general discussion

[long socket]

give me a example this time

[open whale]

A relation is a set of ordered pairs

So for example

a is an element from {0,1}

b=a²

So b is an element of {0,1} as well

The relation b=a² is written formally as {(0,0),(1,1)}

If a is in {0,1,2}

Then b is in {0,1,4}

Their relation is {(0,0),(1,1),(2,4)}

[long socket]

so in a equation of x = y + 1

y = {1, 2 ,3}

x = {2, 3, 4}

[open whale]

And {(1,2),(2,3),(3,4)}

[long socket]

but what is the notationif y is all the numbers in R?

[open whale]

We use rules to construct the set

{(y,y+1)|y is in R}

[long socket]

I sorta can read it

but it its nice to hear the worded version

[ebon depot]

hey

to the last parliament, nice job in turning this into a forum channel

lets see if texit works

$1+2=3$

[vivid gulch]

Hebrews 12:1-2

[ebon depot]

nice

[ebon depot]

so its like for every y, the set contains (y, y+1)

is that good?

[long socket]

what does the comma mean in this context?

[ebon depot]

so basically this means like if our x value is a, then the y value is a + 1

im changing y to a here so as not to confuse it with the y axis

the notation (a, b) is called an ordered pair

[long socket]

like coordinate pairs

[ebon depot]

yes!

exactly

except we are dealing with parametric form or parametrization here, where a is a parameter

[long socket]

hey hebrew, do you know anything about deriving the integers from the natural numbers?

I heard it uses something call equivalent relations or smth

[open whale]

The set of all (y,y+1) where y is an element of R

[long socket]

Hey @dire sable

Found a way to say a number is smaller than another

if we can represent numbers as some sort of a set

It uses the proper subset relation btw

[dire sable]

Hmm, interesting

Oh yeah, @long socket , there was an interesting video I've been wanting to share with you

https://www.youtube.com/watch?v=myKkhVy74V4

[long socket]

then it makes induction proofs possible

and makes inequalities a reality

just need to find a way to represent numbers as sets

btw, we are focused on doing things on the natural numbers

how do we adjust our stuff to integers?

[dire sable]

If we define zero, we could define integers that way

With subtraction

[long socket]

hm

let n - n = 0

(assuming we defined subtraction)

we can prove that but that's for later

[dire sable]

If we define multiplication we could go with n-2n, otherwise just 0-n

[long socket]

yeah

0 - n = -(n) = -n

let -n be the number that -n + n = 0

now how do we define operations with negative numbers?

[dire sable]

Yeah, and those numbers can be added to the set of natural numbers

I think operations with negative numbers can use the definition 0-n for simplicity

[long socket]

let's do addition first

what is 3 + (-2)?

A lemma we can use is n + (-m) = n - m

so 3 + (-2) = 3 - 2

which is 1

what about (-3) + (-2)?

another thing we can use is (-n) + (-m) = -(n + m)

so that turns into -(3 + 2)

which is -5

[dire sable]

We could also change it too 0-3-2, and use subtraction

[long socket]

nice idea

what else

right

-a + b

what now?

[dire sable]

We could assume multiplication and division are defined for natural numbers and try to apply it to negative numbers

[long socket]

multiplication is buffed addition, but what about division?

[dire sable]

I guess division almost goes a bit into algerbra

I'm not sure how else to define it

[long socket]

really?

[dire sable]

Like if we have x/y=n, then that'd be x=yn solving for n

[long socket]

so

6/2 = n => 6 = 2n

[dire sable]

Yeah

[long socket]

now how do we solve for n?

without division under our belt?

[dire sable]

Would defining division as the reciprocal of multiplication work?

[long socket]

n x 1/n = 1

that's in rational number territory, right?

[dire sable]

Yeah, it should be

[long socket]

then we're stuck then

unless

we use subtraction

division is informally defined as a / b = c, where c is the the number of times a has to be subtracted by b in order for a to reach 0

I said informally because how do we "keep tract" of operations?

[dire sable]

You could use multiplication too, so c is the value b has to be multiplied by to reach a

Also informally

[long socket]

in both definitions, we need a way to "keep tract" of operations

What math trick can we use?

[dire sable]

Maybe we could just leave it as a=bc, and leave c as a variable while providing hardcoded values for a and b

That probably wouldn't work though

[long socket]

well it shouldn't

hold on

let's ask the more experienced mathematicians

[dire sable]

Yeah, that's probably a good idea

[long socket]

I'll see you later Orth

another interesting convo

[dire sable]

Yeah, I'm looking forward to how this question will be answered

[long socket]

Sup @dire sable

is it me or do I feel like we need to memorise every single identity when we reach calc?

[long socket]

also this -> https://www.youtube.com/watch?v=j5s0h42GfvM

[dire sable]

Calc isn’t too bad with identities, the trig identities are a bit more difficult to memorize

At least that’s how it was for me

[dire sable]

Oh yeah, I saw this a while ago

Although the formula isn’t useful at all, I love how it basically demonstrates a way to program with purely math

[long socket]

Wish it can be made faster

[dire sable]

Yeah, that’s what Riemann’s hypothesis is for

[long socket]

oh hello there

[dire sable]

Heyy

[long socket]

I wonder if anything new happened at math

[dire sable]

I can’t think of anything big

Oh, I won’t be able to respond for a bit, I’ll try to as soon as I can

[long socket]

okay then

[long socket]

why is it that you can simplify rational expressions when you multiply then by the lcd?

[open whale]

Wdym?

[long socket]

like the equation t/25 - t/30 = 1

you multiply both sides by 60

and it will simplify to t = 60

I wonder why

[ebon depot]

the fractions cancel out..

[ebon depot]

isnt the lcd 150?

[long socket]

how?

[long socket]

oh right

wrong equation

I meant to say "t/15 - t/20 = 1"

[ebon depot]

ah

[ebon depot]

then in that case lcd is 60 ur right

and yes t = 60

[dire sable]

For this, you would get $\frac{4t}{60}-\frac{3t}{60}=1\Rightarrow\frac{t}{60}=1\Rightarrow t=60$

[vivid gulch]

Orthodox

[dire sable]

The first part of that is gotten because you're multiplying by 1

$\frac{4}{4}\frac{t}{15} and \frac{3}{3}\frac{t}{20}$

[vivid gulch]

Orthodox

[long socket]

<@&727457814523674674> what are some useful equations manipulation?

[ebon depot]

wdym

[long socket]

It seems that I need to master moving stuff around in a equation

X + X/1 = 3/2 for one

The server said that you can shift the denominator of a rational expression

1 + x/x = 3/2

Turns to 1 + x = 3x/2 for some reason

[ebon depot]

ah

[ebon depot]

thats weird

[long socket]

But does it still hold equality?

[ebon depot]

of course not.

oh wait nvm

to be safe, just do what you know

if ur not sure, dont perform the transposition

[long socket]

One, I did the thing

Second, not sure how would I solve for x without using quadratic techniques, which is something I didn't learn yet

[ebon depot]

u dont need quadratic techniques

wait for 1 + x/x = 3/2 is wrong already

x/x is 1

1 + 1 = 2

which is not 3/2

[long socket]

1+x / x

[ebon depot]

AH

[long socket]

Not that

[ebon depot]

(1+x)/x

alright lets do this

,tex $\dfrac{1+x}x=\frac32$

[vivid gulch]

Hebrews 12:1-2 | Vote for PRP

[ebon depot]

so multiply both sides by x

u get $1+x=\frac32x$

[vivid gulch]

Hebrews 12:1-2 | Vote for PRP

[ebon depot]

right?

@long socket

[long socket]

Yeah

I'm here

[ebon depot]

ok great

now u have x on one side and 3/2 x on another

so this time we subtract x from both sides...

$1=\frac12x$

[vivid gulch]

Hebrews 12:1-2 | Vote for PRP

[ebon depot]

this can also be written as $1=\frac x2$

[vivid gulch]

Hebrews 12:1-2 | Vote for PRP

[ebon depot]

therefore the answer is?

[long socket]

Hold on

It's 2

[ebon depot]

yes

as easy as that

[long socket]

The subtract x part

[ebon depot]

just do the same operation on both sides

[long socket]

Lemme try

[ebon depot]

ok

so yeah the only thing u need to remember is that in any equation, u can add, subtract, multiply, divide, etc. the same thing on both sides

[long socket]

How do I deal with the mixed rational expression?

[ebon depot]

just get rid of that fraction

thats why we multiplied by x on the first step

gets?

[long socket]

For what reason?

[ebon depot]

it makes it easier

like

u can solve normal linear equations

[long socket]

And if I subtracted normally, I get 1 = (x)(3-2x/2x)

[ebon depot]

but u need to realize that rational equations can be reduced to linears

[ebon depot]

wait

can you show your steps?

one by one

[long socket]

Tbf, I don't know what to do with the mixed rational expression

@ebon depot

[ebon depot]

ok u did this part correct

but pls write it as 3/2 x not x 3/2 so we can follow notation properly

ok so u subtracted x right

u have 3/2 of x and 1 of x being taken away from it on the right side

3/2 - 1 is?

[long socket]

1/2

[ebon depot]

ok

[long socket]

Can you rephrase this?

[ebon depot]

so u have 1 = 1/2 x

[ebon depot]

ok lets make this concrete

i have 1 and 1/2 apples

i take 1 away from it

i then have only half an apple left

right?

[long socket]

Yeah

[ebon depot]

so yeah thats how u get 1 = 1/2 x

so u can already get x = 2

gets?

[long socket]

I'll try with a different equation

Hold on

[ebon depot]

ok

yep practice makes perfect

[long socket]

You might want to latex this

Something I running into is when I try to break down the equation, I just get another rational expression, not a nice and simple whole one

Like before

@ebon depot

[ebon depot]

ok lets do this

$\frac{1+x}x=\frac32$

[vivid gulch]

Hebrews 12:1-2 | Vote for PRP

[ebon depot]

multiply by x:

$1+x=\frac32x$

[vivid gulch]

Hebrews 12:1-2 | Vote for PRP

[ebon depot]

subtract x:

$1=\frac32x-x$

[vivid gulch]

Hebrews 12:1-2 | Vote for PRP

[ebon depot]

simplify:

$1=\frac12x$

[vivid gulch]

Hebrews 12:1-2 | Vote for PRP

[ebon depot]

multiply by 2:

$\boxed2=x$

[vivid gulch]

Hebrews 12:1-2 | Vote for PRP

[long socket]

Hold on

I can see how it works

In elementary math, they taught about how improper fractions can be converted to a mixed fraction

Guess that kinda applied to rational expressions, fractions with letters

Since p and q are numbers, you can use that fact pretty easily

@ebon depot guess I learned something

[ebon depot]

yes

3/2 = 1 1/2

[ebon depot]

nice!

[long socket]

How else can I convert rational equations to linear equations?

[ebon depot]

just multiplying by the denominator

ig thats all u can do

[dire sable]

Huh what

x doesn't equal 2

[dire sable]

What you guys did from here is right

But this doesn't work

@long socket @ebon depot

$x+\frac{1}{x}=\frac{3}{2}$

[vivid gulch]

Orthodox

[dire sable]

Multiply by x on both sides

$x*(x+\frac{1}{x})=\frac{3}{2}*x$

[vivid gulch]

Orthodox

[dire sable]

For the left distribute

$xx+x\frac{1}{x}\Rightarrow x^2+\frac{x}{x}\Rightarrow x^2+1$

[vivid gulch]

Orthodox

[dire sable]

So we have $x^2+1=\frac{3x}{2}$

[vivid gulch]

Orthodox

[dire sable]

Multiply by 2 on both sides

$2x^2+2=3x\Rightarrow 2x^2-3x+2=0$

[vivid gulch]

Orthodox

[dire sable]

Hmm, there are no real solutions

Interesting, I might've just misread it

But

$x+\frac{1}{x}=\frac{3}{2}$ has no real solutions

[vivid gulch]

Orthodox

[dire sable]

@long socket @ebon depot Sorry for pinging you twice, but the first one was the beginning of it, here's the end

[hollow ice]

bruh this one is obvious, since you can just do x+1 = 3, x=2 and get x=2

[ebon depot]

oh yeah i forgot to point that out

[ebon depot]

ok

he never wanted quadratic techniques tho 💀

[dire sable]

Yeah, that’s true, although it wouldn’t really have been possible without it

You did a good job explaining the rest of the problem though, I feel like I learned more about fractions as well❤️

[ebon depot]

👍🙂

[long socket]

@dire sable Sup Orthodox

How are you and your math journey?

[dire sable]

It’s been slow unfortunately, I got caught up in other schoolwork. Still been able to have fun in math class though, we’re going over integrals now.

How’ve things been going for you?

[long socket]

Almost done with linear equations in two variables

[long socket]

Sweet, Claculus

I think I wanna write some informal things about natural numbers

[dire sable]

Oh, that could be fun

[long socket]

Then use that to make a game of some sort

A math game no less

Albeit it's going to be hard explaining proofs and set theory

[dire sable]

Yeah, but it’s probably going to be a fun challenge

[long socket]

Like before, what do you think with be the final boss of that game?

[dire sable]

If it’s a person, probably Godel, but if it’s an equation or concept, maybe like power sets

Or introduce some calc just for fun

[long socket]

Not him again

[long socket]

Calc without negative numbers?

[dire sable]

Maybe the game could introduce negative numbers

[long socket]

Still am figuring out the formal definition of negative numbers

[dire sable]

Although it maybe like for the set of negative numbers and positive numbers, the exists one number from each set that add to 0

[long socket]

Okay

And what happens when I use numbers that aren't opposites of each other?

Like 1 + -2?

[dire sable]

It only needs to exist, it’s not for all pairs

[long socket]

I meant what happens when I pick two numbers from each set and add them with the fact that they aren't opposites

And how do we even know that a number is the opposite of another number?

[dire sable]

The definition makes it so the within the negative numbers set there will always exist the opposite number. It’s kinda like the definition will run through all the possibilities to see if the definition holds

And by doing that it provides the negative numbers

[long socket]

But wait

We didn't define the operations to work with negative numbers yet

I can define the operations with natural numbers using my homemade Presessor Theorm

[dire sable]

I think the way the operation would work could be defined within the definition, as they are opposites

[long socket]

In my time dealing with the Z set, I use symbol - to kinda make the Z set possible

Btw, does it make sense to you?

That using a symbol in justifying ways is axiomatic?

[dire sable]

Yeah, that makes sense

Since it’s been rigorously defined already

[long socket]

Hm?

[dire sable]

The Z set has been defined, so using it for further definitions works well

[long socket]

Can you define it now?

[dire sable]

I’m not sure how to define it

[long socket]

I guess the set of all N numbers plus zero plus all N numbers with a minus sign in front

Notice the symbol being used

[dire sable]

Yeah, so it’s be a union of the sets

Or could do 0-x with x being within the N set

[long socket]

Also, I want to show you something

[dire sable]

Oh, okay

[long socket]

Pretty neat hashing function

Right?

With a bit more set theory, we can make whatever hashing function we want

@dire sable

Just in case

[dire sable]

Oh, sorry, I was just looking at it

[long socket]

What do you think?

[dire sable]

Yeah, it seems like it’d work very well

[long socket]

It's been a while since I made something like this

Time does stuff to our minds

[dire sable]

Yeah, in both good and bad ways

[long socket]

The last time I've made some cool math stuff was the presessor therom

[dire sable]

Last time for me was pointlessly looking at collatz conjecture

[long socket]

Don't worry buddy. Someday, you might make a dent in math even if it's just a little bit

Math is built about very tiny bits of stuff that is useful

[dire sable]

Yeah, that’s true

[long socket]

My goal in math is of course to build it up from scratch

Using nothing but ZFC and logic

More or less

[dire sable]

Oh, that reminds me

[long socket]

Hm?

[dire sable]

https://i.redd.it/4o4y99b2v6x91.png
Pythagorean theorem in purely logical form

[long socket]

Oh my

Can you share the link?

To the paper?

[dire sable]

It's on quora, so I'm not sure if it's real, but it seems to give a pretty in depth explanation

https://www.quora.com/How-do-I-represent-the-theorem-of-the-right-triangle-Pythagoras-in-the-Gödel-code-number/answer/Senia-Sheydvasser?ch=15&oid=296647555&share=5b6f0d29&srid=LqLdE&target_type=answer

[long socket]

Ahh yes

Godel numbers

Anyways, gotta bounce

See you soon

[dire sable]

Ah, yeah see ya, it was nice talking

[long socket]

@dire sable you there?

[dire sable]

Yeah what's up?

@long socket

[long socket]

It's morning here

Fine

Well, how about we deal with the natural numbers?

[dire sable]

Ohh

I won't be able to talk long, sorry

But for natural numbers, would it be possible to define them by a unit?

[long socket]

What unit?

Also as a fun challenge, we don't use geometric definitions

[dire sable]

I mean like kinda just declare that 1 exists, and build from there

I'm not sure if that's okay to do though

[long socket]

The axioms of natural numbers already did that

Zero exist and every number number that isn't zero has a successor

[dire sable]

Hmm, that makes sense

[long socket]

@dire sable

In the name of godel

This works somehow

My definition of negative numbers

[dire sable]

Oh hmm, it seems like some of this could also be used to define inequality operators

[long socket]

Defining -a + -b is tricky tho

Also what do you mean by that

[dire sable]

Like if a-b is outside of the set of natural numbers, it indicates that a<b

[long socket]

I guess so

Distributive Property in clutch

Believe that the minus sign is a number(it is just -1) and you get that

[dire sable]

Yeah, that tends to work pretty well

[long socket]

Suck on that equivalence relations

I can just algebra just fine

For rational I may need a bit more help

[dire sable]

Yay for more operations

[long socket]

And without much set theory too

[long socket]

@dire sable

Sup there

I'm dealing with linear inequalities and happen to run into fractions that I can't get rid of

The math problem is 2x + 5y < 7

Y = (-2x/5) + (7/5)

I graphed the thing and it seems to need transformation if I want to use b with 5y

[dire sable]

What do you mean?

Wait

,w graph 2x+5y>7

[vivid gulch]

Sending query to Wolfram Alpha, please wait.

Something went wrong!

An unexpected error occurred while processing your command!
The error has been reported and should be fixed soon.
If the error persists, please contact our friendly support team at our support guild!

Exception

aiohttp.client_exceptions.ClientConnectorError: Cannot connect to host www6b3.wolframalpha.com:443 ssl:default [None]

[dire sable]

Hmm

Well, @long socket you would graph y<-2x/5+7/5, such that it would be shaded under the line

The shaded area represents the solutions

[long socket]

I meant finding the b of that equation

It's in slope-intercept form

And the b part is a fraction

Makes it a pain to graph

[dire sable]

Oh, yeah, that’s true

But there’s no way to get rid of the fraction unfortunately

Oh, I need to head to school now, sorry

[long socket]

See you soon then

[hollow ice]

It's bullshit

I've read the article

There's a reason why it's BS

The Godel encoding really just uses a notation to express the direct equivalent of the expression

Godel forms are not even necessary

One can simply use 2nd-order logic notation

[dire sable]

Honestly, I don’t really understand this, but thanks for letting me know! It’ll be interesting to look back on this if I gain more experience in the matter

[hollow ice]

Ah. Basically, there's primarily 1st and 2nd-order logic notation. 2nd-order is much more complex, but it lets you do more things.

[dire sable]

Whoops, sorry I didn't see your reply. That seems interesting, thanks for the description

[long socket]

Can you give a example to me and my fellow colleague?

[long socket]

How would one define a relation? @dire sable

Likewise, what is the formal definition of a property?

[dire sable]

Hmm, maybe for a relation, a comparison between properties of different parts of a set

While a property would be unique and/or non-unique characteristics of something that come together to form something that is unique

@long socket

[long socket]

Hm

It's a bit tricky to formally define

Say I have a relation y = x +1

So y is a member of the set Y

And x is a member of the set X

X is just N

Y is all of N's members plus 1

So 0 is in X but not Y

What can you see here?

[dire sable]

Could the properties of a set be defined in relation to other sets that are already rigorously defined?

[long socket]

What do you mean by that?

[dire sable]

In the example you gave, it's defining Y based off of N

[long socket]

Any two sets can work when defining a relation

As long as there is a way for members of one set relate to members of another set

[dire sable]

Ohh, so general relations between sets

[long socket]

Yep

[dire sable]

Hmm, could it be defined by iterating through elements of a set?

[long socket]

How would you iterate through a set?

A "for all" statement?

[dire sable]

Yeah, I think that would work

Like

For all X, X_i+1=Y_i

[long socket]

That works

[open whale]

Good morning!

[long socket]

Hello OP

[open whale]

Lol

[long socket]

Just discussing about relations

[open whale]

Just joining the conversation

Ah

[long socket]

Go on

[dire sable]

Oh, hey Yoav

[open whale]

You'd go on, I think, as you were talking

[long socket]

Anyways

If I gave you two sets

A is 1, -1, 2, -2, 3, -3

And B is 1, 2, 3, 4, 5, 6

Can you find the relation?

[dire sable]

Would you be able to say it's a union between the absolute value of A, and the absolute value of 2A?

[long socket]

Challenge, no set operations

[dire sable]

Oh, hmm

[long socket]

Yoav, can you define relations and properties for us?

[open whale]

A relation is a set of ordered pairs with the first element of each pair being from some set, and the other element from another set

Properties not so much

Unless you mean symmetry or transitivity and so on

[long socket]

So a relation is just a set then?

Given x, how would you find y in that relation set?

[dire sable]

I haven't been able to figure this out, I was thinking A+3, but then there's be a 0 instead of 3

[dire sable]

Yoav, correct me if I'm wrong, but I think that could be arbitrary

[open whale]

What are x and y?

[long socket]

A element in the set X and Y respectively

[open whale]

ah

well, you can't find y because this isn't necesserily a function

[long socket]

But can you find the relation?

I didn't say find the function that maps X to Y

[open whale]

not any particular one

you can know it is from the set

$\qty{(x,y)\middle|y\in Y}$

[vivid gulch]

yoavmal

[lethal wigeon]

Im learning calculus at age 13

[wicked junco]

Thats nice. I started at 15.

[open whale]

cool! make sure you focus on the intuition and not drown in the definitions

[lethal wigeon]

ok I will

[lethal wigeon]

Thats cool also thanks

[inner nexus]

Is there a formal definition/name of this animation on desmos: https://www.desmos.com/calculator/pteljfcfig
I thought that it kind of looked similar to a double pendulum
I made the graph cause I was bored

[long socket]

hello people, what is the formal definition of the "element of " relation?

[open whale]

No idea, but i'd imagine it's belonging or something like that

[open whale]

I'd say this looks very similar to Fourier sums

[inner nexus]

O ok thx I’ll take a look at that. But I’ve never rly heard of Fourier sums. Is it an advanced concept?

[open whale]

Depends on how you approach it

Effectively, it's about summing sine waves to get different functions

[shrewd haven]

does the determinant of conics have any geometric significance?

i feel like it does but

i cant figure out how

[charred wolf]

https://math.stackexchange.com/questions/3425703/what-is-the-physical-significance-of-the-determinant-delta-in-the-general-equ

is this necro?

i mean there had been no activities here since last year

[open whale]

Sorta? We had different channels before, they're just archived now

Wait

I fell for it

💀

[lethal wigeon]

Can anyone help me learn calc 2 its so hard

[open whale]

intuitively or formally?

[static rune]

is pair of straight lines a smooth curve?

[open whale]

depends on how do you treat it

[static rune]

elaborate

[open whale]

a smooth curve has a very specific definition

,w smooth curve

[vivid gulch]

Results provided by WolframAlpha

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

Requested by yoavmal#5800

[open whale]

the issue here is

[static rune]

yeah its derivative is defined everywhere

[open whale]

how do you pair it smoothly?

you need a function that maps to those points

[static rune]

so yeah its derivative is not defined on the point of intersection

[open whale]

again, it's complicated

if you treat it as a set of two lines

so you have

{line A, line B}

then you can easily make a function that does that

whereas saying a union of the set of all points in both lines

then, it is not smooth

[static rune]

ohhh

alright

[static rune]

well

is pair of straight lines a conic section?

[open whale]

depends, it can be

[static rune]

when can it be?

[open whale]

when all points in the two lines also satisfy an equation of the form $a(x-b)^2-c(y-d)^2=1$

[vivid gulch]

yoavmal

[static rune]

why?

do they have to satisfy this equation too?

[open whale]

because that's how conic sections work

[static rune]

very helpful bro

[static rune]

so other conics also satisfy this equation?

[open whale]

sometimes

well, not others

but conic sections are parts of a 3d form

[static rune]

what

[static rune]

so like ellipse circle parabola

?

[open whale]

yes

[static rune]

but what is that equation?

[open whale]

$x^2-y^2=z^2$, with linear transformations accordingly

[vivid gulch]

yoavmal

[static rune]

what

you are expecting too much from me

[open whale]

google "conic sections"

sorry, google "en passant"

[long socket]

Is it possible to calculate the values of trig functions using only algebra and geometry and not calculus

[static rune]

lmao

[static rune]

for some

[open whale]

yeah

but it's usually very tedious, and won't work for the majority of values

mainly because we do use calculus... to define trig functions

even in the geometric form, we need to define the length of a circle arc, and this is done using calculus

[topaz cobalt]

What kind of maths ultimately ended up interesting you?@open whale

[open whale]

Googology, which is really big yet finite numbers

One way to make a really big number is using the tree game

The longest game for a given number, we'll use n, of unique colours, is denoted TREE(n)

TREE(1)=1, TREE(2)=3

And TREE(3) is so big you cannot express it in physical terms conventionally

TREE(4) is even incredibly unimaginably larger

One number even larger than that is your mom's weight

Just kidding I like fractal geometry

[topaz cobalt]

Heh cool

[crude plume]

https://archive.org/details/numencasus

Hey @topaz cobalt

Here it is!

[topaz cobalt]

Yep, I'm reading it too

[crude plume]

Ok dude

@normal arrow

[normal arrow]

i'll skim through it, i have homework to do

[crude plume]

Ok you can see it whenever you have time!

@topaz cobalt are you done?

[topaz cobalt]

Nope, besides it will take me some time to figure out since I am sick lol

[crude plume]

Ok tell me the remarks when you get the concept!

[topaz cobalt]

Will do

[normal arrow]

wait wait wait

so like

you're proposing a new way to write numbers?

[crude plume]

Actually the main purpose is partitions

[normal arrow]

wait

[crude plume]

But this could be a choice

This wasn't my intention

[normal arrow]

so, like, a number notation designed for one specific use?

[crude plume]

Yeah but it also has additions , multiplication ect

So We can also see it as aa new way to write numbers in the case of big partitions

[normal arrow]

"...powerful enough to revolutionize the mathematics." is kind of a clickbait then

[crude plume]

😅😅

Sorry dude

My mistake

I will edit pdf

[normal arrow]

it doesnt "revolutionize the mathematics", it kinda just solves one little problem

[crude plume]

I accept

[topaz cobalt]

Never be afraid of overselling yourself

[normal arrow]

this is overselling with a capital O

in bold

underlined

[crude plume]

Actually It was meant to write"It will revolutionize the mathematical world of Partitions"

[normal arrow]

font size 100 pt

[topaz cobalt]

Damn

[crude plume]

Just tell me how the concept is!
I am ready to solve that mistake

Is it helpful?

[normal arrow]

so it says that uh

"This formula is best in the case of huge numbers such as 737747, ..."

[crude plume]

Yeah it's right!

Try to use it in huge numbers

It is best then!

[normal arrow]

so

hm

[crude plume]

Tell fast

[normal arrow]

you scale this method to numbers like 737747?

[crude plume]

Use manual method!

[normal arrow]

which is?

[crude plume]

Written in pdff

[normal arrow]

oh

oh i found it

isnt manual method the same as N.C. method but like only a one specific row

that yields only partitions of a specific scale

am i missing something?

[crude plume]

Actually yes

You are missing some notations not mentioned in this pdf

Because these are just basics

Those notations minimise the time to a great extent!

Please can you tell me if it is helpful or not!

[crude plume]

Yeah because the chart method can be used in smaller numbers but it will be useless in huge numbers!

[normal arrow]

so you have no method of getting an amount of partitions for large numbers?

[crude plume]

We have it!

But that isn't in pdf

Because this pdf is just about basics

And the methods in this pdf will also work for huge numbers but if we want to find ""all partitions"" then it would be just some time taking!
Using advanced NumenCastical Functions will minimise the time by 95%

[crude plume]

Here I meant that we should use manual method in huge numbers because drawing N.C chart in case of large numbers will be time taking NOT WRONG!

[normal arrow]

well then whats the point of the thing if it doesnt offer a faster or more convenient way of getting the results

[crude plume]

Yes it minimises the time by 95%

I said above!

I just want you to rate my work please

@normal arrow

I don't think "minimising the time by 95%" is a small deal!

The best things I liked in NumenCasus are:--
Even a 2nd grade student can easily understand NumenCasus basics!
And also it is always accurate

[normal arrow]

im thinkin

that maybe you should look into the uh

generating functions

if you're lucky enough, you can find a closed form solution for amount of partitions of any number

[crude plume]

I know a formula for number of partitions already

n² maybe!

Because it is still in research

[normal arrow]

no?

4 has 5 partitions:
4
3+1
2+2
2+1+1
1+1+1+1

[crude plume]

But the main thing I am asking to you is to rate my concept a little so that I can get that I am doing right or not!

[normal arrow]

eh

you're trying to make a system being able to be used by even 2nd grade pupils but really

its only goal is solving a single problem

[crude plume]

Ok

[normal arrow]

its only worth developing a system if it allows solving a whole group of problems

like p-adics

or it makes computation easier/faster

for things like these, you should pretty much just say

"here's a closed form solution and few fun facts"

[crude plume]

Should I continue researching on it to find its other functions or It is worthless!

[crude plume]

First of all the main thing you aren't understanding about NumenCasus is "Using it as a new system can be a choice, but it isn't a new system!" It can also be used like a Identity, theorem or formula.
I already said that "It may be a new system for writing positive integers but it is used in partition numbers only" and I don't think we can say it a whole new system then!
Just take it as a linear function!

[normal arrow]

try to find a closed form solution and prove it

[crude plume]

Ok

[ionic violet]

Can somebody tell me what I should do? I had a 2 tutors in elementary, and 2 now. I am in 7th grade and have a good teacher who everyone understand but me. I had an 89 until today I got two 50’s and now have a 0. I think I might have dyscalculia. I don’t really study besides from when I was in elementary and now my 2 tutoring sessions, and I’ve never been that good. I just need help on what to do, I don’t want to fail. Any advice?

[topaz cobalt]

Study without outside stimulus

Maths is hard to learn, but it's doable

Also you can see a doctor to get diagnosed

But the first step is to study by yourself

Anything that stops you from is the issue

[short dock]

hey!

anyone good at math here? i am working on finding a formula for prime numbers and i would love to work with someone !

[normal arrow]

there is one already

so you're kind of too late

unless you want to make a better one

which will require, like, a very deep understanding of mathematics

its basically related to an unsolved problem

[open whale]

how would you define a formula?

it's important to know what exactly are you trying to achieve

[real sable]

Hey

[real sable]

Zeta functions right? That’s kind of what I’m talking about solving

[real sable]

Plug in “n” get the nth prime

[short dock]

essentially a function

[open whale]

well, i have the "formula" for you

it's a function so that

f(n) = nth prime

i think what you're really asking is

if there's a way to calculate it

[short dock]

yea

@open whale yes, since there is no known approach i am looking at areas of math that i think will help narrow down a solution

@open whalenumber theory obviously, statistics, and calculus for things like zeta functions

[open whale]

As far as I recall there is no formula

[normal arrow]

there is one

https://www.youtube.com/watch?v=j5s0h42GfvM

[vale egret]

whats the best way to learn quadratic inequalities

[open whale]

starting from absolute value inequalities

[short dock]

@normal arrowthat uses a floor function tho....not quite something im looking to find

[normal arrow]

trust me, if it can generate primes, then having a floor function is not a big deal

a bigger deal is making it easy to compute

and not some scary nested sum

[golden yoke]

hi

[golden yoke]

im new here because I had a stupid thought that I know im not smart enough to figure out lol

so basically I made a 3d scatterplot of all rgb values (255x255x255) and the thing is very trippy.
it has some very strange properties depending on the perspective you look at it from, which got me wondering about a 2d grid question I saw before.
if you have grid of squares like a chess board, the whole thing is a big square but how many squares are contained within (regardless of size)? so in the chess board example a single space would be a square but 4 spaces 2x2 would also count for the question because it is also a square and the 4 squares inside of it would count separately as well.
if you see where i'm going with this the next question is obviously: how many cubes are in that 255x255x255 cube I made?

here's the cube and a small glimpse of the trippiness inside for reference, if it helps XD

[normal arrow]

gay cube

[normal arrow]

can we come up with some fancy function that can solve intersections between quadratic and exponential like Lambert-W solves linear = exponential

[normal arrow]

nvm, found a paper

https://arxiv.org/pdf/1501.00138v2.pdf

[open whale]

generalize it to taylor, really

[open whale]

How does one phrase and prove Taylor's theorem, over the complex numbers?

Hmm

I'd suppose i'd want to clarify Taylor's theorem first

Sorry, differentiability

$f'(z)=\lim_{h(\in\mathbb Z)\to0}\frac{f(z+h)-f(z)}{h}$

[vivid gulch]

yoavmal

[open whale]

Soo

$f'(z)=\lim_{h\to0}\frac{f(z+h)-f(z)}{h}$

[vivid gulch]

yoavmal

[open whale]

Now, how do I convert this to real and complex methods

$f'(z)=\lim_{a+bi\to0}\frac{f(z+a+bi)-f(z)}{a+bi}$

[vivid gulch]

yoavmal

[open whale]

$f'(z)=\lim_{a+bi\to0}\frac{(f(z+a+bi)-f(z))(a-bi)}{a^2+b^2}$

[vivid gulch]

yoavmal

[open whale]

$f'(z)=\lim_{a+bi\to0}\frac{a(f(z+a+bi)-f(z))-b(f(z+a+bi)-f(z))i}{a^2+b^2}$

[vivid gulch]

yoavmal

[open whale]

Now I suppose we can decompose f to g+hi

Hmm, what if I do it earlier

$f'(z)=g'(z)+h'(z)i$

[vivid gulch]

yoavmal

[open whale]

$f'(z)=\lim_{d\to0}\frac{g(z+d)-g(z)}{d}+\lim_{d\to0}\frac{h(z+d)i-h(z)i}{d}$

[vivid gulch]

yoavmal

[open whale]

$f'(z)=\lim_{d\to0}\frac{g(z+d)-g(z)}{d}+i\lim_{d\to0}\frac{h(z+d)-h(z)}{d}$

[vivid gulch]

yoavmal

[open whale]

So g,h must be differentiable

But, that isn't enough, no?

There has to be some connection between them

$f'(z)=\lim_{d\to0}\qty(\frac{g(z+d)-g(z)}{d}+\frac{h(z+d)-h(z)}{d}i)$

[vivid gulch]

yoavmal

[open whale]

Well surely it's equivalent

[open whale]

Then it must be that if f is not differentiable, at least one of those isn't

Right, well, what if I express each as a function on two variables?

$f(a+bi)=g(a,b)+h(a,b)i$

[vivid gulch]

yoavmal

[open whale]

Now I can take partial derivatives

So $\pdv{a}f(a+bi)=\pdv{a}g(a+bi)+i\pdv{a}h(a+bi)$

[vivid gulch]

yoavmal

[open whale]

No, not very helpful

What about sequences?

$z_n\neq z, \lim_{n\to\infty}z_n=z$

[vivid gulch]

yoavmal

[open whale]

$\lim_{n\to\infty}\abs{f'(z)-f(z_n)}=0$

[vivid gulch]

yoavmal

[open whale]

Oh, this can work

No, oops

$\lim_{n\to\infty}\abs{L-\frac{f(z_n)-f(z)}{z_n-z}}=0$

[vivid gulch]

yoavmal

[open whale]

No, well,

I'll work with g only for now

$\lim_{n\to\infty}\frac{g(z+d_n)-g(z)}{d_n}$

[vivid gulch]

yoavmal

[open whale]

Hmm

Lets say at 0

$\lim_{n\to\infty}\frac{g(a_n+b_ni)-g(0)}{a_n+b_ni}$

[vivid gulch]

yoavmal

[open whale]

$\lim_{n\to\infty}\frac{(g(a_n+b_ni)-g(0))(a_n-b_ni)}{a_n^2+b_n^2}$

[vivid gulch]

yoavmal

[open whale]

So this goes to infinity unless the left side converges

To 0

$\lim_{n\to\infty}\frac{g(a_n+b_ni)-g(0)}{\sqrt{a_n^2+b_n^2}}$

[vivid gulch]

yoavmal

[open whale]

Nah, we lost information

Ah, but this it all real, so it's fine

[open whale]

Hmm

$f'(z)=\lim_{a+bi\to0}\frac{a(f(z+a+bi)-f(z))-b(f(z+a+bi)-f(z))i}{a^2+b^2}$

[vivid gulch]

yoavmal

[open whale]

$f'(z)=\lim_{a+bi\to0}\frac{a(g(z+a+bi)-g(z)+h(z+a+bi)i-h(z)i)-b(g(z+a+bi)-g(z)+h(z+a+bi)i-h(z)i)i}{a^2+b^2}$

[vivid gulch]

yoavmal

[open whale]

$f'(z)=\lim_{a+bi\to0}\frac{ag(z+a+bi)-ag(z)+ah(z+a+bi)i-ah(z)i-bg(z+a+bi)i+bg(z)i-bh(z+a+bi)i+bh(z)ii}{a^2+b^2}$

[vivid gulch]

yoavmal

[open whale]

$f'(z)=\lim_{a+bi\to0}\frac{ag(z+a+bi)-ag(z)+ah(z+a+bi)i-ah(z)i-bg(z+a+bi)i+bg(z)i-bh(z+a+bi)ii+bh(z)ii}{a^2+b^2}$

[vivid gulch]

yoavmal

[open whale]

$f'(z)=\lim_{a+bi\to0}\frac{ag(z+a+bi)-ag(z)+ah(z+a+bi)i-ah(z)i-bg(z+a+bi)i+bg(z)i+bh(z+a+bi)-bh(z)}{a^2+b^2}$

[vivid gulch]

yoavmal

[open whale]

$f'(z)=\lim_{a+bi\to0}\frac{(ag(z+a+bi)-ag(z)+bh(z+a+bi)-bh(z))+(ah(z+a+bi)-ah(z)-bg(z+a+bi)+bg(z))i}{a^2+b^2}$

[vivid gulch]

yoavmal

[open whale]

$f'(z)=\lim_{a+bi\to0}\frac{(ag(z+a+bi)-ag(z)+bh(z+a+bi)-bh(z))}{a^2+b^2}+i\lim_{a+bi\to0}\frac{(ah(z+a+bi)-ah(z)-bg(z+a+bi)+bg(z))}{a^2+b^2}$

[vivid gulch]

yoavmal

[open whale]

ok i'm on computer

i may have done wrong calculations

$\lim_{n\to\infty}\frac{f(z+a_n+b_ni)-f(z)}{a_n+b_ni}$

[vivid gulch]

yoavmal

[open whale]

$\lim_{n\to\infty}\frac{g(z+a_n+b_ni)-g(z)+h(z+a_n+b_ni)i-h(z)i}{a_n+b_ni}$

[vivid gulch]

yoavmal

[open whale]

ok

i have a better idea

if $f$ is differentiable in some region $s=\qty{z\middle|\abs{z-z_0}\leq r}$ around $z_0$

[vivid gulch]

yoavmal

[open whale]

and also $f(z_0)=f(z)$ for some $z\in s$

[vivid gulch]

yoavmal

[open whale]

then there is some $c\in s$ such that $f'(c)=0$

[vivid gulch]

yoavmal

[open whale]

is that correct?

[open whale]

that would require proof

how does the regular proof go? it uses weierstrass's extremum theorem

how do i convert that to complex numbers?

i need...

well, $\abs{f(z)}$ takes a minimum and maximum in the closed region

[vivid gulch]

yoavmal

[open whale]

can i say that if $|f(z)|$ is an extremum point and differentiable, then $f'(z)=0$?

[vivid gulch]

yoavmal

[open whale]

that requires fermat's theorem

how is fermat's theorem proven then?

,w fermat's theorem

[vivid gulch]

Results provided by WolframAlpha

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

Requested by yoavmal#5800

[open whale]

lmao

,w extremum theorem

[vivid gulch]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by yoavmal#5800

[open whale]

hmm

what if i try prove it myself

if it is an extremum point, lets assume without loss of generality that a maximum point

in the reals

$x_0$ is a maximum point if there is some environment $\abs{x-x_0}<\varepsilon$ such that $f(x)\leq f(x_0)$

[vivid gulch]

yoavmal

[open whale]

how would that work in the complex numbers then?

wwell, no, i want to prove it first off

the derivative

so if there is some $\varepsilon$ that satisfies this, then $\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}=0$

assuming the function is differentiable, of course

[vivid gulch]

yoavmal

[open whale]

well, by definition, $\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}=0$ means that for all $\varepsilon>0$ there is $\delta>0$ such that for all $\abs{x-x_0}<\delta$, $\abs{\frac{f(x)-f(x_0)}{x-x_0}}<\varepsilon$

so

[vivid gulch]

yoavmal

[open whale]

so we can get

$\frac{\abs{f(x)-f(x_0)}}{\abs{x-x_0}}<\varepsilon$

[vivid gulch]

yoavmal

[open whale]

now, lets say x>x0

then we get

well, first off

$\frac{f(x_0)-f(x)}{\abs{x-x_0}}<\varepsilon$

[vivid gulch]

yoavmal

[open whale]

this is still true

$\frac{f(x_0)-f(x)}{\abs{x_0-x}}<\varepsilon$

and for convenience

[vivid gulch]

yoavmal

[open whale]

how do i show that

uhhh

well, lets assume the derivative is not 0

then, without loss of generality, $\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}>0$

[vivid gulch]

yoavmal

[open whale]

so there is some environment such that

$x_0<x<x_0+\varepsilon$

[vivid gulch]

yoavmal

[open whale]

such that $f(x)>f(x_0)$

[vivid gulch]

yoavmal

[open whale]

and this is a contradiction

[open whale]

is that valid?

no, well, it loses all meaning since extremum points are uhh

nonexistent for most functions in the complex plane

no?

no, actually

hmm

maybe i can find a point that does it without the extremum theorem?

hmm, well, first the environment

it has to be between z0 and z

so

$\abs{c-\frac{z_0+z}{2}}<\abs{z-z_0}$

[vivid gulch]

yoavmal

[open whale]

$\abs{\frac{c-z_0+c-z}{2}}<\abs{z_0-z}$

[vivid gulch]

yoavmal

[open whale]

$\abs{\frac{c-z_0}{2}+\frac{c-z}{2}}<\abs{z_0-z}$

[vivid gulch]

yoavmal

[open whale]

alright, lets say i understand something here

lets take sine

cosine is better since it's at 0

$\cos'(0)=0$

[vivid gulch]

yoavmal

[open whale]

how do i show that it's true in the complex plane

well, the analytic continuation of cosine

$\cos(z)=\sum_{n=0}^\infty\frac{(-1)^nz^{2n}}{(2n)!}$

[vivid gulch]

yoavmal

[open whale]

so all i need to do is prove differentiation of polynomials

is valid there

but no, i want to do it without knowing the formula

lets say for a polynomial, how would i prove it

what if i simply state

there is some environment $\abs{z-z_0}<\varepsilon$

[vivid gulch]

yoavmal

[open whale]

so that for all z in this environment

$\operatorname{re}(f(z))<\operatorname{re}(f(z_0))$

[vivid gulch]

yoavmal

[open whale]

and also

$\operatorname{im}(f(z))<\operatorname{im}(f(z_0))$

[vivid gulch]

yoavmal

[open whale]

right, i can work with this

is this correct though?

lets take it with cosine

,calc cos(0)

[vivid gulch]

Result:

1

[open whale]

,calc cos(0.01i)

[vivid gulch]

Result:

1.0000500004167

[open whale]

aha, it's not

right, because cosh is not

it's a minimum point

huh

maybe it can be a minimum point on one and a maximum point on the other?

does that count possibly?

so then i can say

$\abs{f(z_0)}>\abs{f(z_0+\varepsilon)}$ and also $\abs{f(z_0)}<\abs{f(z_0+\varepsilon i)}$

[vivid gulch]

yoavmal

[open whale]

can i say that if it satisfies that

then $f'(z_0)=0$?

[vivid gulch]

yoavmal

[open whale]

obviously no

since this is only two axis, it could go wacky all over the rest

however, it must be differentiable

so maybe not too wacky

what if i try to apply polynomial stuff here again

like, i know 0 is a minimum point of x^2

so what properties does z^2 have at 0

$\dv{z^2}{z}=2z$

[vivid gulch]

yoavmal

[open whale]

so the derivative in this case is a scalar of the actual point

it is always pointing away from the point itself

what if i take a more complex function

where this sin't true

such as

$z^2+z^4$?

[vivid gulch]

yoavmal

[open whale]

its derivative is...

$2z+4z^3$

[vivid gulch]

yoavmal

[open whale]

can factor out the z

$2z(1+2z)$

[vivid gulch]

yoavmal

[open whale]

lets say i plug in 0.1i

i get

$0.2i(1+0.2i)$

[vivid gulch]

yoavmal

[open whale]

$=0.2i-0.04$

[vivid gulch]

yoavmal

[open whale]

this is clearly not pointing in the direction itself

but it is very close to doing so, actually

so i'd imagine the limit does that

yep

[open whale]

it's insignificant here, the main thing is the 2z still

and if i do -z, it pulls towards the point

and it'd be a maximum point itsead

ok, i can work with this

so

i want to find a point who's derivative is always pointing away in all directions

how do i say that without differentiating proper

$f(x)\cdot x<f(x_0)\cdot x_0$ i suppose?

[vivid gulch]

yoavmal

[open whale]

there is some environment that satisfies this

no, that isn't right

it should be

$f(x)(x-x_0)<f(x_0)(x-x_0)$

[vivid gulch]

yoavmal

[open whale]

this creates a difference between left and right

this isn't right, but it's closer

and it would create the appropriate angle in complex numbers

so now the absolute value matters

i want |f(x)|<|f(x0)|

but only in the reals

uhhh

hmm, derivative is

well

$\dv{f(z)}{z}-kz\to0$

[vivid gulch]

yoavmal

[open whale]

if this goes to 0 we're good

for some k