general discussion

[open whale]

Hello! this is our most general maths chat: discuss anything you want here, so long it's not about help.

[open whale]

@hollow ice

[hollow ice]

Thanks

[open whale]

@vivid gulch

,tex x^2

[vivid gulch]

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

[open whale]

Ok it works here, nice

[wicked junco]

$x^2$

[vivid gulch]

Schlaumau

[long socket]

any good books to read about "building" math from the ground up?

[dire sable]

These are pretty good:
https://artofproblemsolving.com/store

[long socket]

what are these?

[dire sable]

AOPS runs several math competitions, and these books are pretty comprehensive

@long socket

[long socket]

This looks pretty neat

But I don't see proofs or anything like that

[dire sable]

What do you mean?

Ohhh

[long socket]

any good books to read about "building" math from the ground up

[dire sable]

I get it now, sorry

I thought you meant like building up general math knowledge

But yeah, I know what you mean

I've always thought that that would be a pretty interesting concept, and have always actually wanted to write a book on it

I don't know of any though, sorry

[long socket]

a runner-up would be books about making proofs

[dire sable]

Hmm, I know of a couple books that might interest you, but none on that topic

[long socket]

what are they?

[dire sable]

There's one called "Is God a Mathematician" by Mario Livio. It's not really about religion, but it does a deep dive into the origins of mathematics, and how they reflect the universe. I found it really interesting as it approached math from a philosophical perspective. It's more of a layman's book though, so I'm not sure if that's what you want

There's also one that I can't remember the name of, something about Math A-Z, it had a bunch of interesting theorems and proofs.

I'll see if I can find it

Here we go, "The mathematical universe : an alphabetical journey through the great proofs, problems, and personalities", by William Dunham

I found the title in an overdue book email

[long socket]

for real?

[dire sable]

Yeah, I had like 5 random books I checked out and that was one of them

Wait, I'm going to see if I can post a screenshot without doxxing myself

Ah, only 4

Hopefully the barcodes and dewey decimal numbers can't be used to find me

[long socket]

say, how good are you at math?

I just got the hang of algabra

[dire sable]

I'm okay, but I'm nowhere near as good as some of the people on this server

[long socket]

Ring Theory looks to be the tool that can help define subtraction of natural numbers for me

[dire sable]

Oh, I just looked it up, that's cool

[long socket]

If I want to define subtraction like addition, I need to prove something

For every natural number x that isn't 0, there exist a natural number y such that the successor of y is x

basically x - 1

[dire sable]

Yeah, although that wouldn't work for 0 since natural numbers don't include negative numbers

[long socket]

yeah

[dire sable]

If you include all integers though, you might be able to prove it based on adding negative integers

[long socket]

but that has to be derived from operations with just natural numbers

also edited the thing

[dire sable]

Hmm, that's difficult

[long socket]

building math from logic and set theory isn't easy

[dire sable]

I think it's be numbers y that aren't 0

[long socket]

one of the peano axioms said that 0 exist and its fair to say that succ(0) is 1

[dire sable]

Oh, that's interesting

[long socket]

with that thing plus recursion, addition is defined

you can also define subtraction if you first made up the inequality symbols to make sure that n - m(n < m) isn't allowed

but how?

I swear to godel that we need to prove that N is a ordered set

[dire sable]

First of all, I'm going to try to use that phrase in everyday conversation now

Also, maybe it'd be possible to prove it by induction

[long socket]

the what?

[dire sable]

So, if it's true for x, and assume it's true x=k, then prove it's true for x=k+1

[long socket]

I meant what are you trying to prove

[dire sable]

The y=x-1

[long socket]

okay then

let P(x) be that thing

and P(k and k is 1) => 0

that's our base case

now we need to prove P(k + 1)

hold on, I forgor how to do induction

[dire sable]

P(k+1)=>y, which is true

I think that might do it, but I'm not entirely sure

[long socket]

P(k + 1) spits out k, right?

[dire sable]

Yeah

[long socket]

now to write this thing formally

(I bet there is already a journal about this already)

For every natural number x that isn't 0, there exist a natural number y such that the successor of y is x
Proof - let k be 1
P(k) = 0
P(k + 1) = k
QED

(idk if I wrote this this correctly)

[dire sable]

Yeah, I think that should work

Here, I found something on it

https://www.math.utah.edu/~bertram/courses/4030/Nats.pdf

[long socket]

and i'm guessing the same author also have a journal about integers?

[dire sable]

I'm not sure, I could take a look

Here's a nice article on proofs

https://cs.stanford.edu/~jtysu/proofs.pdf

[long socket]

meh

[dire sable]

Oh, hmm

Maybe because it's a PDF

Wait no, the other one's a PDF too

[long socket]

wait, it worked

[dire sable]

I can't wait to take discreet mathematics, it seems like it'd be a very interesting course

[long socket]

how's the search for the integers gold mine?

[dire sable]

I haven't found one unfortunately

[long socket]

http://www.math.hawaii.edu/~pavel/syllabi_old/aluffi_321/NZ.pdf

help me

its ring theory all over again

[dire sable]

Oh, that looks like a nice one

Hey, I need to head out now, but could you please ping me if anything comes up? It's really interesting talking with you

[long socket]

sure, no problem

but im curious as to why im interesting to talk to

[dire sable]

I've been thinking about similar topics for a while, but I haven't really met anyone else who's interested in these things before, so it's nice to talk about it

The things you propose make me think, and I'm learning a lot looking for articles about them

And I'm learning a lot talking with you

[long socket]

math isn't about "solve this", its about "how do you solve this" and "why this is the solution to this"

and also "make up stuff" too

imaginary numbers is a product of "make up stuff"

[dire sable]

Yeah, we're taught "solve this" in school, which is I think why so many people get scared off by math

But if we were taught more about the foundations of math, I think more people would be interested

[dire sable]

Yeah, that's true

[long socket]

also stuff like logic and reasoning

because you can't run if you don't know how to walk

[dire sable]

It's really amazing when concepts existing purely in math can actually be applied, like with imaginary numbers, and with knot theory

[long socket]

It feels like we are more like explorers than inventors

[dire sable]

Like imaginary numbers are used in concepts relating to electricity, and knot theory is being used for modeling DNA

[dire sable]

Yeah, the question of is math an invention or a discovery is really fascinating

I would say it's a bit of both

[long socket]

some of the new stuff in math is found by finding a definition for the undefined instead of leaving it in

Say, you are good in discussing math as you foster thinking by your insight

given that you are talking to people that listens carefully, not just hear your words

[dire sable]

Yeah, I think getting new perspectives on things is very important, that's why it's nice talking to you

You've made me think of things I wouldn't have normally

[long socket]

like what?

[dire sable]

I usually think of natural number subtraction as axiomatic, I hadn't thought to try and prove it

[long socket]

Well, we can't talk forever

its nice discussing with you again, mathematician

hope we can talk again soon

[dire sable]

Yeah, I hope so as well

[long socket]

@dire sable found out that you need something called a equivalent relation to define the integers

basically, Z is the set that defines the result of n - m(m > n)

[dire sable]

Oh, hmm

[long socket]

https://www.youtube.com/watch?v=v3MYQtAF4H8

[dire sable]

I’m a bit busy, so I can’t really look at it now, sorry

[long socket]

oh alright

do tell me when your free

[dire sable]

But I’ll try to get to it sometime this weekend

[long socket]

sure thing

[dire sable]

Of course, it might be a bit though

I’ve had a lot going on recently

Thanks for letting me know about this though

[long socket]

no problem

[long socket]

@dire sable wild idea, a game about algebra

algebra seems like the perfect math topic to make a game of

[dire sable]

Yeah, that could be really interesting

Or maybe one on the complex plane

[long socket]

For the sake of the game, I'll restrict stuff on the real domain

there are a lot of topics in algebra and I just need to figure out how do we make it into a problem solving game

any ideas?

[long socket]

mathematics is the ultimate problem-solving game but hella abstracted

[dire sable]

Yeah true

[long socket]

let me know if you have ideas on how to make this bearable for the gamers

[dire sable]

Are you thinking board game or video game?

[long socket]

video game

[dire sable]

Hmmm

Maybe collecting variables somehow

Could try and do Diophantine equations, but honestly I don’t quite understand them

[long socket]

before they do like equations and stuff, they should be exposed to the important properties and concepts

in the beginning, swiftly introduce and test the properties of addition, subtraction, multiplication, and division

in the domain of integers I guess

and the story of the game would be to explore the history of math and its mathematicians

done right, and this might be a decent educational game

[dire sable]

Oh yeah, can’t wait for the section on Euler, it’ll probably take up like half the game

[long socket]

Euler is the pi of mathematicians

he shows up everywhere

[dire sable]

I like how they started naming things after the second person to discover them after Euler because too many things were named after him

I think that was Euler at least

[long socket]

what do you think would the end of said game be?

[dire sable]

Maybe meeting a modern mathematician

Who will like send them off to learn more on their own, maybe providing a link or something

[long socket]

or godel himself

with a final message about the future of maths

[dire sable]

Yeah, Godel would be really cool to add into the game at the end

[long socket]

"Idk if math truly is imcomplete, my soul have seen the impossible become possible.
Take care, maybe you will discover something one day, mathematician"

(one of the few millennium problems got solved recently, so this might hit a bit harder later on in the years)

[dire sable]

Maybe send them on their way by making them solve a millennium problem

I mean, they probably wouldn’t but it might be interesting for them to think about it

[long socket]

this game might have a crap ton of quotes if anything

[dire sable]

I mean, quotes are always effective

[long socket]

another one, what would "boss" problems look like?

of course, we go easy on them and not just let them find the quadratic formula for example

[dire sable]

solve collatz conjecture

Maybe walk them through the quadratic formula within the boss fight so it’s not too difficult

Or like have different mini bosses be different parts of the formula

[long socket]

solving math intensifies

what other things qualifies as a boss fight?

[dire sable]

Maybe derivatives using limits at some point

[long socket]

thought this is algebra, not calc

[dire sable]

Oh yeah, forgot about that

Hmm, maybe small proofs like .9 repeating equals 1

[long socket]

that's a limit, isnt it?

what else?

[dire sable]

You could do it algebraically

.9 repeating =x
9.9 repeating =10x
9.9 repeating-.9 repeating = 10x-x
9=9x
1=x

[long socket]

you just chop off the decimal once you multiplied it by ten?

anyways, how would we weave the story of maths into this game?

and do you think such a game can teach math in a neat way?

[dire sable]

It’s because the 9 is repeating, so there would be another 9

[dire sable]

Yeah, I do think it’s a good idea

It’d be a bit difficult to implement, but I think it’d be possible

[long socket]

the first concept is of course variables

"there exist a symbol x that stands in for a number in a set of real numbers"

(dare you to find a simpler explanation)

(hold on, we need to explain the bare basics of set theory to introduce variables?!)

[dire sable]

We could just start with the different levels of organization for numbers

Maybe make it like a social hierarchy within the game

[long socket]

"social"?

[dire sable]

Like natural numbers are the rules, real numbers are the lowest class

Make the numbers like people, but not quite people

[long socket]

like a metaphor then

[dire sable]

Yeah

[long socket]

after variables, exponents are a bit easier

just need to explain visually its properties

in ancient cave drawings of course

[dire sable]

Naturally

[long socket]

I still have to know why x to the zeroth power is 1

[dire sable]

Uhh, I knew this at some point, I'll check quickly

Ohhh, if you're doing x^y, then it's 1*x y times, so if it's zero times then it's just 1

Could also be considered from the perspective of permutations

[long socket]

after exponents, polynomials

and this is where stuff goes a bit harder

[dire sable]

Could go a bit into physics for that

Though that might make it more confusing

[long socket]

to explain operations between polynomials, squares are going to be used a lot

then special products and factoring

and then comes linear equations

to make this part math training app, randomized problems with a solution algorithm needs to be designed

a bit hard technically

[dire sable]

We could probably use an already existing CAS program

[long socket]

good point

now how do we fit randomized problems into the game flow?

(also, problems that needs you to apply math can also exist in the game)

[dire sable]

We could randomize the numbers, although I'm not sure about game flow

[long socket]

we should also provide whiteboards but not a calculator

[dire sable]

Maybe just hope they have scratch paper since writing on technology can be a bit difficult

But yeah, no calculator

[long socket]

explain and/or defend your stand

[dire sable]

Maybe if it's a slightly longer problem, but not for problems that shouldn't require it

[long socket]

the graphs parts is going to be gamified to the death

[dire sable]

Of course, that's what makes graphs fun

[long socket]

you heard of making parabola equations to make mario jump in the correct arc in desmos?

[dire sable]

I haven't, that seems interesting though

[long socket]

but what about the rest of the topics?

[dire sable]

Maybe collection type games

[long socket]

seems too typical as a educational game

but then this is for people who likes figuring stuff out

and for people that wants(or forced) to learn math

so we should make it fun for those two groups

the rest isn't our problem

[dire sable]

Gears would be interesting to incorporate, I'm not sure how though

[long socket]

gears?

[dire sable]

Yeah, they tend to have a lot of implications in math

[long socket]

that would fit in applied algebra

actually, maybe we use the pure maths for explainations, harder problems, and "boss fights"

the fun part lies in the applied math

[dire sable]

Oh yeah, that's a good idea

[long socket]

and if this game is kind of a open-world adventure game, if you prove your worth as a mathematician to someone like Gauss, he would give you his formula(which is a neat nugget of information)

[dire sable]

Yeah, and maybe you could discover some older formulas like from ancient empires in archeological sites.

[long socket]

imagine seeing i^2 + j^2 + k^2 = -1 in a bridge in the game

or a piece of paper about calc in netwons grave

we can go real nuts with this

[dire sable]

That would be really interesting to incorporate

[long socket]

once you see Godel tho, ohh boy

[dire sable]

That's when things get interesting

[long socket]

What about when you are presented a ominous barbershop with a recursive infinity symbol as its sign

[dire sable]

There will be many moments when things will get interesting

Pretty much the entire thing, because all of math is interesting

[long socket]

Im guessing a good bit of those interesting things are related to infinity and recursion

because those two cracked maths long ago

[dire sable]

Yeah, along with Godel himself

[long socket]

don't forget Turing

[dire sable]

Ah yes

Of course

Oh, I should probably get back to work, it was nice talking though

But yeah, the game is definitely a good idea, it'd be really interesting

[long socket]

good luck to that

no problem

[open whale]

So

@craggy meteor

Do you know how integrals work?

@craggy meteor

So, do you know how integrals work?

[craggy meteor]

ok

i somewhat know them work

[open whale]

They go from point A to point B

But how do they go there?

[craggy meteor]

🤔

[open whale]

Usually you'd think, in a straight line, right?

[craggy meteor]

yes, but in this case they go in a function like

uh

[vivid gulch]

PooPoo, not CEO of comedy

[craggy meteor]

i got the a and b reversed

anyways

[open whale]

But, lets say

Can I go at different speeds?

Like, can I go really fast at the start, then slowly at the end?

[craggy meteor]

How would that impact the integral?

[open whale]

It wouldn't

So long we do it appropriately

[craggy meteor]

But what will change then?

[open whale]

Exactly nothing, that's the point

[craggy meteor]

[open whale]

Lets see why that is the case

An partition

Is when I cut the line into equivalent sections of length

Then, I take some point inside each section

And multiply f(point)

By the length of the section

[craggy meteor]

This?

[open whale]

dx

Exactly

So we multiply f(point) by dx

and then we sum for all partitions we did

$\sum_{n=1}^kf(a_n^*)\cdot(a_{n+1}-a_n)$

[vivid gulch]

yoavmal

[craggy meteor]

This look like the reimann sum i think, is it

[open whale]

Where a(n)* is some point inbetween a(n) and a(n+1)

[open whale]

Now take the limit to infinity

$\lim_{k\to\infty}\sum_{n=1}^kf(a_n^*)\cdot(a_{n+1}-a_n)$

[vivid gulch]

yoavmal

[open whale]

Now, we could make it so the start is really dense and the end isn't

All that matters is we multiply the sections by the appropriate lengths

So one part may have tons of points

And one part won't

But since we multiply each section by the corresponding dx

It's fine

Infact, we could take this to the extreme

And say we could wait in the spot or even go backwards

So long we eventually go all the way to point b

It eventually all cancels out

And the integral is the same

Why is this important?

Because it turns out we can use that for complex integrals

[craggy meteor]

So instead of integrating you use Riemann Sum?

For complex function?

[open whale]

Yep

[craggy meteor]

Huh

[open whale]

What we can actually do to describe the concept I just said is

$\int_a^bf(\gamma(x))\md\gamma(x)$

[vivid gulch]

yoavmal

[open whale]

This allows us to have a function do all sorts of weird stuff, and still get the same result

But it can also take complex values

And this statement holds true

So really, $\int_a^bf(\gamma(x))\md\gamma(x)=k$

[vivid gulch]

yoavmal

[open whale]

k is the same number, no matter what gamma is

The key importance is $\md\gamma(x)$ is not always the same, and can infact take any complex direction

[vivid gulch]

yoavmal

[open whale]

So if you go in the positive imaginery direction, it will later cancel out when you go back in the negative direction towards the reals

Are you with me, @craggy meteor ?

[craggy meteor]

Yes

[open whale]

So uhhh

[craggy meteor]

Why are you using incomplete gamma function🤔

[open whale]

This gamma function is a path

It's the path we take as we integrate

Lets try to make it a complex integral proper

$\int_a^bf(\gamma(z))\md\gamma(z)$

[vivid gulch]

yoavmal

[open whale]

Or, if we're lazy

We'll just drop the a and b

$\int_\gamma f(z)\md z$

[vivid gulch]

yoavmal

[open whale]

It's just done with respect to some path

Now, I lied earlier

It isn't always the same number for different curve

For example

If I integrate 1/z

I can get different results if I go in different paths

This is ground breaking

Since we can integrate from a point to itself

In a loop

And naturally, expect the result to be 0

And instead get a different number

In such cases, it becomes clear that the closed path integral would be an essential tool

So we simply have a unique notation for it

$\oint$

[vivid gulch]

yoavmal

[open whale]

The circle is denoting a closed curve

$\oint_\gamma f(z)\md z$

[vivid gulch]

yoavmal

[craggy meteor]

Can you give me an example of calculating a closed path integral against an integral

[open whale]

Yes

What is the integral of 1/z from 1 to 1?

Or uhh

[craggy meteor]

0?

[open whale]

$\int_1^1\frac{1}{x}\md x$

[vivid gulch]

yoavmal

[open whale]

This is clearly 0

But

It can also be 2πi

If it's $\oint_\gamma\frac{1}{z}\md z$

[vivid gulch]

yoavmal

[open whale]

Why?

Because we can use the fundamental theorem of calculus

To get that $\dv{\ln(z)}{z}=\frac{1}{z}$

[vivid gulch]

yoavmal

[open whale]

Now, if we choose our curve to be the unit circle

Oh wait, before this part

[craggy meteor]

?

[open whale]

And as such $\int_a^bf(x)\md x=\ln(b)-\ln(a)$

[vivid gulch]

yoavmal

[open whale]

As you may know

$\ln(re^{i\theta})=\ln(r)+i\theta$

[vivid gulch]

yoavmal

[open whale]

So if we go about the unit circle

We integrate a portion of it

We get that the integral from point at angle θ1

To point at angle θ2

Is just iΔθ

The difference of angles, times i

Yes?

[open whale]

If we take this, from e^iθ1 to e^iθ2

We get i(θ2-θ1)

If we agree this grows continuously (and it does)

We eventually get that from θ1=0 to θ2=2π

From 1, to 1

The integral it 2πi

It can also be 4πi

And -2πi

If we go enough rotations in the right direction

Coolest part is

We actually don't get this if we don't go around 0

If we make a loop that doesn't engulf 0

The contour integral is just 0

You'll later get to conclude that the only way to get a none 0 closed path integral, is by integrating around an asymptote

Sooo

Do you see why $\oint$ exists?

[vivid gulch]

yoavmal

[craggy meteor]

somewhat

my brain is exploding inside

[open whale]

We get all sorts of weird formulas this way

For example $\dv[n]{f(z)}{z}=\frac{n!}{2\pi i}\oint_\gamma\frac{f(u)}{(u-z)^{n+1}}\md u$

[vivid gulch]

yoavmal

[open whale]

That's not true actually

[craggy meteor]

really?

||Damn google tricking me||

[open whale]

It's

[craggy meteor]

log(x)?

[open whale]

$\ln\abs{x}+\begin{cases}c_1&x<0\c_2&x>0\end{cases}$

[vivid gulch]

yoavmal

[open whale]

Since there's no connection between the right and the left part, this is the result

In complex numbers

There's a connection

So what you'll get is

$\ln\abs{x}+\begin{cases}2k\pi i&x>0\(2k\pm1)\pi i&x<0\end{cases}$

Where k is an integer

[vivid gulch]

yoavmal

[craggy meteor]

Is this calculus lesson done yet

probably

[craggy meteor]

@torn lava can you message

[torn lava]

yes

[open whale]

Sleep on it and reprocess it

[craggy meteor]

please no

forgive my sins yoavmal

stop this eternal hell

[open whale]

sin(z)

[long socket]

@dire sable hey there bud

When I finish high school, im going to major in math

what do you think?

and what about you?

[dire sable]

Yeah, that’s probably what I’m going to do too

I’m thinking of double majoring math and physics

But yeah, if you major in math it’d also be helpful to do work with computer science as well

Since the two fields are so closely linked

[long socket]

Not a bad choice, since im going to be a programmer anyways

why physics?

[dire sable]

I'd like to go into theoretical physics

[long socket]

fair enough

but what degree I wonder?

I think a master degree is a bit too much for me

so i'll settle on a bachelor's

[dire sable]

Theoretical physics is hard to do professionally, so I was thinking of getting a PHD and working as a professor

I'm not sure though

[long socket]

and how would you support yourself while your at college?

mine's going to just code for money

[dire sable]

That's a good question

I'd probably try to go for scholarships, and maybe do some coding work on the side

[long socket]

after college, what are your job options?

for me, its ether be a programmer at some tech company or be a programmer in some game studio

and math skills will make my programs hella optimised

[dire sable]

I'm probably just going to teach, and do research

[long socket]

at least be a decent teacher that everyone loves

[dire sable]

I hope I can be

[long socket]

btw, how are stuff there?

I wont discuss heavily on math atm since I have to do some school work

[dire sable]

How's stuff where?

[long socket]

on your end

[dire sable]

Oh, um, eh

Yesterday was not the best, but today was a bit better

How're things with you?

[long socket]

just chilling

watching some math vids and gaming

like a student is supposed to relax in the weekends

[dire sable]

Oh, what games?

[long socket]

a moba that is similar to LoL

[dire sable]

Nice, I haven't really played any games like that

[long socket]

wait, where are you on the education ladder?

[dire sable]

I'm in highschool right now

[long socket]

and what are your fellow's reaction to math?

[dire sable]

A few other kids like it, but my school is mostly kinda behind in math

What about your classmates?

[long socket]

in one line: they are shocked when 1x is just x

basically, they don't even mastered the basic properties of basic operations when we were supposed to do factoring

[dire sable]

Wow, yeah, that's about where my classmates are at

I feel like people wouldn't be like that if math was taught better

[long socket]

which is isn't

math can get hella hard when you dont even know about the simple but neat properties

like x is x^1

so x(x^3) is x^1+3 = x^4

[dire sable]

Yeah, being able to do basic operations is necessary for math later on, so a strong foundation needs to be built

But it's not sufficiently established for the most part

[long socket]

what would you propose?

[dire sable]

I don't know, it's something people have been struggling with for a long time

[long socket]

not to mention the** fear** of math

[dire sable]

Yeah

[long socket]

but alas they say: "what's the point?"

"why learn math?"

"im going to be some internet star or some creative artist or anything that doesn't need math"

tbf, they kinda have a point

[dire sable]

Internet stars make way too much money, it's not reasonable for what they do

But that's the world we live in

[long socket]

the ones blessed by the system by whatever means have their profits grow while the rest have their profits be mostly fixed

[dire sable]

Yeah, which is kinda like how everything else works too

Some people make lots of money while others have fixed profits, which are sometimes not enough to live on

[long socket]

well, this turn quite off-topic

if you like, we can debate on DMs

[dire sable]

A little bit, perhaps

Actually, I should probably get back to work, sorry, but I'd love to take you up on that offer sometime

[long socket]

no problem

we chat for many reason

including as a way of relaxation

hope we can chat again

[dire sable]

Yeah, I look forward to it

[open whale]

@short trail

[short trail]

Yes

[open whale]

so

lets use the one rule we know

$a(b+c)=ab+ac$

[short trail]

Hm..

[vivid gulch]

yoavmal

[short trail]

Ok

[open whale]

so

$(a+b)^2$

[vivid gulch]

yoavmal

[open whale]

lets break it down

$(a+b)^2=(a+b)(a+b)$

[vivid gulch]

yoavmal

[open whale]

correct?

@short trail

[short trail]

Yeah

[open whale]

ok

$\color{red}{(a+b)}\color{black}{(}\color{green}{a}\color{black}{+}\color{blue}{b}\color{black}{)}$

[short trail]

Hm. ....

[vivid gulch]

yoavmal

[open whale]

yes?

we now break it down using our rule

[short trail]

Yep

[open whale]

$\color{red}{(a+b)}\color{green}{a}\color{black}{+}\color{red}{(a+b)}\color{blue}{b}$

[vivid gulch]

yoavmal

[open whale]

yes?

[short trail]

Wait.

How*

Whyd you add an extra a and b

[open whale]

brb

[short trail]

Ok-

[open whale]

back, sorry

you're still here?

[open whale]

that's how the rule goes

[short trail]

Yes

[short trail]

Oh?

[open whale]

$\color{red}{a}\color{black}{(}\color{green}{b}\color{black}{+}\color{blue}{c}\color{black}{)}=\color{red}{a}\color{green}{b}\color{black}{+}\color{red}{a}\color{blue}{c}$

[vivid gulch]

yoavmal

[open whale]

does that make sense?

[short trail]

Yes
I think

[open whale]

in context of this

the first (a+b) splits

[short trail]

Oh

[open whale]

so there's (a+b)a

and (a+b)b

yes?

@short trail

[short trail]

Yes

[open whale]

if it makes sense, try doing the same proccess with (a+b)a

how do you simplify it using the same rule?

[short trail]

Sorry my discord lagged for a sec

(a+b)b
?

😬..

[open whale]

no, i mean

try opening the brackets using the same rule as earlier

$a(b+c)=ab+ac$

[vivid gulch]

yoavmal

[open whale]

so according to this

$a(a+b)=\dots$

[vivid gulch]

yoavmal

[short trail]

Opening the brackets?

Wouldn't that be (a+b)c then?

(sorry If I'm very very very annoying i feel like i forgot everything I've learnt in past years)

[open whale]

regarding this, it's all good

[short trail]

Ok ;-;

[open whale]

and uhh

lets say it this way

[short trail]

Mhm-

[open whale]

what's $2(1+3)$?

[vivid gulch]

yoavmal

[short trail]

8

[open whale]

how did you calculate it?

[short trail]

2×(1+3)
2×4
8

[open whale]

ok

so

the rule we're using says

[short trail]

Hm-

[open whale]

we can actually solve it another way

[short trail]

Oh?

[open whale]

2×(1+3)
2×1+2×3
2+6
8

[short trail]

Wait- what

I'm sorry but how does the 2 come after the 1 in the second line?

[open whale]

that's exactly the rule

three people

and two people more

having two cookies each

is like three people having two cookies

and then two people having two cookies

ye?

[short trail]

So the rule is adding a 2? 😬

[open whale]

we can first add and then multiply

but we can also

first multiply each term

and then add up

$a(b+c+d)=ab+ac+ad$

[vivid gulch]

yoavmal

[open whale]

$a(b+c+d+e)=ab+ac+ad+ae$

[vivid gulch]

yoavmal

[open whale]

etc etc

[short trail]

Oh I'm kinda getting this

[open whale]

so if we have a group of people

each of them has something

we can add all the people and then see how many we have in total

or, we can see how many each has in total, then add these up

[short trail]

Oh yes makes sense

[open whale]

so

$(a+b)^2$

[vivid gulch]

yoavmal

[open whale]

is just a+b people

and each has a+b stuff

in other words

a has a+b stuff

and b has a+b stuff

[short trail]

Ok

[open whale]

$(a+b)^2=(a+b)(a+b)$

[vivid gulch]

yoavmal

[open whale]

$(a+b)(a+b)=a(a+b)+b(a+b)$

[vivid gulch]

yoavmal

[open whale]

yes?

[short trail]

Yes

[open whale]

ok

lets break it further

what's $a(a+b)$?

[vivid gulch]

yoavmal

[short trail]

(b+a)
?

Is that correct

Wait no

I read it wrong

[open whale]

ok, i'll write it differently

$a(a+a)=aa+aa$

[vivid gulch]

yoavmal

[open whale]

$a(a+k)=aa+ak$

[vivid gulch]

yoavmal

[open whale]

what's $a(a+b)$

[vivid gulch]

yoavmal

[short trail]

aa+ab
?

[open whale]

yes

exactly that

and what's $b(a+b)$?

[vivid gulch]

yoavmal

[short trail]

ba+bb

[open whale]

ok

lets write that

$a(a+b)+b(a+b)=aa+ab+ba+bb$

[vivid gulch]

yoavmal

[short trail]

Got it

[open whale]

now lets simplify

what's aa?

[short trail]

a²
?

[open whale]

yes

precisely

and what's bb?

[short trail]

b²

[open whale]

so

$(a+b)^2=a^2+ab+ba+b^2$

[vivid gulch]

yoavmal

[open whale]

do you see?

[short trail]

YES

[open whale]

aha

[short trail]

I feel so good after understanding that
Tysm-💀

[open whale]

ok

so the final result is

$(a+b)^2=a^2+2ab+b^2$

[vivid gulch]

yoavmal

[open whale]

now, lets try applying the same logic

step after step

for $(a+b)^3$

[vivid gulch]

yoavmal

[open whale]

it will be a bit longer

[short trail]

Is it
(a+b)³=a³+3ab+b³
?

Oh?

[open whale]

that would be amazing

but no

[short trail]

Oh ;-;

[open whale]

lets try applying it just like we did earlier

what does it mean to be

cubed?

[short trail]

Something multipled by itself thrice?

[open whale]

ok

[short trail]

So
a³= a×a×a

[open whale]

so

instead of a

write

[short trail]

Hm-

[open whale]

(a+b)

[short trail]

(a+b)³=(a+b)(a+b)(a+b)

[open whale]

ok

so lets phrase it a bit differently

[short trail]

Hm-

[open whale]

$(a+b)^3=\Big((a+b)(a+b)\Big)(a+b)$

[vivid gulch]

yoavmal

[open whale]

yes?

[short trail]

Yes

[open whale]

ok

now comes the fun part

we already know what this is

so we can put it right in

yes?

dose that make sense?

[short trail]

Yes

[open whale]

ok

so do simplify it

we now have

$(a+b)^3=(a^2+2ab+b^2)(a+b)$

[vivid gulch]

yoavmal

[short trail]

oH

Interesting

[open whale]

can you open the brackets now, then?

[short trail]

Wdym by that?

[open whale]

use $a(b+c+d)=ab+ac+ad$

[vivid gulch]

yoavmal

[open whale]

in here

[short trail]

Hm

[open whale]

try it

don't fear to make errors

that's the only way you'll get to understand it properly

and trust me

once you understand it

it becomes VERY easy

like, i can simplify these extremely well, extremely fast

[short trail]

Ok um

[short trail]

a³+2ab+b³

?😀

[open whale]

did you apply the rule?

try taking the big chunk

[open whale]

that is

the part we already simplified

a^2+2ab+b^2

and multiplying it by each part of (a+b)

[short trail]

But i already did multiply a² and b² right

[open whale]

not quite

i meant

open the (a+b) brackets

multiply the entier chunk by a

[short trail]

OH

[open whale]

and then multiply the entire chunk by b

[short trail]

Can u do this one for me
It slightly confusing;-;

[open whale]

nope

you gotta do it

until you get to the right answer

you can take simpler cases

but

in the end, you gotta try derive it yourself

[short trail]

Mhm---

@open whale (btw sorry i went afk for a while)

Is it
a³+b³+3a²b+3ab²
I doubt it but-

[open whale]

Getting there!

Oh wait sorry yes!

It's 100% correct

There's some obvious symmetry now

a³+3a²b+3ab²+b³

https://cdn.discordapp.com/attachments/792861622724591616/1024250429896392775/IMG_20220927_145445_940.jpg

Lets go one by one and see how the identities work

[short trail]

Oh

Ok

[open whale]

$(a-b)^2$?

[vivid gulch]

yoavmal

[short trail]

(a-b)²= a²-b²-2ab
?@open whale

[open whale]

Almost

It's +(-b)²

Not -(b)²

[short trail]

Oh ok

[open whale]

So it becomes

a²-2ab+b²

Now try to find what (a+b)²-(a-b)² equals

[obtuse oar]

umm... do anyone here know complete square format?

[vivid gulch]

ﾉ.ﾘ.ﾑ.ﾑ.り.ん

$x^2+8x-10=0$

[obtuse oar]

$x^2+8=10$

[vivid gulch]

ﾉ.ﾘ.ﾑ.ﾑ.り.ん

[obtuse oar]

$x^2 + 8x + 5^2 = 10 + 5^2$

[vivid gulch]

ﾉ.ﾘ.ﾑ.ﾑ.り.ん

[obtuse oar]

$(x+5)^2 = 35$

[vivid gulch]

ﾉ.ﾘ.ﾑ.ﾑ.り.ん

[obtuse oar]

how is tht done?

[short trail]

(a+b)²-(a-b)²=(a+b)²(a+b)²-(a-b)²(a-b)²

??

[craggy meteor]

$(x+5)^2=x^2+10x+25$

[vivid gulch]

PooPoo, not CEO of comedy

PooPoo, not CEO of comedy

[craggy meteor]

i forgot the x💀

fuck

Ok, now use quadratic formula to solve this

[vivid gulch]

PooPoo, not CEO of comedy

PooPoo, not CEO of comedy

PooPoo, not CEO of comedy

[craggy meteor]

@obtuse oar the solution with steps

[open whale]

Sorry, I had to go

[short trail]

It's ok

[open whale]

So you kind of mixed it up

Basically

You first need to find $(a+b)^2=a^2+2ab+b^2$ and $(a-b)^2=a^2-2ab+b^2$

[vivid gulch]

yoavmal

[open whale]

Then when you subtract one from the other...

[long socket]

@dire sable Sup dude

two things

I found out what limits are

and I also found out just how hard is factoring polynomials by grouping

[dire sable]

Nice, limits are fun

Factoring polynomails can be difficult and first, but once you get the hang of it it's kinda fun

[long socket]

you still there?

[dire sable]

Sorry, I need to get to school

[long socket]

This is about the math game idea

[long socket]

Ohh

do tell me of your time patterns in DMs so that I can annoy you less

[dire sable]

Nah, no annoyance is happening

I'll definitely let you know my time patterns though, could you do the same?

[long socket]

sure thing

[mortal olive]

Does this make sense.

[open whale]

yes, at t=0

The faulty step is the substitution

There can exist $t^m=\frac{x^m}{m+1}$

[vivid gulch]

yoavmal

[open whale]

However it isn't the same t for all m

If we choose a different m you'll see that

$t=x/2$

[vivid gulch]

yoavmal

[open whale]

But also $t=x/\sqrt{2}$

[vivid gulch]

yoavmal

[open whale]

This only works at t=0 and nowhere else

[long socket]

@dire sable hey bud, important question

what is a variable?

[dire sable]

That’s hard to describe, I’d say something indicating a value

[long socket]

a variable is like a symbol right?

and a symbol is a something that represent something else

[dire sable]

Yeah, a symbol that works as a placeholder for something else

[long socket]

and the symbol itself is important

one of its importance is that they are unique

x isn't y

in terms of its name

but the value x holds can hold the value y holds

different names carrying the same value

also in a formula for example, the names doesn't really matter

e.g you are given x = y + 1

the x might be a and the y might be b

but the equation is the same

[dire sable]

Some functions don’t even even require multiple variables, as you can simply state f(x)

[long socket]

in math, functions have mostly one argument

in contrast to programming where its more common to have two arguments

[long socket]

the x part of f(x) can be anything else

variables can be placeholders for other variables

first order logic baby

does the definition of a variable require some set theory?

[dire sable]

I’m not sure, it might be interesting to try and apply it

[long socket]

a variable is a symbol

a symbol who's value is a part of some number set

or the set of all variables

a variable is also like a pointer in terms of programming

[dire sable]

Oh yeah, I wonder if a variable could hold the empty set

[long socket]

it can?

variables can hold entire sets instead of any one elements in their given domain

it can't hold itself according by modern set theory

it can hold parts of itself tho

I wonder what puzzles arise from this

[dire sable]

I mean, can a variable be defined by the empty set?

[long socket]

well, hm

let's use a analogy

right now, x the variable doesn't exist

and then, bah! out of nowhere we made x

what does x look like?

[dire sable]

Could x be defined by any set?

[long socket]

let's see

empty set - x is just empty

set of all letters - x can be a or b or c and so on

set of natural numbers - yes

set of its proper subsets - yes

set of all variables - yes

what else?

[dire sable]

What about if x is undefined

[long socket]

what do you mean?

[dire sable]

If x could be defined by an empty set, it could be undefined due to that

I'm not sure though, I don't know much about set theory

[long socket]

but then, can set theory provide good puzzles or problems in a game enviroment?

[dire sable]

Sorry, I don't quite get what you mean by that

[long socket]

edited

its supposed to be provide

not prove

[dire sable]

Oh yeah, it definitely could, although I can't think of any applications off the top of my head

[long socket]

its a form of reasoning, a skill greatly needed for the rest of my math game

math is the ultimate puzzle after all

[dire sable]

Yeah, using set theory could greatly expand the types of puzzle that could be made

[long socket]

Quick Question, do I show the barbershop paradox early in the game as some sort of forshadowing?

[dire sable]

What would you be foreshadowing?

[long socket]

the idea of self-reference

and maybe the zfc axioms

and also infinity

[dire sable]

Ohhh, yeah, would provide an interesting call back

[long socket]

should it be vivid, or should it be like a nightmare?

since that paradox breaks maths after all

[dire sable]

I was just thinking about it maybe being in the background, so it's not too emphasized

Maybe like a hidden advancement

[long socket]

Im guessing the normal ending of the math game is the endless loop of trying to find the solution to those very hard problems

and the true ending accepting that math is broken and moving on

[dire sable]

Maybe also have one where it's finding out math is broken, but ignoring Godel and still trying to find answers

[long socket]

that sounds like a bad ending

[dire sable]

I dunno, I think of that as being hopeful

[long socket]

what do you think is a bad ending here?

[dire sable]

statistics

[long socket]

the "final boss" in this game is where the soul of Godel traps you in a endless nightmare

[dire sable]

Like when people start p hacking

[long socket]

either you ignore godel and get stuck in the nightmare(bad ending), escaping from the nightmare without taking what godel said to heart(normal ending), and accepting that math is broken and godel shows mercy and ends your suffering(true ending)

in the nightmare, you have to face off against famous paradoxes, very hard problems, and some stories here and there

[dire sable]

I feel like that seems a bit like giving up on math though, I dunno

[long socket]

smart people do get insane after all

godel is just helping you to deal with that insanity

[dire sable]

Yeah, that's true

[long socket]

and math has a effect on its people

math is the center of everything

if we found out that math is broken, then some important dreams people have gets cut off

[dire sable]

It's kinda like the search for a unifying theory in physics

[long socket]

I say this game can have a real impact on people if done right

and having a horror like factor in a (decent)educational game is unheard off

not baldi's basics, its not educational

[dire sable]

That'd be really fun to incorporate

Oh, I should probably get to sleep, sorry

[long socket]

get some rest

okay then

sleep well

[dire sable]

Thanks, see you around

[mortal olive]

Yea 👍

[long socket]

Sup @dire sable

Do you mind making some set theory puzzles with me?

[dire sable]

Sorry, I can’t talk right now, but I’ll think about them and let you know if I can think of any good ones

[long socket]

sure thing

[long socket]

another one of those head starching questions @dire sable, how would we define the four inequality symbols?

hm . . . using the subtraction operation?

[dire sable]

Yeah, that’d probably be easiest

[long socket]

how else can we define if one is smaller or bigger than the other?

[dire sable]

Maybe by showing it on a number line

[long socket]

that's a geometric definition

what I want is a formal one

like the pressesor theorem I showed you before

[dire sable]

Oh, in that case yeah, subtraction is probably the only way

I can’t think of any others

[long socket]

okay then, how do we go about defining that?

[dire sable]

If zero and negative numbers are defined, we can just subtract and see if the resulting number is part of one of those sets

Might be hard defining negative numbers without inequalities though

I need to get to school now, but that might be something to think about

[open whale]

@proven gate

Let us define the hyperreals as the sequences of rational numbers such that they do not diverge

The mandelbrot polynomial is convergent in this system

Over a larger subset of the mandelbrot set

Excluding the Julia set of the mandelbrot set

[long socket]

Anyone knows about a video about the history of algebra?

[open whale]

No, but it all starts in cantor

[long socket]

how so?

[open whale]

Set theory

[long socket]

Hm, how would one formally define a variable with set theory?

[open whale]

No idea, but algebra isn't about variables

[long socket]

what do you mean?

[open whale]

Algebra is about set, group, field theory, ntc

Etc*

Not about manipulations

[long socket]

isn't that abstract algebra?

[open whale]

If you want to call it that way, sure, but that's what algebra is

Manipulations of equations are just demonstrations of algebraic systems

Also, about variables

using set theory

x is an element of a set X

[long socket]

so for example:

x and its a element of let's say R

and what happens when x is a expression?

[open whale]

It is an element of a certain set, and has a certain relation with another element from another set

A relation is a formal concept using sets as well

[long socket]

and a function is a kind of relation

what about equations?

[open whale]

That is also a relation

g(a)=f(b) is a superset of a=f(b),

And a=f(b) is a functional relation between a,b