geometry-and-trigonometry

[upper karma]

and wondering how

nothings can make somethings

and then turn into nothings again

hence the zeno dialogue

https://billypilgrimage.tumblr.com/tagged/zeno

[quasi lion]

According to wikipedia, a space is a set with a structure.

[upper karma]

"structure"

so like rules?

[quasi lion]

yes

[upper karma]

"taxonmy of spaces" sounds like a crazy progmetal band

sorry i was away

but a projection

is something like a shadow

or even simpler

imagine if we smash

something 2d

onto 3d

sorry 3d onto 2d

that can be applied

from n dimensions to n

n dimensions to m*

wow can't type today

would every point in 3d be mapped to exactly one 2d?

or could*

or do you always have ones that can't be reached

every 2d point

would be reached

but some 3d points

will go to the same 2d point

like if you're 2 feet above a point on earth

or 3 feet above a point on earth

you'd still go to the same point on earth

(okay the earth is spherical, but imagine it's flat for this discussion)

i get what you mean

the projection doesn't have to work that way

but that's quite a "natural" one

hmm

so some information

about the original point is lost

now there are ways to do it without lost information

but it will no longer be a projection

and it won't have a lot of nice properties

like we could map say

(2.3462, 9.8813)

to

[quasi lion]

So what's the question?

[upper karma]

i'm intersted in pure math for sure, but MORE interested in the application of math of physics

to*

2.938486123

(weaving digits)

but that's often not useful

the "physical" problem

is literally

[quasi lion]

"i'm intersted in pure math for sure, but MORE interested in the application of math of physics" lol, pure maths and applied maths are very different

[upper karma]

i'm aware

ok ok let me start from the top

@quasi lion why do you say pure and applied math are very different?

hello club, my name is omar lol

[quasi lion]

They are

[upper karma]

i'm interested in all kinds of higherdimensional things

they inform one another quite a bit

and forms of transport

and bending spacetime etc

@upper karma agreed

[quasi lion]

If you say so

[upper karma]

what are some things you'd consider squarely in the domain of pure math and not applied math?

[quasi lion]

Topology is pure math

[upper karma]

yet it has all sorts of applications

[quasi lion]

Really?

[upper karma]

bruh i'm pretty sure when reimann was thinking about infnite sums

[quasi lion]

Like what?

[upper karma]

general equilibrium theory

in economics

he wasn't thinking about work and stuff

or even

[quasi lion]

Do you have a link to that?

[upper karma]

minkowski spacetime?

anyway the physical challange is

how to engineer a device that can rapidly take you from one point in space to another

so how to make a wormhole

there's a lot involved

here's a very simple example

https://en.wikipedia.org/wiki/Local_nonsatiation

but the one aspect i'm really interested in is the coordinate system

so suppose you ALREADY had the machinery needed to bend space and do all that

[quasi lion]

-> Physics discord https://discord.gg/qyZE2E

[upper karma]

you'd have to specify a location

i'm already there

but this is math based question

[quasi lion]

Really?

[upper karma]

yea!

just hear me out

the "whole" problem of making the machine and all that is physics and math and engineering and all

but this one ASPECT of the problem

the coordinate thing

is a math problem

because we're dealing with 3 dimensions and 4 dimensions and who knows how many more dimensiions

some say 11

some say 21

some say 6

either way

as far as i know, i live in 3

3 are open to my perception

[quasi lion]

I don't think so to be honest and I am in engineering in specialization in energy.

[upper karma]

https://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem

this was used in Nash's work on game theory

and I would say the result is topological

[quasi lion]

your links are fairly cool, thanks @upper karma ❤

[upper karma]

@quasi lion what makes you say that?

[quasi lion]

It's more of a physical challenge than math

[upper karma]

i mean it certainly is

i agree entirely

[quasi lion]

Now I don't know how to define pure maths anymore

[upper karma]

hahaha

I'm not @loud viper haha

[quasi lion]

@upper karma You were right

[upper karma]

I mean the distinction is quite vague

[quasi lion]

Shit, I wrongly alt-tab

[upper karma]

haha no worries

have you ever read any mathematical fiction?

like once you get into the surreal world of mathematical fiction

you know

you've entered a new realm of human experience

where the distincition between "math" and "physical" and "real" just kinda.... melts

tatke lewic carroll for example

but that's kinda another discussion

you certainly have to figure out how to engineer the device

Borges

to manipulate the energy

is pretty good for that kind of thing

BORGES!!!! my hero

The Library of Babel

is worth a read

[quasi lion]

I don't think it's that easy

[upper karma]

what is (or is not) that easy?

https://www.youtube.com/watch?v=H82BU2RMFAE

an homage i made

to borges' library of babel

[quasi lion]

"you certainly have to figure out how to engineer the device to manipulate the energy"

lol no

[upper karma]

ok ok

like i said

yes

someone somewhere at some point

cool!

needs to engineer the machine

to do the thing

to achieve the stuf

BUT

once that's done

however it looks

or whatever that device iis

a machine to do what?

to hop spacetime

teleport

[quasi lion]

Ye, this is pretty unlikely

[upper karma]

dude

I don't know anything about physics really

[quasi lion]

I think that maths and physics aren't that exciting

[upper karma]

so I can't speculate

i appreciate your rational concerns

actually

let me share with you htis

math is quite exciting!

https://arxiv.org/abs/gr-qc/0511086

if you can make it through that paper

then you'll know

i tried, but am not quite there yet

but i'm interested in this one aspect of the whole problem

of specifying the coordinates

because any kind of mapping or motion needs some coordinate system

latitude longitude

north due north grid something

so lets say that i'm at x=2.32352345 y=23.23423425 z=2354234234

and i want to get where you are

at x=something else y=something else z=something else

i could calculate the displacement/distance and use my machine (car,plane,boat,jet,whatever)

[quasi lion]

Finally a math question 😄

[upper karma]

to travel the space

hahahaha

[quasi lion]

Yes, but that's a physical problem

[upper karma]

lets say i want to take a shorter faster path

in physics don't we treat "space" in a coordinate system?

so it is "math" in a way

[quasi lion]

Indeed

[upper karma]

in a very real way

the "matter" or "real world" is more complex than just that pure math coordinate system

[quasi lion]

Yes

[upper karma]

but that level of simplicity is still useful in thinking and measuring and determining

so that's kinda where i am

[quasi lion]

Yes

[upper karma]

i want to get from

x1,y1,z1 to x2,y2,z2

but i want to do it in this shotest way possible

so lets say in normal 3d space i calculated the distance to be 40000000

in some higher dimensional space there might be a route

that's actually MUCH shorter and faster

[quasi lion]

Displacement is generated by the vector (x2-x1,y2-y1,z2-z1) in traditional physics

[upper karma]

mhm mhm

but if i'm using these higherdimensional highways

i need to know that when i come out

i'll actually be at

x2 y2 z2

so that's what i'm trying to understand

[quasi lion]

How can you be in higher dimensional if you used only 3 coordinates?

[upper karma]

well that's the thing

so we in our 3d perception are most familiar with 3coordinates

but in a universe with more than 3 dimensions

you would indeed use more coordinates

but we're only concerned with those 3d space

which just made me think though

a line is usually the shortest path

[quasi lion]

I don't understand how you can do higher dimension calculus with 3d space

[upper karma]

so i'd need to figure out how to make straight lines between points in higherspaces

no think of it like tthis

are you familiar with the story of flatland"?

[quasi lion]

A line isn't a problem in higher spaces

yes, I know flatland, even though I haven't read it

[upper karma]

ok

so for ease of thinking and conversation

lets pretend you and i are on the plane

but we're clever engineers/physicists/mathematicians

and we know there's the 3rd dimension around us

and we want to exploit it

so

in the 3d world

every 2d point is somewhere in it

[quasi lion]

No

A point isn't in a dimension

[upper karma]

no?

[quasi lion]

A point can be either set in a world with 2 dimension or 3 or 4 or whatever

But you can't say "in the 3d world, there are 2d points"

[upper karma]

think of

x y z

in x y z

you have the whole

x y plane

[quasi lion]

Yes, it's a plane

[upper karma]

ok sorry for misusing the words then

yea so the whole plane you and i are on

x y

is in the x y z

[quasi lion]

yes

[upper karma]

and we want to get from x1 y1 to x2 y2

this is kinda falling apart in my mind because the shortest path between two poiints in flat space is a line

so you really wouldnt have a shorter path that used the 3rd dimension to get to your desired point on the plane.... but anyway

for the sake of explanation

lets continue with the example

[quasi lion]

The shortest path with non-wormhole physics is a line.

And the shortest plane is in our x y plane

So in conclusion, if you want to go to x y with the shortest path, you must stay in the same plane (x,y), so without moving in the z axis.

[upper karma]

true

[quasi lion]

wasn't that your question?

[upper karma]

so i'm on the same page

when you say z plane

[quasi lion]

Actually I am wrong

I meant the same plane (x,y), so without moving in the z axis.

[upper karma]

yea!

but you do have a point in the larger question

[quasi lion]

I don't understand the question

[upper karma]

we'll save that

this scenario we're talking about right now in x y z

[quasi lion]

yes

[upper karma]

is a smaller example

of the real thing i'm interested in

but so we can be on the same page

and so i can clear my thinking

i'm starting smaller

[quasi lion]

just ask your question

[upper karma]

ok

i want to know how to specify coordinates in a higherdimensional space so that they will take me to the location i desire in a lower dimensional space within

as if i were taking a shortcut

by jumping up dimensions and then returningg back

i don't even know if that's the proper way to phrase the question

i'm jus trying

just*

[quasi lion]

You can't jump through n-dimension space to m-dimension space with a vector

[upper karma]

why?

so the idea is projections
which are idempotent maps
so f(x) = f(f(x))
where f maps from n dimensional space
to m dimensional space
where m < n
and it's a linear map
so f(x + y ) = f(x) + f(y)
and f(a*x) = a * f(x)
when a is a scalar
and x and y are n-dimensional vectors

quoting @upper karma

i don't fully understand that whole statementt so i'm trying to work through it

okay so

[quasi lion]

You are in the point A = (a,b,c) of space 3D. You want to go to point B = (x,y,z,t) of space 4D. There exists no vector u so that A + u = B

[upper karma]

i think I get what you're asking

if you consider two points on a projection

before the projection happens

[quasi lion]

You lose information

[upper karma]

those points are even further apart

[quasi lion]

And it's a mapping

[upper karma]

typically

[quasi lion]

It's not a vectorial addition

[upper karma]

but you're talking about embeddings

of 2d spaces

in 3d

where points could be closer

in 3d than 2d

(similar analogues exist for higher dimensions)

so if you have a subset of a space

S

a distance in S

may be further

by that i mean if you're restricted to paths in S

it can take longer

than if you can move freely about

the larger space

exactly

another idea that is related is the idea of "identification"

in topology

it's faster for me to drill through earth

than to fly around the surface

"identification"

basically lets you glue two points together

to make shortcuts

anyway I need to go sleep, but yes

these things are pretty well understood

in pure math I belive

believe

in differential geometry

[quasi lion]

I have no clue what's going on, you were talking about projection, then idk

[upper karma]

topology

and differential topology

dope

i'm talking about embedded spaces

yea i'm ALMOST done with the prereqs

you've given me much to feed on

thanks!

most of it is just fancy multivariable calculus

alright i'm heading to sleep

peace

good night

see you somewhere in the library

@quasi lion

we can keep going if you want

[quasi lion]

I think you didn't answered the question xaqxit

He asked you the displacements

And you gave a solution with embedding which is an embedding

Or maybe I didn't understand the question

[upper karma]

idk if it's he or she or what but they gave us a general direction

the embedding

is what we were talking about

how the x y plane is in x y z

[quasi lion]

I thought the question was : I am in point (a,b,c) and I want to go to (x,y,z,t) by the vectorial addition of a vector u? What's u? And then I said there exists no u so that it matches.

[upper karma]

no

i want to get

[quasi lion]

Ok then I misunderstood

[upper karma]

from a b c

to x y z

but i want to take a path

that uses

a forth or fifth or sixth component

[quasi lion]

Honestly I don't understand

Maybe I am not knowledgeable enough

[upper karma]

mmm it's ok

[quasi lion]

Oh wait

[upper karma]

i think you do understand

[quasi lion]

I have an idea

[upper karma]

just getting lost in the typing

and the words

lets use this image as reference

you see how you have that whole curved plane

and then you have the bridge between them?

so that outside plane is our x y plane

[quasi lion]

THIS WORKS ONLY IN MATHS : You have a space A in 3D with your points (x,y,z) and (a,b,c) and a space B in dimension n. So you need a mapping f that builds the extra dimensions around other vectors orthogonal of the space A. Then you need another mapping g that would be a projection of B in A.

So that g(f(a,b,c)) = (x,y,z)

[upper karma]

you use the word "maths"

where are you from?

[quasi lion]

France

[upper karma]

je suis omar

je no se pas french though

i'm american

[quasi lion]

No need to speak french

[upper karma]

just sayin hi

[dark sparrow]

je ne parle pas français

[upper karma]

american/egyption

egyptian*

[quasi lion]

I am lost in what you are trying to ask

[upper karma]

i can see that

it's alright

just look at the picture for a moment

your idea i think is really close

but dont assume you need to build something

or add something

so you know how the whole 2d plane

[scarlet quail]

je no se pas french though

[upper karma]

is IN the 3d space yea?

[quasi lion]

Yes

[upper karma]

so in a 4d space

[quasi lion]

A plane is in a space

[upper karma]

the whole 3d space is IN it

and likewise

up and up

[quasi lion]

No, a 3D volume is in the 4D space

[upper karma]

yea

here we use space openly

[quasi lion]

not a 3d space

[upper karma]

so you could say 2d space

or 3d space

or 5d "space"

just a general word

meaning the whole thing

but you're right

2d is tthe plane

3d is the volume

4d is i don't know

7d is i don't know even more

so we just say

"space"

and that wiki gave a technical definition

something with structure

[quasi lion]

A set with a structure, yes

[dark sparrow]

let me guess

[upper karma]

guess

[dark sparrow]

the technical definition was for a vector space

[upper karma]

nah

not even

i mean sure

but no hahaha

[quasi lion]

@dark sparrow Please help me, I am stuck here, he wants me to build a space travel machine and I need to take a shower 😢

[upper karma]

"but you're talking about embeddings
of 2d spaces
in 3d
where points could be closer
in 3d than 2d
(similar analogues exist for higher dimensions)
so if you have a subset of a space
S
a distance in S
may be further
by that i mean if you're restricted to paths in S
it can take longer
than if you can move freely about
the larger space"

HAHAHA

@quasi lion too funny

what time is it there?

[quasi lion]

7h59 but I haven't slept yet because my sleep schedule is off

[upper karma]

ooo

2am here

[quasi lion]

Just ask your question to @dark sparrow

I don't think I understand your question though

[upper karma]

i can pose it another way

lets say

that you and i are now friends

i want to come visit you

what's the fastest way i can get from here in america

to you in france?

thatt you know of

supersonic jet?

[rustic spire]

Let R be the region consisting of the points (x,y) of the Cartesian plane satisfying both |x|-|y| <=1 amd |y| <= 1. Sketch the region R and find its area.

[hard atlas]

I'm imagening a circle of radius 2 around the center as the shape, but I'm not sure 😛

[dark sparrow]

no way

[upper karma]

err nope

yeah as anto said

[dark sparrow]

@rustic spire can you sketch the region x - y <= 1 in the first quadrant?

[hard atlas]

Ahh ok ^^

Oh wait I just had a brainfart

I thought of both x and y as points themselves xd

[dark sparrow]

waiting for Da3m0nX

[hard atlas]

My brain is totally messed up today 😛

[dark sparrow]

(we aren't just going to do it for you, btw @rustic spire)

[foggy oxide]

the figure its two trapezoids right ?

[vale edge]

Hi. I was wondering if someone could help me clear a doubt. Say I have a vector that can be represented as a linear combination of three (or more) vectors. None of the parent vectors are parallel or anti parallel. Is that linear combination unique for that particular derived vector, or can the resultant vector be represented in form of multiple linear combinations using the same parent vectors?

All vectors are coplanar.

Thanks 😃

[quasi lion]

If the parent vectors aren't orthogonal, the combination is not unique.

[vale edge]

Got it. I suppose this goes without saying, but this is only applicable to three or more vectors?

[quasi lion]

If you can write one of the parent vector as a linear combination of the other parent vector, then the combination is not unique

[upper karma]

If the parent vectors are a basis its unique

What matters is having a basis

[quasi lion]

Yes, but not only that, @vale edge , it also generates the space

Oh you deleted the post 😦

[vale edge]

Yeah, it was obvious.

Sorry.

Thanks guys.

@quasi lion If we were to go proving that statement, how would we do it?

[quasi lion]

Prove what statement?

[vale edge]

That we'll need orthogonal vectors to have a unique linear combination for a certain resultant vector.

@quasi lion ^

[quasi lion]

That's just the definition of orthogonal

orthogonal = can't have a linear combination between the parent vectors

So it's trivial

[vapid kettle]

just dropping in now but that doesn't seem like the definition of orthogonal :p

[quasi lion]

All parent vectors are orthogonal = there is no parent vector with a linear combination of other parent vectors = unique combination = base

[vale edge]

So the parent vectors are orthogonal here.

Not the resultant

[quasi lion]

it is rio

[vapid kettle]

do you mean linear relation?

[quasi lion]

Because orthogonal means that the projection is zero

[vapid kettle]

cause it sounds like you're defining linearly independent

^

[vale edge]

orthogonal is (i think) just vectors that can't be expressed as linear combinations of each other.

[vapid kettle]

that's linearly independent

[quasi lion]

I don't understand then

[vale edge]

wikipedia says dot has to be zero.

[quasi lion]

if u is orthogonal to v, then the projection of u in v equals zero

[vapid kettle]

yeah one can think of orthogonal as perpendicular

[upper karma]

Orthogonal works for what you want but its too specific you can have linearly indendant without orthogonal

All basis have unicity of decomposition but not all basis are orthogonal ones

[quasi lion]

Hold on

Do you have an example @upper karma ?

Oh, yes I am stupid

Yes, ok

[vale edge]

Can someone tell me how would I prove that orthogonal vectors are required to have unique linear combinations for a particular vector?

[vapid kettle]

mm that isn't true

I believe you're looking for linearly independent vectors

[vale edge]

Oh yeah. Sorry. My bad. linearly independant vectors

But they are coplanar and not parallel.

So, are they still linearly independant?

I have a vector that can be represented as a linear combination of three (or more) vectors. None of the parent vectors are parallel or anti parallel. Is that linear combination unique for that particular derived vector, or can the resultant vector be represented in form of multiple linear combinations using the same parent vectors? All vectors are coplanar.

[upper karma]

If they are all coplanar and you have a vector that is combination of 3 its not unique

A base of a planis 2 vectors

If you have more than that there wont be unicity

[sacred vessel]

@upper karma help pls 😅

[upper karma]

What about ?

[sacred vessel]

x×4=3 5/6cm^2 i need x also this is taliking about a rectangle

I need to find height

[upper karma]

I have no idea what you are trying to say here

[sacred vessel]

X is height

And 4 is widht

[upper karma]

so ?

[sacred vessel]

I need to know height

[upper karma]

You have 4X=35/6 and need to find X ?

[sacred vessel]

No its something times 4 equals 3 and 5 6ths

Cm^2

And i need to find the something

[upper karma]

4X=3+5/6 ?

[sacred vessel]

No i aded the x theres no x in there in its own

I just need to find the cumber that will make the thing corect

Number*

[upper karma]

Well yeah that's why i wrote an equation

Try making an equation of your problem

If you dont see how make a drawing of the rectangle and the values you have

[sacred vessel]

K one sec

A=4

S=3 5/6

And i need h

[upper karma]

so what relation is there between those 3 numbers ?

[sacred vessel]

Thei are used to find the area of the rectangle

Because s is area

[upper karma]

can you write it down

[sacred vessel]

Write what down?

[upper karma]

The relation between S a and h

[sacred vessel]

S is area a is widht h is height

Thats theyre relation now i need h

Please

[upper karma]

S=.... ?

Can you write an equation

[sacred vessel]

Something*4=3 5/6

I need something

[upper karma]

Can you write this with S, a and h

[sacred vessel]

Ohh yeah sure

H*a=S

[upper karma]

so

H= ?

[sacred vessel]

I dunno thats what im asking

[upper karma]

You know division right ?

[sacred vessel]

Yeah but it comes out weird

[upper karma]

H*a=S gives you H=S/a

[sacred vessel]

1/6?

No 1 1/6

[upper karma]

no

how did you get that?

[sacred vessel]

Oh :p

Wait

I get it

[upper karma]

you want to do (3+5/6)/4

[sacred vessel]

Thank you

[upper karma]

best way is to write 3+5/6 as a fraction imo

[sacred vessel]

Kk will do thanks

[sacred vessel]

Need to solve the upper one for x

<@&286206848099549185>

[dark sparrow]

do you know the pythagorean theorem?

[sacred vessel]

Yeah

[dark sparrow]

12^2 + x^2 = 13^2

is it clear where that came from?

[sacred vessel]

Yes

[dark sparrow]

can you solve it for x?

[sacred vessel]

Yes i think so

[dark sparrow]

well, do that

[sacred vessel]

Kk

X=13^2-12^2=1^2 right?

[dark sparrow]

a^2 - b^2 is not (a-b)^2, and no

13^2 = ?
12^2 = ?
13^2 - 12^2 = ?

[sacred vessel]

Ohh

I think i get it

1 sec

25?

[dark sparrow]

13^2 - 12^2 = 25, yes

so now

if x^2 = 25

what is x?

[sacred vessel]

25÷2

[dark sparrow]

no

that would be the correct answer if the question was "If 2x = 25, what is x?"

[sacred vessel]

Oh right

[dark sparrow]

=tex x^2 = 25 \ x = \ ?

[sacred vessel]

X÷x?

[dark sparrow]

...

[sacred vessel]

Im really slow on math

Sorry

[dark sparrow]

what number gives you 25 when you multiply it by itself?

[sacred vessel]

12.5?

Oh bo

No

5

Lel

[dark sparrow]

finally

[sacred vessel]

Sorry

Im stupid

😀

[dark sparrow]

x = 5

that is your answer

[sacred vessel]

Thank you for putting up with me

😀 😀

[sacred vessel]

<@&286206848099549185> by using the Pythagorean Theorem i need to find the real lenghts of the straight sticks of this triangle

[brazen roost]

x = 2

[sacred vessel]

On both?

[brazen roost]

Ye

[sacred vessel]

How did you get it

?

[brazen roost]

Like you said

Pythagoras

[sacred vessel]

Yeah but i mean im studying so i need to know how you explain it

Using the theorem

[brazen roost]

=tex 9x^2 + 16x^2 = 10^2

wait no

[brazen roost]

Add the coefficients together

=tex 25x^2 = 10^2

[brazen roost]

Take the square root on both sides

=tex 5x=10

[brazen roost]

Divide both sides by 5

=tex x=2

[sacred vessel]

Thank you

[brazen roost]

QED

[sacred vessel]

Can we do another one @brazen roost

?

[brazen roost]

Sure

[sacred vessel]

Okay so this i know the paramiter but i dont know the lenght of the sides also i need to know after the fact if its standing up straight or whatever

Heres the thing

The midle one

The perimater is 12 cm

[brazen roost]

This one is rather easy

=tex 0.3x + 0.4x + 0.5x = 12

[sacred vessel]

And thats it?

[brazen roost]

=tex 0.12x = 12

[brazen roost]

=tex x=10

[sacred vessel]

Ohh

[sacred vessel]

0.12 *10

I get it

[brazen roost]

Wait

nvm

[sacred vessel]

Soo

0.12*10?

[brazen roost]

no

12/0.12

[sacred vessel]

What?

[brazen roost]

Wait

[sacred vessel]

Im sorry im slow with this part of science

[brazen roost]

What are you asking again?

It's 2am here

[sacred vessel]

12/0.12

I dont get that one

[brazen roost]

Multiply the numerator and the denominator by 100

So actually

[sacred vessel]

Ohh so (100*12)÷0.12

?

[brazen roost]

multiply 0.12 by 100 too

[sacred vessel]

Ohh

Okay

[brazen roost]

=tex x=100

[brazen roost]

my bad

OH WAIT

0.3 + 0.4 + 0.5 = 1.2

@sacred vessel

[sacred vessel]

😛

Kk

And after that everything the same

Multiply by 100

Divide by 1.2*100

?

[brazen roost]

sure

It's just to remove the decimals

[upper karma]

=tex \textup{Given } \bar{z}=\frac{1}{z} \Leftrightarrow \left | z \right | = 1 \
\textup{Prove that if } \left | z_1 \right |=1 \textup{ and } \left | z_2 \right | =1 \textup{ then } \frac{z_1 + z_2}{1+z_1z_2} \textup{ is a real number }

[upper karma]

<@&286206848099549185> pls

[final prairie]

Well if |z_1|=1 and |z_2|=1 then z_1 and z_2 are on a circle of radius one centred at the origin

[upper karma]

Yes

[hazy field]

let f(x, y) = (x+y)/(1+xy) for complex x, y
verify that f(1/x, 1/y) = f(x, y)
the solution follows from that the conjugate of f(x, y) is f(1/x, 1/y)

[final prairie]

Meaning they have form z_1=cos(x_1)+sin(y_1)i z_2=cos(x_2)+sin(y_2)i i think

[upper karma]

Oh that's a fancy method

Functions!

[final prairie]

Lol yeah

[upper karma]

heteroing you're a legend

[final prairie]

Anyways if you calculate f(z_1, z_2) in my method you (should) get a real number so thats one solution :p

[upper karma]

Could you please explain your method a little more clearly, @final prairie. I am facing difficulty in understanding it.

[final prairie]

Nevermind it’s inefficient and not nice

[upper karma]

So I'm trying to do number 18 I'm supposed to sketch what 18 says what would be an example or can someone draw me something that represents question number 18

And what does Picture Match that I drew mAtch number 18

[obsidian flicker]

Have you tried google?

[upper karma]

Thank you

I understand now

[umbral rivet]

:3

[crude kraken]

@umbral rivet Prove that for any n > 5, it is possible to divide a square into exactly n parts, each of which is also a square.

[umbral rivet]

Ooh

Ok

I think I member that

360/5 =

Uh

72 degrees

Wait

You said square

.-.

[crude kraken]

I did say square.

v Is a square

You should know how to identify a square if you wanna help people with geometry 😛

[umbral rivet]

x.x

So

You would just keep dividing it into pieces :3

[crude kraken]

yes

but observe initial question

Prove that for any n > 5, it is possible to divide a square into exactly n parts, each of which is also a square.

[umbral rivet]

🤔

n = number of squares?

[crude kraken]

...

[umbral rivet]

:c

[crude kraken]

I-

gets white board

n in math means number

[umbral rivet]

cries

[crude kraken]

"For any n > 5" means all numbers from 6 to infinity in the context

[umbral rivet]

I don't understand the question

:0

:c

I believe

This is the answer

[crude kraken]

:Thonk:

[umbral rivet]

Now please tell me why I'm wrong :3

[crude kraken]

okay so

[umbral rivet]

[crude kraken]

1. the squares in that square are not of equal size

2. that's not proof, that's just a case of 16

[umbral rivet]

:T

O-oh

[crude kraken]

congrats

[dark sparrow]

your method only works for perfect square numbers

[umbral rivet]

Numbers?

There are numbers in this question? 👀

[crude kraken]

you have proven that one n > 5 out of infinity can be divided into smaller squares

Prove that for any n > 5, it is possible to divide a square into exactly n parts, each of which is also a square.

Yes, there're numbers

5

[umbral rivet]

Oh

So

5 cases of squares being divided?

[dark sparrow]

no

[umbral rivet]

I don't understand the question x_x

[dark sparrow]

exalted wants you to prove that you can cut a square into any number of smaller squares, provided that number is at least 6

[umbral rivet]

Hm

[crude kraken]

not necessarily of the same size

Examples with 4, 6, 7, and 8 (And also this is the answer)

[umbral rivet]

Hm

The top left image

[oblique raft]

Wait, now I am confused. You just showed examples of you dividing the square up to n=8? How is that proving for numbers greater than 8

[umbral rivet]

Only has 4 squares

[crude kraken]

That's the question.

Turn that into a full answer.

[dark sparrow]

i have an idea, but i'm not gonna spoil it yet

[umbral rivet]

;_;

[crude kraken]

Is crying your final answer?

[dark sparrow]

...because i don't have a full solution yet anyway

[umbral rivet]

Oh okay

So

You want us to like

Make a formula right?

:3

[crude kraken]

No

[umbral rivet]

3:

[crude kraken]

"Prove that for any n > 5, it is possible to divide a square into exactly n parts, each of which is also a square. "

All you gotta do is prove it

[umbral rivet]

I mean

Just keep dividing the square :3

[crude kraken]

For instance, 17.

[umbral rivet]

? :3

You can keep dividing each of the squares into 4 smaller squares infinitely

And keep going

[dark sparrow]

ah

that'll give you divisions for all numbers of the form 3n+1

[final prairie]

induction

[dark sparrow]

but that doesn't cover numbers like 5 or 8

[umbral rivet]

Huh

Ummm

[dark sparrow]

you start with 1 big square, and at each step you're turning one square into four, thereby adding 3 squares to the count

going by your algorithm

[umbral rivet]

I think we need to figure out how the 2 pictures on the left were made

[crude kraken]

Yes, go on

[umbral rivet]

:0

[dark sparrow]

actually

you don't need to turn one square into four

you can also turn one square into nine

[umbral rivet]

👀

[crude kraken]

Angel you continue to taunt me with emojis

[dark sparrow]

which adds 8 to the count

[umbral rivet]

Hehe

[crude kraken]

I will taunt you with geometry

mhm

[umbral rivet]

don't bully me

[crude kraken]

But also, one square to 4 adds 3 to the count

and that's the important part

[dark sparrow]

yeah

[umbral rivet]

I'm lost :3

[crude kraken]

6, 7, and 8 are all consecutive numbers, of which there's three

[dark sparrow]

take one square

cut it into four pieces

[crude kraken]

so therefore, because you can always add 3 to the count

you can divide the square into four until you get close enough

[umbral rivet]

What

[crude kraken]

then just add 6, 7, or 8 to the count

[umbral rivet]

Which specific picture do you mean? x.x

There are three 5's present

[crude kraken]

So with my earlier example 17, you would divide a square into four, divide two of the corners into four, and divide one of the corners into the 7 square

[umbral rivet]

🤔

[crude kraken]

actually I'm gonna write a program to make these squares...

brb

[oblique raft]

I hate trig

[dark sparrow]

how come? D:

[umbral rivet]

:0

Is trig used for anything besides angles? :3

[dark sparrow]

definitely

[oblique raft]

Circles

[umbral rivet]

Circles? O.o

How would you use trig with circles? :3

[oblique raft]

Unit circle?

[dark sparrow]

that

circular motion, too

[umbral rivet]

What's a unit circle? :3

[oblique raft]

O

[umbral rivet]

Oh

🤔

[dark sparrow]

well

[oblique raft]

So, basically you put triangles inside circles and do math with it....

[dark sparrow]

a "unit object" usually refers to an object whose size, in a way that makes sense in context, is 1

[umbral rivet]

:o

[dark sparrow]

the unit circle is the circle with radius 1

[umbral rivet]

That makes the diameter 2 :3

[dark sparrow]

true

anyway

[umbral rivet]

:D

[crude kraken]

and the circumference pi

[dark sparrow]

and basically you measure angles using arcs on the circle

@crude kraken 2π

[oblique raft]

Yeah, and you stuff triangles inside of it. It also happens that the x coordinate on the circle corresponds to the cosine of the angle that takes you to that point, and y corresponds to sin of the angle

[dark sparrow]

i'd say more but i don't have access to any pics atm

[umbral rivet]

I thought circumference was 2pir

[oblique raft]

It is

[umbral rivet]

:3

Oh

2pi is the same thing

x.x

I mean

Wait

Im confusing myself now

[oblique raft]

I'm confused as to what you are getting at lol

[umbral rivet]

Never mind :P

I was trying to correct @dark sparrow 's message 👀

But I'm wrong :3

[oblique raft]

Ic ic

[umbral rivet]

👁 🌊

[dark sparrow]

sqrt(-1) times the speed of light

xP

[oblique raft]

lol I just got it

[dark sparrow]

anyway, the unit circle is a way of extending sin and cos to angles beyond the familiar range of 0 to 90 degrees

[umbral rivet]

:3

[dark sparrow]

and at this point, it starts making sense to think of angles as rotations

(also, radians become a sensible unit for measuring angles)

[oblique raft]

Huh, I always thought of it as rotations....

[dark sparrow]

yeah sure

angles are now signed though

ccw rotations are positive and cw rotations are negative

[oblique raft]

Yeah...

[dark sparrow]

the positive direction for rotation is the one that takes you from the x axis to the y axis, generally

anyway

yeah

also at this stage radians become a sensible unit for angle measure

[oblique raft]

I don't like how some sin/ cos values are negative

[dark sparrow]

that corresponds to negative coordinates on either axis

like, once you're past π/2 you're west of the y axis, ofc your x coordinate will be negative

[oblique raft]

Yeah, I still like to think of trig in terms of triangles, and having a negative trig value would imply that one of the lengths of a side is a negative

But I was told the negative indicated direction... so I accepted it

[vale raven]

I saw a conversation about dividing squares

[dark sparrow]

angles past pi/2 don't really name sense for triangles

[oblique raft]

We are on triangles now; the superior shape

[vale raven]

I have a gif about it

[oblique raft]

I just picture triangles placed backwards. I'll send a image to show what I mean

[vale raven]

The superior shape is obviously the circle

[umbral rivet]

That's a cool gif

I like triangles the most because I understand them the most 👀

They're just half of a square

[vale raven]

Rectangle~

[umbral rivet]

And squares are just... squares

Rectangles O.o

No one uses those

[oblique raft]

That is how I thought about it. Which is why I was confused on negative trig values @dark sparrow

[vale raven]

Triangles

half of a rectangle

[umbral rivet]

That is true

[dark sparrow]

@oblique raft yeah, i think of those sides as signed

[foggy oxide]

:D

[vestal pine]

hello!

sin^2(2x)

there is a trig identity for this apparently

i know there are for squares and for double angles, but this is a combo.

over the internet it seems people just know how to convert from this to 1-cos(4x)

how comes?

i mean the idea is to integrate it, so we're looking for something simple to integrate

[upper karma]

cos(4x)=cos^2(2x)-sin^2(2x)

[vestal pine]

is there a pattern or something i can use to remember those idendities?

it seems that i have to know what cos(4x) is

[vestal pine]

there are lots of combination...... i mean, how am i supposed to know all possible combinations?

ok

[vestal pine]

are you a teacher? do you use these things daily in your work?

[azure storm]

Pretty useful in physics

[vestal pine]

i mean i don't understand how can ppl remember these things so easily...

oh ok

[azure storm]

i only remember a few of them (mostly after i've used them), but most of the time i just recheck to be sure

[vestal pine]

ok

i'm just studying because i want to know math and do more complex stuff later

dont have any exams

so i guess it's ok not to know ALL this stuff by heart. this will comes with practice and time

[azure storm]

knowing how to find them is best imo

[vestal pine]

still i don't understand how you went from sin^2(2x) to those ident

yea i wanted to practice rebuilding the triangles thing...

but not today XD too tired

well wait.

those are 4

[azure storm]

I usually rush them with exponentials just to check signs etc.

[vestal pine]

there are all the others cos^2(x) cos(2x) cos(x/2).... etc

or you are saying starting from those identities i can derive the others?

well i guess yeah i can....

i still dont see the pattern you werer trying to explain me before sorry

[dark sparrow]

there's a way simpler explanation

cos(2t) = 1 - 2sin^2(t)

[vestal pine]

your icon is bad apple @upper karma

[final prairie]

! <- simpler exclamation

[dark sparrow]

2sin^2(t) = 1 - cos(2t)

sin^2(t) = (1 - cos(2t))/2

that's it

@vestal pine does this make sense?

nope

menoz's is a Julia set

[vestal pine]

@dark sparrow i'm readingeverything

[dark sparrow]

Mandelbrot set has a butt

[vestal pine]

julia yes. I didn't know it was julia set at the time, i was messing around with a code i've written trying to do the mandelbrot set

[dark sparrow]

okay so

does what i wrote make sense?

[vestal pine]

yes, i prefer anto's solution for its simplicity XD but i'll try to study yours. I mean i get it but i realise now I' very tired and i can't really absorb more ot this stuff,,,,,

wait a sec

anto

you in the end have sin^2(t)

but i have a double angle

sin^2(2t)

[dark sparrow]

yeah

[vestal pine]

it might be because i'm tired.. but, this means i can ignore the double angle? 🤔

like... is not a double angle then. is just an angle

her's ? who?

[dark sparrow]

mine

[vestal pine]

so i can treat the angle as a normal angle in this case

it's not that if there's a multiplier next to the angle i have to find a indetity for it

[dark sparrow]

yeah i mean, sin^2(7x) could be rewritten as (1 - cos(14x))/2 just fine lol

[vestal pine]

all this makes sense 😑

thanks!

regarding indentities i found this page, it seems that lists the more importants

http://www.math.com/tables/trig/identities.htm

i guess more used

[dark sparrow]

taking a look now

[vestal pine]

does that makes sense or feel there some big parts missing?

i like wiki page because i can spot all the ones i've studied and they are in a nice order.

[dark sparrow]

mm

[vestal pine]

just comparing stuff around to see the more used..

[dark sparrow]

the webpage you linked is pretty good in terms of content imo

[vestal pine]

cool

[dark sparrow]

there's this one cute little identity that has a nice geometric proof behind it

that i'm p fond of

tan(x/2) = sin(x)/(1 + cos(x))

i don't have access to any drawing medium right now

[vestal pine]

snipping tool on windows

[dark sparrow]

i'm on my phone

and i'm on vacation anyway so no pc

[vestal pine]

on the sand !

jk

[dark sparrow]

i can describe how to draw the pic tho