Mathematics - 2017-04-06 (page 4 of 6)

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Akiva Weinberger

7:01 PM

(Three Letter Acronym)

You can't comb a hairy ball flat.

what's your point "lad"

click the link from @MikeMiller

And sorry, lads and lasses.

You're an Oakland Raiders fan? That's my dad's team!

I already like you @skillpatrol

Las Vegas Raiders in two years pal ;-)

This I did not know!

7:07 PM

They gotta build us a stadium first...

...who knows maybe we'll win a Super Bowl for Oakland as a going away gift!

That'd be awesome! I'm sick of the patriots.

2

Hello. I am reading this: http://math.stackexchange.com/questions/1266906/example-2-sec-24-in-munkres-topology-2nd-ed-how-to-show-this-set-to-be-a-li?rq=1
I am wondering, aren't the two cases in the first proof unnecessary?
The cases I am referring to are x_1 = x_2 and x_1 < x_2. In either case, couldn't we choose the second coordinate to be the midpoint of r_1 and r_2 and let the first coordinate be x_1?

Shut up @skillpatrol your team is a loser.

@skillpatrol That hairy ball theorem is pretty intuitive to think about, but obviously difficult to prove. I first heard about it in Richard Feynman's first book. I thought it was a joke at first, until right now. Then I read the wiki on it and it said "You can't comb a hairy ball flat..." Almost fell out of my chair.

@skullpetrol excuse me?

was I talking to you?

7:16 PM

Your team is a loser and they are traders of the Oakland faithful.

We cheered for the raiders through 13 losing seasons.

@MikeM: I saw that earlier.

Oh, skull is schizophrenic today.

@Dodsy how can you like a trader?

I don't know what to say.

I don't know what's happening.

Welcome to the chatroom :-)

Ted I didn't know you wrote books?

7:23 PM

@Ted What's a good heuristic reason that a directional derivative operator should be $C^\infty$-linear in the direction vector field? Just observe for $\Bbb R^N$?

@TedShifrin i need some help understanding $X=\Bbb R /[0,1]$ from yesterday. we said that it is the space $\{A , \{x : x\notin A\} \}$ , i want to understand how open sets here looks like. i know that $U\subset X$ is open iff $p \ ^{-1}(U)$ is open in $\Bbb R$ where $p(x) = [x]$.

@Ted You totally do look like Richard Dreyfus.

I just watched Shot in the Dark again last night, so when I think of Dreyfus I think of the chief inspector.

Can somebody just do my chemistry homework for me? Because I don't get it, and I don't like it.

@AkivaWeinberger just saw your msg, what's TLA ?

The Periodic Table

Haikus are awesome / Chemistry's even better / So pull up a chair

7:32 PM

Huh. it was a response to "every combination of 3 letter" ? @AkivaWeinberger :P

Akiva Weinberger

@Liad Three-letter acronym

Yup

It describes itself

it is amazing how they like 3 letters in compu.

@MikeMiller Maybe think in terms of a particle flowing in a river? The speed of the particle should vary with the water flow

they ruined it with GNFA

@Danu That requires that fluid mechanics be intuitive :P

7:37 PM

Not really; it does require you to establish that directional derivatives should apply to a situation like that.

How are you nowadays Semi?

eh.

Ever figured out that Riemann surface you were talking about months ago?

Not really. Moved on to something else.

Hmm :(

(or maybe you're happy to :D)

Though mostly I've been in the loop of: "What should I do first, research or grading? oh, I can't decide" and then procrastination

which...yuck.

7:39 PM

Mhm...

I'm grading for 2 courses next semester

and teaching a bit, for Riemann surfaces

The problem I'm doing in research right now may wrap back around to Riemann surface stuff again, hopefully.

oof

(hope I won't be horribly underskilled)

What kind of grading?

problem sets

(The bane of my existence is lab report grading.)

ah. that's not too bad.

7:40 PM

1 undergrad topology of surfaces course

and the Riemann surfaces

That could take some time to grade.

well, I know exactly how many hours per week I'm being paid for

I've decided I'm not going over that, at least not for the grading.

True.

For the teaching component of Riemann surfaces I might.

That's a good mindset.

7:41 PM

They're just apying me 6 hrs/week for the undergrad course

but probably 20 to 40 sheets per week

That'll be fun

Ah. Our setup is typically 20 hours per week.

Ah

But that's averaged out through the entire semester, so the peaks tend to be sharper than that.

It's just a side job here

like 330 euros per month

Anyways. I just finished handing stuff back to one of my sections, so I don't really want to think about grading right now.

7:43 PM

@MikeM: I love Shot in the Dark. Yeah, I would say it's obvious from $\Bbb R^n$.

Oh hey, the Pink Panther. Haven't watched one of those in years.

I do remember the ending of Shot in the Dark, though.

The ending is great.

@Liad: Imagine it this way. Take the real line and crush the interval $[0,1]$ to a single point (think of sliding from the right, moving $1$ to $0$, $2$ to $1$, $4$ to $3$, etc.) What is the resulting space?

I managed to get started writing my thesis today

I'm pretty good at doing the Peter Sellers pseudo-French accent, too, @MikeM.

hi @Danu

7:44 PM

Hi Ted

Hi chat

Hi Astyx

Hey Ted.

Appropos of not much, I should watch Airplane again.

Don't call me Shirley. .... Yes, I talk jive!

7:47 PM

Has anyone seen the movie Proof? What do you think they were trying to say she proved? The Riemann Hypothesis or something? She does talk about the "L" function I think at one point.

I understood why the correct notion of a generalized flag manifold is $G/C(T)$ where $C(T)$ is the centralizer of a torus $T$.

Chump don't want no help, chump don't get no help.

Yeah, @Dodsy, I saw it years ago, but I've forgotten.

@TedShifrin when you say it that way it looks like $\Bbb R$ stays the same

I remember this really dumb movie called the Oxford Murders, I think, which features Fermat's last theorem

7:48 PM

@danu One problem I'd been dealing with lately is pretty much purely mathematical.

@TedShifrin I can never find any good math movies to watch, except pretty lame lectures on Kanopy. So now I have to rewatch movies like proof or the man who knew infinity.

@Ted I showed a friend of mine. I think I'll show him the sequel that also includes Dreyfus.

@Danu Oh yeah!!!!! That movie was so terrible!

Baker-Campbell-Hausdorf tells you about $\log(e^{tA}e^{tB})$ for operators $A,B$.

Don't forget Jill Clayburgh in It's My Turn, too, with its wonderful commutative diagram chase.

Right, @Liad. You need to see intuition for this stuff. Now can you prove that it's actually homeomorphic to $\Bbb R$?

7:49 PM

However, the Dynkin series usually only makes sense in some radius of convergence about $t=0$.

@TedShifrin I've seen the intro, something to do with the snake theorem or something?

right, @Dodsy :)

What about the asymptotics at $t=\infty$?

I don't feel a need to see my field in film unless there's actually a story to tell.

Snake lemma*

7:50 PM

doesn't the qoutient map will be homeo' ? @TedShifrin

Well, in A Beautiful Mind they covered up a lot of the true story.

I dunno any of the films mentioned

A beautiful mind is a great movie, but after a couple viewings becomes taxing and hard to watch.

Ugh, it's sending messages and not telling me.

No, the quotient map is not a homeomorphism. It collapses a whole bunch of points to one point, @Liad.

7:51 PM

Right.

@Dodsy: CHEMISTRY! What are you working on now?

(more precisely, I want asymptotics of $\log(e^Ae^{Bt})$. I know that the spectrum of $B$ has strictly negative real part, so I think the limit should be 0.)

Ah yes, you're right I should get to work. I am trying to learn and understand hybrid orbitals and electron promotion....

How do you see that this is the space that we get? @TedShifrin , im trying to get this.

(but how to say more than that...)

7:53 PM

@TedShifrin Absolutely horrible movie IMO

@Danu What is your favourite math movie?

@Liad: Here's a big hint. Think of the function $f\colon\Bbb R\to\Bbb R$ defined by $$f(x)=\begin{cases} x, & x< 0 \\ 0, & x\in [0,1] \\ x-1, & x>1\end{cases}.$$

Draw the graph and think about it.

The 'math movie' I refuse to watch is The Imitation Game.

Because it apparently mangles the history quite badly.

Very little math in that movie.

Yeah, it's not quite a 'math movie.' But it does have that same tack of 'troubled/misunderstood genius'.

7:54 PM

@Danu: To my mind, they hid one of the most important ingredients of Nash's mental illness — his gay/bi-ness.

@Dodsy I have none. Pi is kind of doable, but if it's not about math. Just features some numerology.

Stand and Deliver, maybe?

@TedShifrin That sounds as if you're saying that him being gay/bisexual made him mentally ill :P

Is that what you're saying?

@Danu I've had to start reading math books to satisfy my cravings. Currently reading E.T Bell's book.

That was an inspirational movie, @Semiclassic, although I hate the fact that he made everything math completely mechanical.

7:55 PM

Fair.

@Dodsy Be warned: It's known to be full of fiction.

@Danu: I'm suggesting that trying to deny it did not help his mental health at all.

9

Q: What resources are available for lives of recent mathematicians besides E.T. Bell's Men of Mathematics?

I am about halfway through reading E.T. Bell's Men of Mathematics, and I absolutely love it. I'm a mathematician, and I enjoy learning about the lives behind the names that I know and use so often. (I also rather like the distinct opinionatedness and bias). I consider Men of Mathematics to be a ...

biographical-details mathematicians reference-request

@TedShifrin Ah, the denial. Hmm, I honestly don't know too much about Nash's case.

I've recently been watching lectures of Villani, however, and he seems to love Nash.

@Danu As I've heard! Although it is very well written, I don't take the content seriously. I do very much like the stories though, and it does discuss some mathematics here and there.

The original book is pretty forthright about it, but Nash and his wife were furious with Sylvia Nasar for what she wrote.

7:57 PM

The Extraordinary Theorems of John Nash - with Cédric Villani

It's a bit different, but anyone interested in the history of astronomy should read Gingerich's book on Copernicus's 'De revolutionibus orbium coelestium"

Layman level, but super enjoyable as background, for me.

it is really really good.

Plus I didn't know Nash worked on PDE's, honestly.

He did an amazing amount of truly deep mathematics.

7:57 PM

@TedShifrin Interesting. I'm very intrigued by his recovery.

Oh I've seen that video before.

I never heard the Gay Nash Theorem.

Perhaps this was the case, but I don't focus much on sexual orientation.

In the vein of what I was saying above

Owen Gingerich - Saturday Morning Physics - 11/03/07

@Semiclassical Sounds like something only a physicist could come up with :D

@Danu @TedShifrin Remind me of convention: When we take a Lie derivative, we take the derivative of $\varphi_t^*(T)$, where $T$ is your favorite tensor. Am I pushing forward or pulling back?

Depends on the variant nature of $T$.

You pull back forms, you push forward vector fields backwards. :)

8:00 PM

@MikeMiller Pulling back (pushforward with the inverse for a vector)

Erk

@TedShifrin It's a diffeomorphism. You can push or pull any kind of tensor.

@TedShifrin I didn't realize you were famous, Ted!

@Dodsy: Those of us who are gay, or female, or not a straight white male, often think about such things.

Ted is internet-famous!

8:02 PM

So you could always push forward with the inverse. (But ok, same as pullback.)

@MikeM: I totally understand.

And then there are always the sign confusions in Lie groups between left- and right-invariance. That changes the sign on the Lie bracket.

Signs on the Lie brackets are so terrible

Hey everyone!

hi

I ignore signs and scalars until I have to.

8:03 PM

The exponential map also gives sign problems (I'm encountering them currently)

As long as I'm working over R or Q it mostly doesn't matter, I can rescale or resign to make things work.

@Danu: Mostly because I was taught/brainwashed by Chern, I always did my principal bundles on the right instead of on the left. This changes signs in the structure equations, etc.

rehi Demonark

@TedShifrin Sorry, I should explain. I accept all races, religions, and sexual orientations- so I often admire without wondering about these small facts. I've had friends of all types and I would stand up for them no matter what. Of course, what I said could have been taken with offense, so I will apologize.

"rescale or resign"

lol

@TedShifrin Exactly... The right or left action problems... Urgh.

I learned it with right actions too, but now the books I'm reading use left actions.

8:04 PM

@Dodsy: I took no offense. There's no reason you should pay particular attention. But I emphasize that a lot of students/scholars feel very alone in the hard sciences ...

@Danu: Yeah, the flow of a left-invariant vector field is a right action. The flow of a right-invariant vector field is a left action. Agh. :P

@TedShifrin So about flag manifolds. I arrive at the definition of a "generalized flag manifolds" as $G/Z(T)$ ($Z(T)$ the centralizer of a torus) rather naturally, by looking at flags and the subgroups fixing them. Is there also a natural way of arriving at the lie algebra POV (defining it as an orbit under the adjoint action)?

@TedShifrin Precisely :(

@TedShifrin Yes, you are right.

GHOST RIDER vs POLICE

@TedShifrin You should have a wikipedia page!

^

8:10 PM

I hardly think that is warranted. I think my dad has one though (he was a composer).

Oh cool!

Have you ever collaborated with any Canadian mathematicians?

Nope.

:C

I've collaborated with ones who have, though.

I have a joint paper with a French mathematician.

@Danu, I dunno. I don't think about these things abstractly at all.

I can't speak any French.

8:12 PM

Hmm. I just feel like I need to mention the Lie algebra POV for completeness, since I think a lot of theory of flag varieties (algebraic geometry stuff) is more in this direction, though I can't be sure :P

Heya =) Anyone want to help me close a duplicate? =)

and if I mention it I want to at least establish equivalence of definitions

I have former colleagues who've done lots with that, @Danu, but not me.

@Astyx Salut, bon moment!

Hi

8:13 PM

Hey @Astyx!

hi

@Dodsy: I even have a published paper in French :P

nice

C'était qui ce mathématicien ?

you could probably find out yourself ;)

8:14 PM

Rémi Langevin (Dijon?) ...

Probably

[math.stackexchange.com/questions/2221292/…}(this])is a duplicate of [math.stackexchange.com/questions/61605/… =)

I give up

I've gotten the vibe that French is probably somewhat more useful to know now than German and Russian, right? The idea being that more content published in the latter two have been translated to English

@N3buchadnezzar: []() syntax

Formules cinématiques dans le cas complexe, @Ted? :)

8:15 PM

Bonjour, je ne sais pas "c'etait", mais je sais "mathematicien"!

Oui, c'est ça.

@Dodsy: CHEMISTRY!

@TedShifrin I tried. there is something wrong with my browser making the text whit when editing. Making fixing mistakes really hard

Oui, ou revoir.

Ah, since that paper with Langevin is on polar varieties, can you tell me what a polarized variety is @Ted?

Something with a nice line bundle, I've inferred from context. But what?

Bye guys and gals, I gotta listen to Ted.

8:16 PM

cya

Effrayant...

Bye

Principally polarized abelian variety?

Nothing to do with polar varieties.

@TedShifrin I don't know.

@TedShifrin Dang!

I just read "polarized variety" in some papers

@Danu: Yeah, it's basically a positive line bundle (which then defines the $\Theta$-divisor)

8:19 PM

"Yeux seulement" :D

@Danu: It's basically what allows you to do intersection theory (the analogue of the hyperplane class on a projective variety).

Hmm, okay.

Does anyone have a link to a page that presents multiple reformulations of the Collatz Conjecture?

@Ted Does intersection number relate to this intersection theory you're talking about?

Yes, Demonark :)

8:29 PM

Nice!

@Danu waz up. . . . .

Algebraic topology, differential geometry, algebraic topology ... are all full of intersection numbers as you're going to be learning.

hey people .

Who the hell is Cows?

hehe

@TedShifrin hi ted

8:31 PM

Would you say that differential topology is a subject that's... I guess somehow has a higher ratio of content versus technicality?

Well

plural pls

"Content" is probably not a fair way of putting it

Yes, Demonark. Of course things will get technical if you get far enough, but in my opinion at the undergraduate level it's the course where you get by far the most bang for the buck.

Um, hi Cows.

Demonark: For example, in G&P they will have you prove the Borsuk-Ulam Theorem. To do that with algebraic topology takes a few semesters of material, basically. And you get all sorts of neat root-counting theorems along the way (generalizing the argument principle in complex analysis) with diff. top.

@TedShifrin Do you disqualify analysis?

Hi again chat

8:34 PM

Welcome back.

@Ted I'm debating how to define a homogeneous space. Should I start with a Riemannian manifold, then say the group of isometries acts transitively? Or a closed subgroup (Besse does this)? Or should I start with a given group and then define its action on the manifold...

What really deep theorems do you see proved without too much technicality in undergraduate analysis? [I agree that Cauchy Integral Theorem is one.]

rehi @Eric

You might see Arzela-Ascoli

that one's really useful and not very technical

But undergraduates don't typically see deep consequences of it. Normal families are not taught in typical undergraduate courses.

But I'm fine with that. :)

@TedShifrin I think I am going tobe watching your course on youtube

8:36 PM

ok that's fair I didn't see any cool applications of it till I started taking graduate courses

@Danu: I probably would use $G/H$ as the definition, because that gives you all the things immediately that I care about (invariant forms, etc.). But I don't think it really matters.

Well I was thinking about implicit function theorem and picard lindelöf mainly

Differential topology is great because you have so much innate intuition for it ^^

Implicit function theorem is just sort of telling you intuitively obvious things. I'm talking about truly surprising, deep topology results in the G&P course.

but they dont satisfy the "no technicalities" criterium D:

8:38 PM

Why is that, @Cows?

criterion, @s.harp :D

What book is G&P?

I'm not asking for no technicalities @s.harp, that's unreasonable

Guillemin & Pollack

Hi, can anyone explain a problem from complex variables?

Having said this, I would include Stokes's Theorem on my list. I actually pretty much proved it regularly in my freshman/sophomore course.

Just ask it, @Paradox101.

8:39 PM

@TedShifrin well, out of curiosity I clicked on your profile, then saw that you mentioned it. I am always looking to improve my calculus skills, and learn more math. The youtube lectures seem like a good opportunity.

Are you reasonably interested both in computations and in proofs, @Cows? How far did you get in math?

Galois theory maybe has good examples of stuff that's deep/not overtly technical. I don't know how much Galois theory the average undergraduate is supposed to learn though.

I meant more, in some significant subset of (undergrad level real) analysis, I've found that a lot of epsilon and convergence gymnastics were dominating the experience, and that turned me off slightly

@TedShifrin I thought polarizations were to help get well-behaved moduli spaces.

It's pretty technical, @Eric, but I guess we should include the non-solvability of the quintic as up there.

That might be true, too, @MikeM. I hadn't thought about that. But the $\Theta$-divisor in a ppav is like having the hyperplane class for a projective variety.

8:41 PM

@Danu It depends on whether or not you think the Klein bottle is a homogeneous space.

Demonark: I happen to like analysis a lot, but I think that with a less-than-inspiring teacher the technicalities overwhelm the beautiful ideas.

user image

There's a book (I think by Arnol'd maybe) that goes through a proof that the quintic is not solvable by roots through a series of problems in group theory and basics of Riemann surfaces that was written for really smart high schoolers in russia

@TedShifrin ok. I know that it's simple enough but I don't get why wolframalpha gives this position in the complex plane for z. Since x and y are both negative, shouldn't it be in the third quadrant?

@TedShifrin Maybe I'm thinking of a different term. For K3 surfaces, I thought a "polarization" was an isomorphism from $H^2(X;\Bbb Z)$ to the standard abelian-group-with-quadratic-form structure.

8:42 PM

so that's a good candidate i think

There are too many automorphisms of the intersection form otherwise and you get a bad moduli space.

@Daminark I happen to enjoy my epsilonics a lot thank you very much

Man my browser crashed and I couldnt connect back to the chat

@Paradox: But then you're taking the square root of a number in the third quadrant, and that will put you in either second or fourth (in this case).

@TedShifrin I took the standard three calculus courses, a course on mathematical structures(we used the text "how to prove it" I think), took some differential equations, linear algebra, and integral equations. I also took a course on mathematical methods for physicists. These are the courses I have taken officially at school.

8:44 PM

I'll go now, bye chat

@TedShifrin However, I have tried to learn differential geometry, and algebraic geometry and topology on my own. It has not been easy, and from interacting with people who took these formally I am fully aware of my deficiencies

@MikeM: We're talking about different things.

@MikeMiller Why?

@Cows: So you probably didn't see multivariable calculus done with linear algebra in it. That's what my course does, so as to make good sense of all the ideas. You might also find my diff geo notes interesting (lots of pictures and even some physics in there).

@TedShifrin OK.

@Ted I just looked it up and Joyce calls this a 'marking'.

8:45 PM

@TedShifrin Does that mean that nth root has something to do with the quadrants? And why would the square root of a number be in the second or fourth quadrant?

@Danu There is no compact Lie group that acts transitively on the Klein bottle; equivalently, no Riemannian metric on it for which the isometries act transitively. You need a noncompact group.

@Eric I guess our tastes differ somewhat on that point. I guess I've just had that "Well, now that we've written up all the important stuff, it's time to put dots on the i's and everything". I can see why you'd appreciate it but somehow that's when I start to like it less

@MikeMiller But I never mentioned compact, right?

Or am I missing something?

@Paradox: Here's the idea. When you take the square root of $z=r(\cos\theta+i\sin\theta)$, you get $\pm\sqrt r(\cos(\theta/2)+i\sin(\theta/2))$.

You said something about a metric.

The isometry group of a compact manifold is compact.

8:47 PM

So look at the point in the third quadrant and take half the angle. If you think of the point in the third quadrant as a negative angle, half of it will land you in the fourth quadrant. @Paradox

Demonark: I actually think a lot of students came out of my course at UGA really liking analysis and did super well in the Rudin course afterward. A lot of this is very teacher-dependent.

Of course, taste is very student-dependent, too.

TRINITY! Save me from hell! i want to remain free!

My style doesn't fit so well the students who like algebra as symbol-pushing.

@Daminark I just think about it pushing epsilons like a game. Of course if you get a teacher who just teaches you how to play the game without teaching you why you're playing, it isn't gonna be a good time.

@TedShifrin: Does your course follow "Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds"?

@MikeMiller I don't know anything about isometry groups. Thanks

8:49 PM

Yeah, @Huy, guilty as charged.

New printing of the book coming out soon, though :P

is it even more expensive?

lol

@Huy like me

The expense is absurd. I have complained for years and gave up.

user image

what a bargain

8:50 PM

Dang

@Eric, @Daminark: When I first started teaching the Spivak course, I tried to motivate a lot of the epsilonics by asking for an error estimate on the area of a rectangle, and I drew the picture of a rectangle with small changes in the sides. I've never seen a math book that draws pictures for the limit of a product proof.

@Huy: I have never had anything to say about the prices of books. When we published our linear algebra book, our editor agreed to keep it at the lowest rung, but after 2 years he was overruled by his boss, and things went through the ceiling. Authors have NO say.

I don't hold it against the subject at all, I find it to be rather important that we're careful, just that when we were toying with $\ell^p$ spaces by hand, I was just like "Yeah yeah yeah", in contrast to the spectral theory, the earlier material on forms/manifolds, and now in difftop + Marianna's topological games

I will just say that there are a lot of crappier books that cost more than mine.

yes, I think we've talked about this before. SAD!

I am on your side, @Huy.

Used copies are in general more cheaply available, but they're more likely to fall apart, because the quality of book production has tanked as well.

8:53 PM

I think a year ago I also found out some of your books are available in the library here. when I have tenure I will read them.

I've resisted publishing my diff geo text, so it remains free, and still used all over the world. So I guess that's OK.

Heya tern!

^^Thank you for that, honestly

@Daminark that might be more because of who's teaching the course. I mean you gotta remember the guy who taught you spectral theory and functional and forms loves spectral theory and forms a lot more than he likes abstract Banach stuff.

(I've had three or four publishers ask me to publish those, Demonark.)

Daminark will be very grateful when he's doing all the problems in them this summer

8:55 PM

@Eric: Of course, teachers have taste and styles. And they influence teaching and course content. My algebra course had a somewhat different flavor than that typically taught by a formal algebraist.

All the problems! I'd like to see someone do every problem, @Eric!!

lol I was being slightly hyperbolic

@Ted I realized that since I will have to discuss homogeneous vs. symmetric spaces, 3-symmetric spaces and more Riemannian stuff, I should probably stick to the Riemannian POV.

I don't recall if we did more than a couple problems in the chapter on curves

I certainly did the vast majority of the problems in the later chapters though

Yeah, Soug is not directly overseeing the bootcamp at all so I wouldn't imagine us doing all of them

:P

(Locally) Symmetric space is a totally Lie-algebraic notion, @Danu. And that informs the geometry.

8:59 PM

@TedShifrin I worked my way from inside and got $r=2$ and $\theta=-\pi/3$. But since it didn't fit in with the position of $-2-i 2\sqrt3$ so I took $\theta=4\pi/6$ and my final answer is incorrect due to this. I took half the angle but after assuming that $-2-i 2\sqrt3$ was in the third quadrant. So should I just stick with $\theta=-\pi/3$ even though it doesn't match up with the quadrant before i take the square root of the complex number?

That's really cool, @Eric. I actually have done every problem.

why do you write $i$ in front of $2 \sqrt{3}$

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