The h Bar

10:01 AM

@Mithrandir24601 I haven't even got my head straight with Quotient spaces entirely yet

Everytime I do something breaks in my head

and I end up being confused over something I took for granted

All I can really say to help is that when I say $\sqrt{-1}$, what I mean is the $\sqrt{-1}$ that gives you $i$ on the complex plane, so that a full 'rotation' gives $\sqrt{-1e^{2i\pi}} = -i$

i.e. it's a just a definition of $0$ phase, if you want to use the term 'phase' :P

Phase

huh

forgive my dumb but

Wouldn't a full rotation just be 1?

Secret

I cannot believe as a room owner in my own room, I cannot trash any messages in the transcript

Mithrandir24601

@Phase Depends what you mean by a full rotation ;)

Or rather, depends where and on what you apply the 'full rotation'

Phase

hang on really confused now

How did you get $-i$

in your message

$e^{2i\pi} = 1$

Right?

Mithrandir24601

10:09 AM

@Phase so what's $\sqrt{e^{2i\pi}}$?

Phase

I guess you could write 1 as $-i(i)$

Mithrandir24601

@Phase Oh no you don't :P

Phase

idk why though

But $e^{2i\pi} = 1$ doesn't it? :(

Mithrandir24601

@Phase OK, what's the square root of $x^y$?

Phase

yeah I get that but

Why can't you evaluate it before square rooting it?

Mithrandir24601

10:11 AM

nope, write it down

Phase

I get that if you keep it symbolically it'll end up being $e^{i\pi}$

Anonymous

@Phase Normally for $\sqrt{-1}$ only principal root is considered. (which you get from De Moivre's) That is equal to $i$

Mithrandir24601

@Phase OK, the actual answer is that they lie on different Riemann surfaces

Phase

@Blue that wasn't what I was referring to tho

@Mithrandir24601 wtf is that

Mithrandir24601

@Phase ... Do complex analysis to find out :P

Phase

10:13 AM

:(

@Blue my point was that you can have multiple complex numbers that aren't identical

but square to give -1

Mithrandir24601

Are you happy that there is an actual answer?

Phase

@Mithrandir24601 no

Because I don't understand it yet

Mithrandir24601

(in which, can we go back to my hacked together method)

@Phase So, write down the square root of $x^y$. Hence, write down the square root of $e^{2i\pi}$

Anonymous

If you square before rooting you lose out on some information. Like 1^2=1 and (-1)^2=1. You don't have a single path backwards

Phase

so if we don't choose to evaluate it before the sqrt, you just get $\sqrt{-1} \sqrt{e^{2i\pi}} = i e^{i\pi} = -i$

@Blue where does the squaring come from though?

I get that you lose information but saying that $\sqrt{-1} = i$ when you have a number system with a continued set of complex numbers doesn't make sense to me

Mithrandir24601

10:16 AM

the point is that $\sqrt{e^{2i\pi}} \neq \sqrt{e^{0}}$, even though $e^{2i\pi} = e^{0}$ as @Blue said

Anonymous

@Phase I normally avoid saying $\sqrt{-1}=i$.

Phase

ikr

Slereah

Do you mean $\sqrt{-1} = j$

Mithrandir24601

@Slereah ::flags:: :P

Anonymous

The $\sqrt{}$ has to be defined carefully before you can say that

Anonymous

10:19 AM

There can be multiple branches

Phase

@Slereah $\sqrt{-1} = iiiii$

clearly

Anonymous

Though the usual rule is to take the principal branch only

Phase

@Blue how do you define a principal branch?

Mithrandir24601

@Blue You're such a mathematician as well :P

Phase

@Blue the example I have in my mind the most solidly is quaternions

Slereah

10:21 AM

$i = dx$

Phase

There's no finite amount of solutions to $\sqrt{-1}$ iirc

Slereah

Obviously

Anonymous

@Phase Just Google for "principal n th root of a complex number". You'll get all the details

Slereah

with $dx$ the multiform of $\mathbb{R }$

Phase

$\int f(x) \sqrt{-1}$

Mithrandir24601

10:21 AM

@Slereah fine. $j=dy$

@Slereah bah!

Slereah

and then $i^2 = dx\cdot dx$

Mithrandir24601

(didn't see that)

Slereah

that is Clifford algebras :p

Phase

@Blue first result was a post on MathSE saying that there is no coherent definition for it : P

Slereah

nobody cares about them except for spinors

Anonymous

10:24 AM

@Phase Sure, but you can define it yourself before writing such starements. Many math books take the root corresponding to n=0 (After applying DE Moivre's) as principal root

Phase

@Blue my point was that in some cases like quaternions, it's not really a good idea to write anything that implicitly means to say that $\sqrt{-1} = i$ rather than any other possibly infinite amount of solutions

as far as I can see at least

Anonymous

@Mithrandir24601 Lol....just trying to avoid getting scolded by the math nazis ;)

CooperCape

Ello ello ello

Mithrandir24601

@CooperCape Mornin'

Anonymous

@Phase I agree. Got to be very careful with those definitions (complex numbers are a bit tricky for that reason- multiple branches and stuff )

CooperCape

10:26 AM

@Mithrandir24601 Wish it wasn't... ;p Was up too late recovering from being lazy all weekend

Anonymous

But dont worry. Most good books define their stuff well

Phase

@Mithrandir24601 what did $e$ and $ie$ realise they had in common when they both laughed at a shit meme

They were both norm(e)

Idk if "norm()" really works there but if I defined it properly idk if it'd still work as a joke. The struggles.

Anonymous

Duh. That's a normie joke

Anonymous

:P

CooperCape

That was... terrible? I'm not even sure where the joke is there.

Mithrandir24601

10:29 AM

@Phase One of them didn't really exist. The other was an imaginary construct used purely to make life easier for people :P

Phase

@Mithrandir24601 Both of them were irrationally angry

It's a shame that JR isn't here to groan at the jokes

Anonymous

@CooperCape Congrats. You aren't a normie :D

Mithrandir24601

@Phase He is. Just musn't be reading the chat :P

Phase

@CooperCape people who like stale memes are called Normies

and Norm(e) read aloud is.. well, Normie

The joke may be shit but it's there : P

CooperCape

Oh... right, that makes sense now? I guess

Phase

10:32 AM

Everyone's a critic smh

CooperCape

Before I thought you were just going "Yeah fair got a magnitude of like 2.7 innit"

Anonymous

JR might be unconscious after drinking buckets of coffee (after reading that paper yesterday). :P

Phase

I spotted the engineer in the room @CooperCape

CooperCape

Ewwww noooooo

Phase

$\pi = e = 3 = \sin(x)$

CooperCape

10:32 AM

Pretty sure I wrote that once here...

Phase

I one upped you

Paraxial approximation is devilspawn

CooperCape

u wot mate

Oh wow

sin(x) crikey

Phase

Croikey*

CooperCape

Last time I checked the range of sin(x) was -1 to 1 fella

Anonymous

@CooperCape Nah...try for complex x

CooperCape

10:35 AM

@Blue I quit.

Slereah

just use natural units

Anonymous

:P it should be a fun exercise

Slereah

$e = \pi = 1$

Phase

Nah mate sin(x) = x for all x, and sin x / n = six, so nsin x is equal to n * 6, and then n looks a bit like pi so you get $\pi * 6 = 3 * 6 = 18$, so now you'll notice that on large scales that's nearly 3. so $\pi = e = 3 = sin(x)$ QED

CooperCape

Omg I keep on getting pranked by UCAS... this is too much.

Phase

10:36 AM

I'll be submitting this to Vixra

Because ArXiv just isn't ready for this kind of research yet

CooperCape

Oh wow Vixra's like really high up there in the world of scientific papers I'm impressed af

Phase

@CooperCape could you tone down the jealousy a bit it's making me feel awkward

Just because you won't be getting a fields medal for proving that all functions = 3

Smh

CooperCape

@Phase Sorry sorry! Just got too excited omw...

Anonymous

Fields medal is too lowly a prize for a discovery of that magnitude @Phase

Anonymous

Aim higher

CooperCape

10:38 AM

What about a function $f(x)\neq 3$

Prove that one...

Nobel prize in economics maybe?

Phase

@Blue all functions = 1000000000

CooperCape

I mean if you can turn everyone's economies to 3...

Exchange rate is 1. Everything is worth 3.

Oh wow you've basically proved communism works.

Phase

Communism: We're all equal
Communism in real life: we're all equal but one guy controls how equal we are

CooperCape

Phase communism: 3

Secret

I think I might need Slereah's help to help digest that paper that mentioned we don't need dark matter nor dark energy

Anonymous

10:44 AM

@Phase So, which math topics are you learning currently ?

Phase

@Blue Right now? I haven't learnt anything rigorously really in the past half a week or so. Drive has been up and down a lot

s a d b o i s

Anonymous

I thought you were doing group theory :)

Anonymous

And analysis

Anonymous

I know nothing of group theory :/

Phase

I've never touched Group theory beyond trivial shit. I did real analysis briefly but lost motivation, and I've been lacklustrely throwing myself at Linear Algebra

Mithrandir24601

10:49 AM

@Phase Ahah! I am the Pope!

Phase

@Mithrandir24601 what is a pope, to a nonbeliever!

CooperCape

is that a kanye reference...

Phase

Mine?

It's a DBZA reference

CooperCape

Pretty sure there's a line in a Kanye song that's like "What's a god to a non-believer"

Phase

Probably, he seems that kinda guy

I always like Parodies with Kanye in because you know they're gonna represent him faithfully

Mithrandir24601

10:55 AM

@Phase Context: One of my CS lecturers had a superb proof that $1+1=3\implies $ I am the Pope

Phase

@Mithrandir24601 you studied CS?

Mithrandir24601

@Phase For a year, yep :)

Phase

Spooky

Where does that fit in the official timeline$^{tm}$

Mithrandir24601

@Phase The CS? It was first year of my undergrad

Phase

Did you go from that to 1st year physics?

Mithrandir24601

11:07 AM

@Phase Nah, I went from that to 2nd year physics :) 1st year CS and 1st year physics were the same thing

Phase

Contrived systems of lies to conceal the flat earth you mean?

Mithrandir24601

CS had the choice of 4 modules - 1 maths, 2 CS and 1 science (I did physics); physics (i.e. NatSci) had the choice of 4 modules - 1 maths, 3 science. the CS physics module was the same as the NatSci Physics module

Phase

I have a question that's basically "explain this irl thing if you can" but idk where to ask it

When my phone is charging, when I bring my finger along one of the edges on the side it feels like it's dragging, but when it's not charging it doesn't

And I'm genuinely wondering why

Anonymous

@Phase What do you mean by dragging ?

Phase

You know when you drag your finger against something and it'll periodically go from static to dragging

But it's not exactly like that, there's just a vibration that feels like it I think

but only when I move my finger along the edge

Anonymous

11:21 AM

If that effect increases near the edges....it's probably some excess charge discharge issue...

Anonymous

If you put your tongue there, you should feel it more I guess because the tongue is more sensitive

Anonymous

Also you know $\sigma \propto 1/r$

Anonymous

$\sigma$ being charge density

Anonymous

Near the edges the r decreases so the $\sigma$ increases greatly

Anonymous

For larger appliances that a why they provide a ground pin in the plugs to prevent such charge discharge through your body (The teeny shocks/vibrations) you feel

Anonymous

11:27 AM

The right word for that sensation would be "tingling" I guess

Phase

but

It only happens when I move my finger

If I keep it stationary I don't feel anything

And it feels virtually equal all the way up the left side

Its only on the left side btw

CooperCape

12:29 PM

Can I use generalized eigenvectors to diagonalise a matrix? I don't have a clue what they are wolfram just output them...

Captain Bohemian

12:43 PM

encyclopedia.thefreedictionary.com/homogeneous+space last time Ben said there are diffeomorphism group and homeomorphism group in addition to isometry group as far as symetry groups are concerned. But I don't exact understand what diffeomorphism group and homeomorphism group mean.

Slereah

diffeomorphisms form a group

The composition of two diffeomorphisms are a diffeomorphism

And there's an inverse diffeomorphism

and an identity diffeomorphism

Captain Bohemian

is isometry derived from isomorphism? and diffeomorphism and homeomorphism just require more than isomorphism?

Slereah

an isometry is a specific kind of diffeomorphism

where the diffeomorphism preserves lengths