativity of addition and multiplication

As well as associativity

[vestal silo]

Okay, so each operation produces a valid result is the general definition

Does the result have to be within the set?

[dense belfry]

Yes

[vestal silo]

Ah

[neat lintel]

can someone tell me if a slot machine with 5 different icons with all same chance what is the chance for 3 of the same in a row

[dense belfry]

This property is called closure under addition and multiplication

[vestal silo]

That really clears it up, tysm

[sleek wing]

addition and multiplication are binary operations on a field K, meaning they map KxK -> K, if they mapped to values outside of K then we could not say they map "-> K"

same is true for any binary operation, they must map back into the set

[vestal silo]

Right, makes sense

[neat lintel]

How do you guys solve hard word problems, stuff that takes quite a lot of logic?

Do you set up the problem, is there a special step-by-step analysis method you use or just practice a dump load of problems

and the skill becomes automatic

I'm trying to become better at word problems, I did good on the calculation part on my tests. But these abstract word problems at the end, they get me twisted

[modest rune]

practice mostly

there are tips and tricks

the first step is to try to get rid of all the extra info

[neat lintel]

I'm all ears

like elephant ears rn, enlighten the heck out of me

[modest rune]

like who cares that the watermelon salesman's name is ryan, or that he is selling watermelons in the first place

The important thing is to figure out the model that is appropriate to the question, and to figure out your knowns and unknowns

Writing those down can often help you go step by step

even if you don't see the full path ahead

[neat lintel]

ok so f ryan gotcha, what's next?

[modest rune]

this is called "following your nose" in math

Unit analysis (dimensional analysis?) is often helpful

like

[little vine]

Practice converting the relationship into algebraic expressions

[modest rune]

if I give you miles/hour

[little vine]

Is what I'd say

[modest rune]

and i ask "how long did it take"

you are probably going to want to use something in terms of miles

and something in terms of miles per hour

and convert that into something in terms of hours

[neat lintel]

mhmm

[modest rune]

yeah there are the basic algebraic things of like

If I say ryan sells twice as many watermelons as grapes

how do i model this with algebra?

[neat lintel]

2x + x

I guess

[modest rune]

Not quite

Or

well i guess it depends what I am asking

if I ask for total sales

yes 2x+x

[neat lintel]

Is that not it? dang

[modest rune]

But I would break this into two steps. First I would let x be watermelons and y be grapes, and then I would change the words into algebra by saying x=2y

[neat lintel]

right, we set it up to x = 2y

[modest rune]

Then you have that info, and you can write it down, and then keep doing this until you've turned all the important info into algebra

then you have to figure out how to turn your algebra into your unknowns

[empty stratus]

Unit analysis is the most OP technique.

[modest rune]

and thats where you have to be a bit clever

[neat lintel]

I'll look into what that is!

[empty stratus]

It's the following principle:

In a physical problem, you can use units of measurements to aid you.

[neat lintel]

Yeah honestly, I think that's a part where I mess up often. How do I turn my algebra to find the unknowns?

[empty stratus]

For example, Newton's Second Law states F = ma

Where F is force, m is mass, and a is acceleration.

[neat lintel]

si

[modest rune]

well this is not a mathematical trick

but 9 times out of 10

you will want to use whatever you just learned in class lol

[empty stratus]

The units of m are, say, kilograms; a, meanwhile, has units meters / second^2

[neat lintel]

lmfao hahah I figured

[empty stratus]

So, when setting up an equation using Newton's second law...

[neat lintel]

mhm

[empty stratus]

You need to check that you end up with quantities with the units kg • m / s^2

[modest rune]

this is where it becomes very important to understand the math you are doing both algorithmically (i.e., you know the steps to solve it) but also conceptually (i.e. what this might mean in terms of the real world or something)

If you know the latter really well

[empty stratus]

When working with word problems, you'll often be given numbers, variable, and constants that have specified units of measurement.

[modest rune]

hopefully the correct approach kinda jumps out at you

[neat lintel]

Right, I guess focus on that

[neat lintel]

I think I need more practice tbh, that might just be the key to improve my analysis

[empty stratus]

So, for example, if the problem asks you to find an expression for a quantity with units:

dollars / hour

[neat lintel]

mhm

[empty stratus]

you can use that to guide your usage of the given constants and variables.

[neat lintel]

Ah I see!

Yeah no, that's not too confusing as I initially thought

I thought it was something complex

[empty stratus]

Indeed, it tells you that you would need to divide a quantity with units "dollars" by a quantity with units "hours".

So, for instance, suppose you are given a formula f(t), where t is time (in hours), and where f(t) is "dollars made"

[neat lintel]

uh huh

[empty stratus]

Then, to get the units "dollars / hour", you would divide f(t) by t: f(t) / t

t / f(t), on the other hand, couldn't be the right answer to a question asking for an answer in units of dollars / hour

Why?

Because t / f(t) has units:
hours / dollar

[neat lintel]

right

[empty stratus]

Likewise, if you have a quantity that has units

dollars / frenchmen

and a quantity that has units:

frenchmen / hour

if you want to get units:

dollar / hour

you multiply the above two quantities by one another

[neat lintel]

ye

[empty stratus]

because the frenchmen units in the numerator and denominator will cancel out.

[neat lintel]

rigggght

[empty stratus]

Would it make sense to add the two quantities?

(dollars / frenchmen) + (frenchmen / hour) ?

[neat lintel]

well ... no

[empty stratus]

Correct!

[neat lintel]

WAIT

no

[empty stratus]

They have different units.

[neat lintel]

I thought that was "dollar" yea no

[empty stratus]

As such, adding them leads to gibberish.

[neat lintel]

right

[empty stratus]

This is dimensional analysis (a.k.a. unit analysis) at work.

[neat lintel]

Ayo.

[empty stratus]

By examining the units of the quantities you are working with, you can discern which operations you can or cannot do.

It also gives you a quick way to check your answers:

[neat lintel]

@empty stratus Thank you brudda. I'm happy you explained it fairly simple and it was interactive (love that lmao, ty). Helps retain ppl's attention

Are you a teacher? (In this case yes, but as professionally I mean)

[empty stratus]

TA

Though I love teaching.

And like being good at it.

😄

[empty stratus]

If your units at the end aren't of the correct type (the ones specified by the problem), you made a mistake somewhere.

[neat lintel]

that's always the case with me LMFAO

🥲

[fervent pebble]

let's say $S = \int_a^b f(x)dx$, $a,b \neq \pm \infty$ and f(x) is continuous on the interval. are we always able to find S?

[fathom swallow]

valley

[fervent pebble]

a and b can be any real number ofc

[dire crag]

Hello, everyone. Does this server have a channel for challenge problems?

[static loom]

I would think that's what it means to be integrable, yeah?

[fervent pebble]

oh, right

i meant continuous

oops

[next schooner]

yes

[fervent pebble]

is this just like obvious or is there a proof

[next schooner]

It isn't obvious, no

There's several ways to prove it

The traditional way is to invoke uniform continuity over the interval to bound arbitrary upper and lower riemann sums by epsilon

The difference of the riemann sums, rather

[fervent pebble]

hm

[next schooner]

Spivak does it a lil differently, from memory

But I never really liked his treatment of it much

[chilly coral]

@fading hull In regards to the limit from zyzzs problem, I'm pretty sure the only requirement to move the limit inside the function is that the function be continuous at x = lim t->0 sint tant /t² = 1, but the floor function is not continuous at x = 1, no?

[fading hull]

Yes you're right in that sense

If you look back at the convo tho I mentioned one more thing

The function is even immediately surrounding x=0

So if the limit exists then x- = x+

Specifically on the interval (-π/2, π/2)

@chilly coral is that cleared up now?

[chilly coral]

Here's my problem though: the function could easily be translated left and right as much as you want. It seems like, if this really works, then we don't need the function to be continuous at the limit of the inside function; rather, the limit of the inside function just needs to exist. This doesn't seem right to me though

Or, perhaps, the limit of the whole function need exist? Depends on if you're requiring the inside function to be even, or the entire function itself

[fading hull]

You have to use symmetry to take this limit

And if the function were translated left or right, you could easily make a change of variable to center the function at 0

Like

Lim x->1 f(x-1) =
Lim u->0 f(u), where u = x-1

[chilly coral]

But therein lies my problem. If this is true, that f(x - h) being even is enough for lim x->h f(g(x)) to equal f(lim x->h g(x)), then it seems continuity isn't even necessary

[fading hull]

not continuity on the whole domain

but piecewise continuity

like consider floor(x^2), what do you get when x = +1/2 and x=-1/2

[chilly coral]

Doesn't seem it has to be continuous at all. We just need lim x->h f(x) to exist, i.e., f(x - h) is even

[fading hull]

that's not what even means

[chilly coral]

Is there ever a case where f(x) is even but lim x->0 f(x) doesn't exist?

[fading hull]

sure

1/x^2

[chilly coral]

I would say the limit exists, it's just infinity

[fading hull]

yeah fair we could say that

i see what you're getting at

[chilly coral]

I guess you could have something like f(x) = {1, x < -1 and x > 1

It's even but lim x->0 f(x) doesn't exist

[fading hull]

like a step function yeah

but that's why we also require that the limit of the inner function exists from both sides

[chilly coral]

It seems like the only exceptions to f(x) being even -> lim x->0 f(x) exists is cases where f is undefined for (-a, a), a ∈ ℝ^+

[fading hull]

like sqrt(x^2 - 1) for instance

yeah which is why we require piecewise continuity in the neighborhood of h

[fading hull]

here's my proposition in full:

1. lim x->h g(x) exists from left and right
2. g(h-t) = g(h+t) for sufficiently small t
3. f(x) is piecewise continuous from one side. this will be the right side if g(h) > 0, and the left side if g(h) < 0

[chilly coral]

2 is implied by 1, no?

[fading hull]

not really, consider an odd function instead

[chilly coral]

Ah, okay

[fading hull]

last line

[chilly coral]

But what about with something like lim x->0 f(g(x)), where f(x) = 1, x ≠ 0, and g(x) = 0?

Requirement 1 is held: the limit of g exists

Requirement 2 is held: g is even

Requirement 3 is held: f is piecewise continuous around x=0

However, f(lim x->0 g(x)) = f(0) = undefined, but it's clear the answer ought to be 1

[fading hull]

i amended 3

[chilly coral]

And what if g(h) = 0?

[fading hull]

hm. i guess in our previous example we also had that f(g(h)) was defined

like floor(1) is defined

but if f(g(h)) is not defined then we take it to be lim x->g(h) f(x)

we're getting somewhere

this is good

[fading hull]

WLOG let's assume that h = 0, this will save a lot of confusion, and can easily be converted through change of variable. if you will accept this it makes everything much simpler

[chilly coral]

Fine by me

It's times like this where having experience in real analysis would be really helpful for me

[fading hull]

if g(0)<0, we require f(0) to be continuous from the left, if g(0) > 0, we require f(0) to be continuous from the right. if g(0) = 0, we require that lim x->g(0) f(x) exists from both sides

[chilly coral]

In my previous example, lim x->g(0) f(x) = lim x->0 f(x) existed, but we still couldn't write f(lim x->0 g(x))

[fading hull]

right, because f(g(0)) is undefined

if f(g(0)) is not defined then we must take it to be lim x->g(0) f(x)

[chilly coral]

Is there ever a case for any f and g where lim x->0 f(g(x)) is not equal to lim x->g(0) f(x)?

[fading hull]

to summarize, assuming that h = 0:

1. lim x->0 g(x) exists from left and right
2. g(-t) = g(t) for sufficiently small t
3. f(x) is piecewise continuous either one side or both sides. if g(0)<0, we require f(0) to be continuous from the left, if g(0) > 0, we require f(0) to be continuous from the right. if g(0) = 0, we require that lim x->g(0) f(x) exists from both sides
4. if f(g(0)) exists, f(g(0)) = lim x ->g(0) f(x)

[fading hull]

if the limit exists then yes that's always true

[chilly coral]

So it seems we have a new requirement: f(g(0)) is defined. Otherwise, it's just like any other limit

[fading hull]

yes

[chilly coral]

Okay, how about this:

lim x->0 f(g(x)), f(x) = {1 if x=0, 0 otherwise} and g(x) = 0

Requirement 1: lim x->0 g(x) exists

Requirement 2: g is even

Requirement 3: g(0) = 0, and lim x->0 f(x) exists

Requirement 4: f(g(0)) is defined

Therefore: lim x->0 f(g(x)) = f(lim x->0 g(x)) = f(0) = 1, yet the answer ought to be 0

[fading hull]

right i was about to say, we don't need requirement 4 at all, it's a limit

the actual value of f(g(0)) doesn't matter

[chilly coral]

But without requirement 4, the issue still persists for my example that lim x->0 f(g(x)) ≠ f(lim x->0 g(x))

[fading hull]

right, because f is not continuous. so the calculation should be:
lim x->0 g(x) = 0
lim x->0 f(x) = 1

[chilly coral]

So f has to be continuous? Seems we came right back to the start

[fading hull]

no it's just that we have to take the inner limit first

[chilly coral]

Then what requirement do we fail? lim x->0 g(x) certainly exists, g is certainly even, and lim x->g(0) f(x) exists. Yet, lim x->0 f(g(x)) ≠ f(lim x->0 g(x)). Either I've made a mistake, or these requirements are not comprehensive

[fading hull]

right, moving limits inside is only valid for continuous functions

[chilly coral]

Isn't your whole point that the function doesn't need to be continuous to move the limit in, and that's why you could move the limit into the floor function despite it not being continuous at x=1?

Otherwise, what are all these requirements for?

[fading hull]

hmm i see what you're saying

the 4th point should be that f(g(0)) = lim x ->g(0) f(x), if f(g(0)) exists

[fading hull]

otherwise we have to do what i did here with two separate limits

[chilly coral]

Isn't that just the definition of continuity?

[fading hull]

f(g(0)) doesn't have to exist though

[gentle bluff]

Does anyone know linear algebra that could help me with my question :((

[fading hull]

#linear-algebra

[chilly coral]

And if it doesn't?

[fading hull]

then that condition doesn't apply

[chilly coral]

So it only applies when f(g(0)) exists?

In other words, f(g(0)) must exist for us to move the limit inside?

[fading hull]

no

what i'm saying is that if f(g(0)) exists, it must be equal to lim x->g(0) f(x)

[chilly coral]

In other words, if f(g(0)) exists, then f(x) must be continuous at g(0)?

[fading hull]

yeah, in order to move the limit inside

[chilly coral]

Can we still move the limit inside if f(g(0)) does not exist?

[fading hull]

yes

[chilly coral]

So, just for clarification, what are all the requirements now?

[fading hull]

to summarize, assuming that h = 0:

1. lim x->0 g(x) exists from left and right
2. g(-t) = g(t) for sufficiently small t
3. f(x) is piecewise continuous either one side or both sides. if g(0)<0, we require f(0) to be continuous from the left, if g(0) > 0, we require f(0) to be continuous from the right. if g(0) = 0, we require that lim x->g(0) f(x) exists from both sides
4. if f(g(0)) exists, f(g(0)) = lim x ->g(0) f(x)

[chilly coral]

Okay, here we go:

lim x->0 f(g(x)), f(x) = 1, x ≠ 0 and g(x) = 0

1. lim x->0 g(x) exists
2. g is even
3. g(0) = 0 and lim x->0 f(x) exists
4. f(g(0)) = f(0) doesn't exist, so the condition doesn't apply

Now we have lim x->0 f(g(x)) = f(lim x->0 g(x)) = f(0) = undefined, yet the limit should be 1

[fading hull]

yeah that's the thing

we're really taking two limits here

$$\lim_{x \to 0} g(x) = L$$
then our answer is given by
$$\lim_{x \to L} f(x)$$

[fathom swallow]

EndTimes

[chilly coral]

But that's true of any g and f

Are we not trying to find the requirements for lim x->0 f(g(x)) = f(lim x->0 g(x))? Why bring up this general property?

[fading hull]

yeah i see what's going on. in zyzz's example we actually did have that f(g(x)) was continuous at 0

f(x) didn't need to be continuous though

but f(g(x)) was

[chilly coral]

f(g(x)) wasn't continuous at 0 though. It was undefined at 0

Moreover, if f(g(x)) is continuous at 0, that means lim x->0 f(g(x)) = lim x->g(0) f(x) = f(g(0)), which is equivalent to saying f(x) is continuous at g(0)

[fading hull]

yeah i'll get back to you on that

i still think this is interesting and that there's something here but it needs some more thought

[chilly coral]

It's 1:42 am where I'm at. I gotta get some sleep anyway

[fading hull]

this was entertaining

gn

[chilly coral]

Gn, if it's night where you are

[alpine zenith]

poggers I finally understood a trig equation without crying

turns out studying really is helpful. who would've guessed

[iron osprey]

u

[neat lintel]

Are there any fun calculus programming projects that one can do? Ones that lean more to the maths side than a physics simulation?

[warm oak]

Some solvers for nonlinear functions use calculus, like Newton method and rational interpolation method

Basically just boils down to adding derivates (actual derivatives, not approximations) in the code

[bright hill]

also, visualizing the maclaurian series or smth is cool

visualization in general is p cool

[alpine zenith]

goddammit
test was trig and algebraic patterns and I completely understood algebraic patterns, but the trig took so long that I couldn't finish the last 10 points of algebra in time 😔

I'm going to weep that algebra was like, free marks, and I lost it because of trig

[wild drum]

was it like right angled triangles trig or

[alpine zenith]

gonna strangle the guy who invented trig

[wild drum]

advanced trig functions?

or like

[alpine zenith]

advanced

[wild drum]

ah

[alpine zenith]

wish I'd left the trig for last and started algebra first but noooooo my dumb brain has to do things in order or it collapses

20% gone

[restive bough]

trig: when in doubt, draw a triangle (or a circle)

triangles are just circles with some bits missing

[lunar spear]

Yes

[bright hill]

...

I hate you

[arctic grove]

lmao

[surreal sapphire]

why are there two standard books by the name of "an introduction to the theory of numbers", one by hardy and wright and one by niven et al

this should be illegal

[azure nymph]

I agree

I always assume everyone is just referring to hardy and wright though

[open aspen]

This is why all textbooks should be referred to be ISBN or Library of Congress index in every day speech

No room for confusion

[leaden torrent]

author names work except specifically for rosen

[azure nymph]

If you just use author name wouldn't Axler also be confusing

[surreal sapphire]

Lang

[azure nymph]

Don't remind me lol

"Read Lang" could probably cover most core subjects

[neat lintel]

@maiden loom here

[maiden loom]

hmm

so

ill try to put that into the equation and see if it work

[neat lintel]

Sure

Also @maiden loom n must be 1 here btw

${-1 \choose k} = (-1)^k$

[fathom swallow]

Pencil/Idris

[maiden loom]

NOOOOOO MY BOOK HAVE RAN OUT OF SPACE

[neat lintel]

F

Interestingly

${-2 \choose k} = (-1)^k \cdot (k+1)$

[fathom swallow]

Pencil/Idris

[neat lintel]

Hmmm

So

${-2 \choose k} = {-1 \choose k} \cdot (k+1)$

[fathom swallow]

Pencil/Idris

[maiden loom]

$f^{[-n]}(x) = \lim_{h\to 0}\sum_{k=0}^{\infty}f(x-kh){{n + k - 1} \choose k}h^n$

[fathom swallow]

byte

[maiden loom]

so uhh

[neat lintel]

Hmmm

[maiden loom]

this doesnt make much sense

[neat lintel]

Shouldn't

h^n be in fraction

[maiden loom]

1/h^-n = h^n

[neat lintel]

Oh yeah mb

[maiden loom]

the problem is

0^0

[neat lintel]

$f^{[-n]}(x) = \lim_{h\to 0}\sum_{k=0}^{\infty}(f(x-kh){{n + k - 1} \choose k}h^n)$

[fathom swallow]

Pencil/Idris

[neat lintel]

Like this right?

[maiden loom]

wait

hold up

[neat lintel]

Is everything after Sigma in summation

Or only some part is

Sure

[maiden loom]

let me double check

[neat lintel]

Sure take ur time

[maiden loom]

i dont think it matter the h^n is in the sigma notation or not

[neat lintel]

Yeah h^n is a constant

Well i gtg now

Cuz college tomorrow

Cya

[maiden loom]

cya

[teal lion]

ikr

[maiden loom]

$f^{[-n]}(x) = \lim_{h\to 0}\sum_{k=0}^{\infty}(f(x+kh){{n + k - 1} \choose k}h^n)$

[fathom swallow]

byte

[maiden loom]

so then $f^{[-1]}(x) = \lim_{h \to 0} h\left(\sum_{k = 0}^{\infty}f(x +kh)\right)$

[fathom swallow]

byte

[maiden loom]

which looks ehhh

anyways now i have made a series for integral , derivative now time to try to make a series for limit because why not

[alpine kindle]

this looks right

i think

[maiden loom]

wait

hold up

isnt that just like adding tiny area of the equation

[neat lintel]

@maiden loom ping me for more discussion if u want anytime today related to yesterday's topic
I found some interesting stuff

[maiden loom]

alr

[maiden loom]

wait hold up

how about

half derivative

💀

[neat lintel]

So uh

Uhhhhh

Should we start lol

[maiden loom]

yes

[neat lintel]

Okay so

For that choose stuff we discussed earlier

[maiden loom]

yes

[neat lintel]

From this for strictly (-n)≥(-2)

U can also check it out

[maiden loom]

oh

[neat lintel]

So from this

Using ur lim formula earlier

Of derivative

n is negative here

[neat lintel]

Just substituted this stuff

[maiden loom]

hmm

[neat lintel]

And since n < k = 0

I switched them

So k=n

And goes to 0

In the summation

So for n=-1 only

The whole fraction with factorial goes away

Funny enough

All such "derivatives" always result 0

[maiden loom]

okay so for this equation i think it's just
${-n \choose k} = (-1)^k{{n + k -1} \choose k) i think

[neat lintel]

Like with e^x

[maiden loom]

wait

[neat lintel]

${-n \choose k} = (-1)^k{{n + k -1} \choose k}$

[fathom swallow]

Pencil/Idris

[neat lintel]

Yeah

[maiden loom]

yes

[neat lintel]

I believe ur notation is actually better

[maiden loom]

wait so

[neat lintel]

Let's

[neat lintel]

Again do this without the restriction and mess

Using ur notation

[maiden loom]

wait let me think

[neat lintel]

Alright

Minor mistake

Refer this

Forgot to remove the "-" sign

In the denominator of the first step

[maiden loom]

wouldnt this become0

because

[neat lintel]

Yeah

That's what really kind of surprised me

So say 0th derivative of e^x is e^x function only

-1th derivative of e^x is 0

-nth derivative of e^x is zero

Which makes kind of no sense

[maiden loom]

because the binomial sequence is actually sum of infinity

but because

the choose function

[neat lintel]

I didn't get u could u please elaborate

Ah that

[maiden loom]

okay so i mean that if k is larger than a it would become 0 i think

[neat lintel]

Uuuhh

You're saying if k>a then a = 0?

[maiden loom]

yeah

[neat lintel]

Hmm

[maiden loom]

wait

wrong

wait

idk

[neat lintel]

Ah I get it

You're saying

[neat lintel]

When k>n

The binom coeff gives non zero values

When n<k

You get 0

[maiden loom]

yes

[neat lintel]

Which is obvious cuz you can't make 5 combinations from 3

For example

Like $^3C_5$ = 0

[fathom swallow]

Pencil/Idris

[neat lintel]

Like that

So yes

[maiden loom]

yes

[neat lintel]

Wait no

[maiden loom]

yeah

[neat lintel]

Forget what I said

Well negative order derivatives are indeed equivalent to integrals, which proves my stuff I posted earlier is wrong

[maiden loom]

oh

[maiden loom]

also i can't get any website to compute this idk why

probably just too hard to compute lol

also

half derivative

also doesnt work with the old equation i made

[neat lintel]

Hmmm

[maiden loom]

so the equation should actually be $\ \ f^{[n]}(x) = \lim_{h\to 0}\frac{\sum_{k=0}^{\infty}(-1)^kf(x-kh){n \choose k}}{h^n}$

wtf

[alpine kindle]

this is wrong

[static loom]

discord mangles latex if you copy from it directly, you can edit the post and copy that way

[alpine kindle]

there are actually binomial coefficients for negative imtegers

[maiden loom]

yeag

[maiden loom]

wait why is it fractional

[static loom]

this picture doesn't look consistent with how they're used

[alpine kindle]

because it's an infinite series

[fathom swallow]

byte

[static loom]

like $$\binom{n}{k}=\frac{1}{k!}\prod_{i=0}^{k-1}(n-i)$$

[fathom swallow]

Merosity

[static loom]

as coefficients on the series expansion on (1+x)^n

[maiden loom]

hmm

[static loom]

plug in -n for n and see for yourself that it's equivalent

[maiden loom]

yeah this make sense

[static loom]

$$\binom{-n}{k}=\frac{1}{k!}\prod_{i=0}^{k-1}(-n-i)$$ pull out the $-1$s from the product $$=\frac{(-1)^k}{k!}\prod_{i=0}^{k-1}(n+i)$$ reverse the order of the product starting at the last $$=\frac{(-1)^k}{k!}\prod_{i=0}^{k-1}(n+k-1-i)$$ this is the original definition I gave with $n$ substituted for $n+k-1$, so $$\binom{-n}{k}=(-1)^k \binom{n+k-1}{k}$$

[maiden loom]

this would probably also work

[fathom swallow]

Merosity

[maiden loom]

time to half derivative

[neat lintel]

My obsidian notes look kinda messy hahah but I'm happy it's actually looking like something

not sure what it's looking like tho lol

[grim pecan]

Hello, why does T-test show that null hypothesis can't be rejected when in percentage of data shows it might reject the hypothesis (60% yes and 40% no)

[ancient flame]

what if I become an aerospace engineer? UCF has a frickin partnership with NASA

lol

[sleek wing]

do eetttt

[ancient flame]

idk maybe

if I don't like business I will consider it

[neat lintel]

1/n = 0????

That case could only be when n is approaching infinity

Is it that case here?

[charred mortar]

Uhhh

Oh yeah in this case yes

It’s because you’re summing over k

Not n

So you’re just got an infinite sum of a constant

Make sure you set your variables right for wolfram

[neat lintel]

Ah sorry about that o7

Thanks!

[charred mortar]

No worries, happens to the best of us

[deep mango]

I should have done aerospace

😵‍💫

Or naval engineering or something

I want to make boats

[vague lion]

suppose i have data on all the frequencies of all the words in a bunch of wikipedia articles

so i have the articles that are like 'Apple', 'Banana', 'Cherry', 'Wheelbarrow', 'Xylophone', 'Yo-yo'

and i have how often words appear, so 'fruit' is 3% of the words in the apple article, 5% for banana and 2% for the cherry article, 0% for all the others

and i also have the data for 'made', 'grown', all the words

how do i find words that appear lots in in some articles but not others? what metric would be a good one to use

_ _
i essentially want to find words with high variance in their frequencies

_ _
but i don't want words that show up in literally one article, 1000 times

_ _
like 'apple' would be like 50% of the words in 'Apple', but 0% everywhere else

_ _
so what metric do i use

[ancient drift]

Most common words on the english language?

[arctic grove]

The.

[bright hill]

words

[arctic grove]

The is the most used word in the english language

ok

the other words are

To.

Be.

[ancient flame]

rip

[neat lintel]

has anyone ever watch the movie stand and deliver?

nice movie on a teacher in cali in a ghetto neighborhood getting his students to pass ap calc

[royal garnet]

can someone solve this pls its just a test its not like homework or anything

[royal garnet]

dose everyone get the joke

just solve it

it will only take

forever

🤣 🤣

[neat lintel]

What is X

[royal garnet]

times

obviosly

well in some cases its not obvios but still

[neat lintel]

Why didn’t they just put π in the summand then

That’s so unnecessary and confusing

[ancient flame]

ugh and using X as multiplication is awful

also solving it wouldn't take forever

you can use the alternating series test

n shit

[neat lintel]

The joke fell flat lavastream

[fervent pebble]

does anyone know a direct proof that sqrt(2) is irrational?

or a proof that doesn't use FTA

[alpine kindle]

fta?

[neat lintel]

Proof by direct? I mean there’s a common one using proof by contradiction…it doesn’t need FTA

[open aspen]

fundamental theorem of algebra?

[alpine kindle]

oh

[fair mural]

what?

oh

[alpine kindle]

how do you use fta for that lol

[neat lintel]

Just assume \sqrt(2)=a/b and contradict that

No FTA needed. Really, there is no need to use such machinery on a proof like that.

[open aspen]

no fundamental theorem of arithmetic

idk why we are guessing and they arent just telling us but thats my guess lmao

[alpine kindle]

they're the same theorem for different rings
nvm

[neat lintel]

How? FT Algebra states C is algebraically closed, FT Arithmetic uses the fact that the integers are a unique factorization domain…

[alpine kindle]

yeah I'm dumb

i just realised that

[neat lintel]

It’s ok

[alpine kindle]

anyway a good proof of the irrationality of the cube root of 2 uses Fermat's last theorem

let cbrt 2 = a/b
then (a/b)^3 = 2 = 1 + 1

but by Fermat's last theorem
c^3 + b^3 ≠ a^3
(c/b)^3 + 1 ≠ (a/b)^3

if we let c/b be 1 then (cbrt 2)^3 ≠ 1+1 which is a contradiction

[frozen merlin]

this is like nuking a fly

but pretty cool

[alpine kindle]

what do you mean
FLT is entirely trivial

[cyan goblet]

fundamental leorem of thalgebra

[neat lintel]

Is it really a nuke, though?

To me it seems to be more of a super hard Diophantine equation that took a while to solve, but does it have a lot of applications/is it powerful elsewhere?

[frozen merlin]

btw this is circular: according to some MO comments, the proof of FLT relies on stuff like 2^1/3 being irrational

[alpine kindle]

all the better

[fair mural]

perfect way to put it

[fair mural]

this is a bad way to phrase it, what is c? that isn’t defined anywhere

just say that (a/b)^3 = 2 implies a^3 = 2b^3 = b^3+b^3

[rancid meadow]

@atomic hornet do u live in boston

what do ppl do for fun here

[atomic hornet]

Not yet. Im moving there in July.

[rancid meadow]

lame. im here till sept

trying to figure out whats fun

[primal seal]

Well you just missed the Celtic games

Although I think they return mid week

[fervent pebble]

do it then?

i don't see how you prove 2b^2 = a^2 means a is even without fta

[primal seal]

What are you trying to prove (or disprove) valley?

[fervent pebble]

i just wanna see one

a proof that sqrt(2) is irrational without the use of FTA

or a proof that sqrt(2) is irrational directly

[primal seal]

FTS being fundamental theorem of arithmetic?

[fervent pebble]

yes

[primal seal]

Ah I see

[charred mortar]

I mean you don’t need FTA for that I think?

[fervent pebble]

how so?

[charred mortar]

Like an even number is just one with a factor of 2

[fervent pebble]

yeah, so a^2 is even

[charred mortar]

Right

Not quite sure why you’d need FTA

[fervent pebble]

that doesn't just give us that a is even

hmm ig you could argue that a^2 is only even if a is even

[errant merlin]

is an odd number squared even or odd?
based on that answer, if something squared is even is it even or odd?

[fervent pebble]

hence

ehhh i feel like this is still sort of icky though

bleh

hm whatever

anyone have a direct proof?

[errant merlin]

hmm not aware of one

[primal seal]

Im writing one

[fervent pebble]

!

[errant merlin]

books and videos love the contradiction one

[fervent pebble]

yeah it's kinda annoying to me

tbh

i'd also love to see a proof that the sqrt of any integer that isn't a perfect square is irrational

that's also not by contradiction

[errant merlin]

i remembered being confused for a while with them having a common factor being a contradiction

[fervent pebble]

wait actually i'd love to see a proof of this that doesn't invoke fta

[charred mortar]

I still don’t see why you need to invoke FTA really

[errant merlin]

same

[charred mortar]

If a^2 is even then a is even

You can show this directly by taking the square of an odd number and showing that it must be odd

[fervent pebble]

wait i think we aren't talking abt the same proof here

[fervent pebble]

for this i mean

[charred mortar]

Oh

Thought you meant root 2 specifically

In the case of any integer then sure

[fervent pebble]

you happen to know 1?

[charred mortar]

Oh no I meant in the case of any integer you probably need FTA

[fervent pebble]

hmm

[charred mortar]

In the case of root 2 specifically you don’t really need it

[fervent pebble]

i think if you did it for every integer you could probably do a case-by-case odd-excluding-prime/even/prime

but that's kinda ugly

i know you can do it for all primes w/out fta

[charred mortar]

Yeah

There is a short proof on wiki, where you apply the rational root theorem to the polynomial x^2 - m

The theorem says that any rational root of that particular polynomial must be an integer

But if m is not a perfect square that is false by assumption

[primal seal]

One that I got to work that you could extend to make work is the following:

Follow the usual proof by contradiction until you get through to the “if a is odd then a^2 is odd” part

Then you need to show that every integer is either even or odd (and not both)

I did this for the natural numbers, but you can probably extend it to Z

It used induction for N

Then you use the contrapositive of the “if a is odd…” Lemma

[neat lintel]

OK then, I'll post a write up here.

[stark charm]

lol

[fathom swallow]

Arr0w_04

[stark charm]

bruh

stop pls

[neat lintel]

No fundamental theorem of algebra or arithmetic was used here.

[primal seal]

“This means that a must be even”

Can you justify this more explicitly?

[neat lintel]

That doesn't use FTA.

[primal seal]

That’s got a lot more behind it though, although yes, never uses FTA

[neat lintel]

There isn't anything more explicit. It is clear that a must be a multiple of 2.

[stark charm]

It is explicit

x = 2n means x is even

[primal seal]

That shows that a^2 is even

[stark charm]

yes

[neat lintel]

It is a famous proof by contradiction. You literally can't get any better than that, it is written the exact same way in hundreds of textbooks.

[errant merlin]

if a is odd, $a=2k+1$, then $a^2 = 4k^2 + 4k + 1$, which is odd

[fathom swallow]

feeds

[neat lintel]

humans are alive here

[primal seal]

I know but to use that you have to prove every integer is either even or odd snd not both

[stark charm]

no you dont lol

this is elementary

[errant merlin]

trivial

[stark charm]

and trivial

[neat lintel]

What? Absolutely not.

[stark charm]

no one ever would need to show that

[neat lintel]

Just use peano axioms wth

[primal seal]

Km just saying, if you want to chase it down to nuts and bolts

Relax

[neat lintel]

You don't need to, though. I am defending my proof.

[primal seal]

No, you don’t need to, and there’s nothing wrong with your proof, so you can unclench your fists. I’m just saying it seemed like he maybe wanted a proof that chases back to uber fundamentals based on how he was asking for it to be shown

[stark charm]

Doubt this

[neat lintel]

OK, challenge accepted.

[fathom swallow]

Arr0w_04

Arr0w_04

[stark charm]

well actually you didnt go and show which set axioms ur using

[neat lintel]

Now I proved that sqrt(2) exists.

How much more fundamental do you want? Peano axioms?

[stark charm]

im sarcasticing

[neat lintel]

To be clear: I was defending my work, which is common place in proofs. If you offer critique or an objection, you can't resist or be afraid of a possible rebuttal.

I mean my "work" is a well known and trivial proof, but still.

[fervent pebble]

this is the kind of proof i wanted

good shit 👍

[neat lintel]

It doesn't show that it is irrational @fervent pebble. My proof just guarantees the existence of such a number in the reals. My previous proof suffices for your claim.

[ancient flame]

dont care

didn't ask

• ratio

[neat lintel]

gmod why

[ancient flame]

LOL

[fervent pebble]

oh well nvm lol

its still good tho!

i like it!

[neat lintel]

My second proof was a response to an objection by los angeles: I guaranteed that such a number existed, and now I can prove that it is irrational.

[ancient flame]

I don't.

[neat lintel]

bathe? Is that the word you left out?

[ancient flame]

I bathe don't.

[neat lintel]

I would take a shower gmod, the fellow humans around you will appreciate the reduced odor.

Anyway

So hopefully that answers everyones questions, objections, etc. Point is, math works and we can move on, yay.

[ancient flame]

but I live alone in a basement

[teal lion]

hello

[primal seal]

I understand, no hard feelings or anything, I was just under the (mistaken) assumption that maybe he wanted something that was down to more primitive concepts like the even/odd partition of Z. We’re all good

[fervent pebble]

that would be cool too

i want everything

!

i think it's awesome that there're tons of ways to prove such a basic fact

i want to see it all

[primal seal]

It would be the same proof arrow outlined, just with some pedantic details filled in (which I originally thought you were seeking)

[fair mural]

how does this show that a^2 > 2 is impossible?

surely you meant - and not + when you said (a+1/n_0)^2?

[charred mortar]

I think it’s fine?

[stark charm]

It is fine

To clarify to quantum they mean it contradicts a being defined as the supremum

or infimum in the next case

[fair mural]

how does that contradict anything, all i see is something practically just saying if a>b then a^2>b^2

[stark charm]

Where do you see this?

This wasnt asserted once in their argument

[fair mural]

i mean it just says that if a^2>2, then (a+1/n)^2>2

how is that a contradiction

[charred mortar]

Although I think in the second half it should indeed be a - (1/n)^2? Since you want to show that a is indeed the least upper bound

So yeah I think quantum is right

[fair mural]

i mean i just looked up a proof of that exact statement

it was a-1/n

https://math.stackexchange.com/questions/1415235/prove-the-existence-of-the-square-root-of-2

[charred mortar]

As of now it seems like the second half just shows that a + (1/n) is an upper bound

Yeah changing it to a minus fixes the second half

As well as a few small changes here and there

But otherwise you get the idea

[stark charm]

They skipped steps

They just needed to assert for case 2 that a is the infinum of the set of reals such that a^2 >2

[charred mortar]

Hm perhaps

[stark charm]

They typoed and forgot to explicitly state other assumption

[brave hollow]

still does not work due to negatives

[charred mortar]

But it still doesn’t really show that a^2 > 2 doesn’t work I feel

[stark charm]

Yeah they need to use a-1/n_0 to find a contradicting element to a being supposed as the infinum

However you can fix their statement by swapping inequalities around

[charred mortar]

Also note that a^2 > 2 implies 2 - a^2 < 0

[charred mortar]

I think all quantum was saying was just to swap the plus to a minus and change some inequalities, i.e. precisely what you’re saying here

Arrow does have the overall gist correct, just a few wonky sign issues and inequalities facing the wrong way

[stark charm]

idk what theybwere saying about a>b implies a^2>b^2

I saw that nowhere

[charred mortar]

Oh yeah that I dunno either

In any case implementing what colee said about the sign and inequality thing should resolve the issues

[stark charm]

the thing is they dont need to work on anything

if i was writing the proof

[brave hollow]

they mean when he skips steps as you say in his proof he is sayin since a + 1/n > a then (a + 1/n)^2 > a^2 > 2 which is nothing special given a > 0

[stark charm]

id only show case where a is supposed as infimum

and then just say mirror argument for a being supremum

[charred mortar]

I mean they still have a minor error

[stark charm]

true

[brave hollow]

also why would they do it for infimum? surely they are showing existence of sqrt(2) in which case they would want to do it for supremum since they have said they want to squeeze a into being a^2=2

[charred mortar]

Ironically though doing what you suggest colee would be both less work and be more or less correct

[stark charm]

thats the idea

they need to show just one

they wrote in their sketch of the proof something different

to squeeze from otherside they would need a new set

They would only need to choose either infimum or supremum

however showing that a-1/n_0 > 2 doesnt disprove a being a supremum

[brave hollow]

why would that prove anything

you meant to square it I think

[stark charm]

yea

actually no

[brave hollow]

either way you want to show that a^2 < 2 and a^2 > 2 would lead to contradiction

[stark charm]

i meant to remove the rhs

[stark charm]

yea

[brave hollow]

yeah so in their first statement they find contrafiction for a^2<2 for a being the supremum

all I was saying is the second step would then be to show that a^2>2 is a contradiction for the supremum

which is why I was confused when you said its a statement about infimum

[stark charm]

i phrased it wrongly

[brave hollow]

Ok something way more important

Someone's cat is in my room and she keeps screaming at me

wtf do I do

[errant merlin]

pat

[brave hollow]

she screams when I try to put my hand anywhere close to her

I am outside my room scared for my life

[wild lantern]

Animal control?

[brave hollow]

yeah idk why people can't control their pets

[wild lantern]

Does she have a collar?

[brave hollow]

ok she went away and no

[wild lantern]

Hmm

[stark charm]

kidnap it

[brave hollow]

bro she scared me to the point I had to leave my room

while I was typing something about that proof

[stark charm]

bruh wth

u scared of cat?

[brave hollow]

yeah she was super mad or smth

[stark charm]

bruh

can u not 1 shot it?

[brave hollow]

I'd rather not fight a cat and leave it to some1 who is trained to handle them

although it didn't come to that

[wild lantern]

I'd prolly do the same

[stark charm]

bruh moment

[wild lantern]

Cats can scratch and bite and shit.

[stark charm]

one punch or kick

and the cat is KOed

[wild lantern]

Why do that if you don't need to?