The h Bar

Danu

5:00 PM

@FreeMind Depends on how much you know already. I don't know you

Sean

Geez what the heck did I miss on here this morning!?

FreeMind

@Danu Fair enough. Just tell me what you would have expected from me if you knew me?

ACuriousMind

@Sean What's "morning" for you?

Danu

@Sean carnage :D

@FreeMind I would expect you to tell me what you know.

FreeMind

@Danu Basic Linear Algebra, Calculus (Almost bamboozled in triple integral and totally integral in n-dimensions), Classical mechanics ( Still lots of gaps to fill in, anyway, I cannot focus on it, it has become boring, I have read the subject several times ).

Danu

5:06 PM

@DanielSank Been hearing MUCH too much of this in chat here lately

FreeMind

What have I forgotten? :D

Danu

@FreeMind You should study some electromagnetism

0celo7

@Danu amazon.com/Special-Relativity-M-I-T-Introductory-Physics/dp/…

There are plenty of them.

Danu

@0celo7 Lol, okay

Sofia

@FreeMind I agree classical mechanics is not challenging. Neither electricity. Electro-magnetism is more interesting, somewhat.

FreeMind

5:10 PM

@Danu Well, I know the ideas and have general over view of it, but the equations happening in Electro are sometimes messy. Vector calculus is a pain :( @Sofia Electro is cool. The math used in it is beautiful and beast!

Danu

@Sofia Have you tried the book on classical mechanics by Arnold?

ACuriousMind

@FreeMind Does classical mechanics mean Newtonian mechanics for you? If yes, learn the Lagrangian and Hamiltonian formulations. A good understanding of Hamiltonian mechanics - in particular the Poisson bracket and the generation of symmetries - eases the dread of quantum mechanics immensely.

Danu

@ACuriousMind Funnily enough, in Amsterdam it's not considered normal to really study Hamiltonian mechanics in any depth before quantum

alarge

@Danu Nothing to argue about here as it seems that apparently we're going with the ad hominem thing today.

Danu

Plus you get into Lie algebra territory which is really too advanced for your typical undergrad

FreeMind

5:12 PM

@ACuriousMind Yeah, one of the guys introduced me to some resources on Lagrangian and Hamiltonian mechanics the last time I was here. I am learning them.

Danu

@alarge I guess it's all just Sayre's law

alarge

@FreeMind Did you do thermo/stat mech yet?

Danu

@alarge Not really relevant for SR and GR directly, but good in general

FreeMind

@alarge Not at all. But I don't think that I will do it in the near future!

ACuriousMind

@Danu Second semester. Theoretical classical mechanics course (part 2), about sixth or seventh lecture. My prof introduces Lie algebras <3

No such thing as "too advanced" for that prof.

Danu

5:14 PM

@ACuriousMind Terrible pedagogically

ACuriousMind

@Danu He was acutely aware of that, but also aware that our university lacks a "graduate level" mechanics course, so he wanted us to know how much stuff there is out there to learn about mechanics without you even thinking about EM or quantum.

Danu

@ACuriousMind This is something crazy I've noticed in Germany... No graduate CM/ED/Stat mech

really crappy

0celo7

@Sofia Said the person who has probably never read Arnold.

FreeMind

@0celo7 Arnold is nice but hard to digest for me at the moment.

0celo7

@Danu Absolutely relevant for Cosmology.

ACuriousMind

5:17 PM

@Danu Well, the theoretical statistical physics course here is graduate StatMech. Unfortunately, it's almost always taught be the worst lecturers we have.

Physics Meta

0

Q: How To Improve Questions and Answers

This a soft question and I can try the chat rooms with it, if that's the consensus for the best place to discuss it. Comments from previous questions I have asked could also sort it out for me. If I post an answer to an OP saying "your problem is currently impossible to answer because we have no...

discussion

ACuriousMind

The only thing I learned in that course was integrating and/or summing up random stuff

Danu

@0celo7 Depends on what kind of cosmology. Not for the GR-oriented kind so much, e.g. FLRW metric etc

@ACuriousMind We have mathematical stat mech, but it's apparently just a lecture on mathematics (operator algebras?)

0celo7

@Danu That's the cosmology you see in GR books. I'm talking CMB, etc.

ACuriousMind

@Danu lol, no, this was a physics course, allegedly

Danu

5:19 PM

@0celo7 Yeah. Did you look at @FreeMind's question? He's interested in GR, not cosmo for its own sake.

@ACuriousMind I'd be interested in that---the undergraduate course I had didn't teach me too much.

0celo7

@FreeMind I highly recommend Zee for an introduction to SR and GR.

Very self contained.

ACuriousMind

@Danu Lemme dig out the "lecture notes" for that. They are as comprehensible as the lecture was.

Danu

@0celo7 I never looked at Zee, because his QFT book is so awful I didn't trust his other books to be any good

0celo7

You need some QM and QFT for the last few chapters, but other than that, all you need is linear algebra, calculus and mechanics.

@Danu He wrote his GR book 11 years after his QFT book.

He used those 11 years very well.

innisfree

Zee's QFT in a nutshell is brilliant!

0celo7

5:22 PM

@innisfree Danu and I consider it unreadable I think.

innisfree

Though the irreverent style might not be for everyone

0celo7

@Danu It's probably not worth reading EG Nut at your level...it's not very mathematically advanced.

Glen The Udderboat

@innisfree Is there a "QFT in a nutshell"? Excellent.

0celo7

@GlenTheUdderboat NO!

IT'S A TRAP

ACuriousMind

@Danu dropbox.com/s/dxufyl8ukrsxe5k/Skript%20-%20Wetterich.pdf?dl=0

Danu

5:24 PM

@innisfree It is tooooo handwavey

0celo7

@innisfree No, the fact that he assumes you know group theory and condensed matter going in makes it not for everyone.

Danu

Urgh, script with a 'k' in German really grinds my gears

0celo7

@Danu You're a primary source: How does one pronounce "Nijenhuis"?

FreeMind

I think Caroll is what I wanted !

0celo7

@FreeMind Carroll is good.

ACuriousMind

5:26 PM

@Danu Excuse me for having a German filename :D

Danu

@0celo7 ...how do you want me to explain it? Haha

innisfree

Zee is hand-wavy, but that's the point. He's tried to provide a more intuitive, less formal approach to learning QFT.

Danu

It's nothing like the Americans would do it!

Also, why do you ask?

@innisfree This is, in my opinion, quite a bad idea

ACuriousMind

@Danu I thought you caught up on chat? :P

0celo7

@Danu Because I want to say it correctly...?

Danu

5:27 PM

The usual approach to QFT is already quite handvavey

@ACuriousMind Damnit!

What'd I miss

0celo7

@Danu Explain it to my inner German.

Danu

I must admit I skipped sections where certain users were absent ;)

Sofia

@ACuriousMind I am not a creature to hold grudges. And of course not on you, and you know that. You know. You want me not to say anymore that you are young? I will try. If that upsets you then I will try not to say.

ACuriousMind

@Danu Ah, you'd just have seen the discussion about Nijenhuis' pronounciation there, and the soundbite of it I posted

Danu

@0celo7 It's not German, they can't pronounce it either :P

innisfree

5:28 PM

Hmm fair enough. I would recommend reading another QFT book in conjunction with Zee.

Danu

@ACuriousMind Link!

0celo7

@Danu Try please.

Danu

I can give you guys a soundbite too

ACuriousMind

@Danu de.forvo.com/word/nijenhuis

Not me saying that, mind you :D

Danu

Ooh...

Right, sounds a little posh but pretty good

ACuriousMind

5:29 PM

@Danu lol, I remember your whistling

0celo7

No freaking way huis -> haus

Danu

@ACuriousMind :D

@0celo7 no. :P

the 'ije' is hard to pronounce to almost anyone

Glen The Udderboat

@ACuriousMind That is absolutely correctly pronounced.

Danu

@GlenTheUdderboat Are you Dutch too? ^^

Glen The Udderboat

@Danu Yep

Danu

5:30 PM

Cool, where from? I'm from Amsterdam

Sofia

Read QFT book in conjunction with Zee? Which Zee? I know ZeeTV, (India), ah interesting movies. I saw once an Indian version of "Thief of Baghdad". Ayye! Splendid.

0celo7

@Sofia Lol what?

Gowtham

lol indeed

ACuriousMind

@Sofia Anthony Zee is a physicist who wrote QFT in a nutshell.

Sofia

@0celo7 "Thief of Baghdad"! What don't you know the story?

ACuriousMind

5:31 PM

Also, lol

0celo7

@ACuriousMind He wouldn't have you know that his first name is Tony. He always writes his name as A. Zee.

Sofia

@ACuriousMind which lol? ZeeTV is very pleasant.

ACuriousMind

@0celo7 What, he's ashamed to be a Tony?

Danu

@Sofia lol lol

Glen The Udderboat

@ACuriousMind Also, I recognise the voice: I'm fairly sure it is Koos Postema.

ACuriousMind

5:33 PM

@Sofia lol = laughing out loud, in this case at your ability to get from QFT to Thief of baghdad in only two sentences. :D

Sofia

@ACuriousMind , @0celo7 @Gowtham @Danu what, you never watch ZeeTV?

Danu

lol means "laughing out loud", but it is just used as a shorthand to denote that you find something funny

0celo7

@ACuriousMind When his colleagues thank him in their books, they call him Tony. But even in the Library of Congress info, he's A. Zee.

Danu

@Sofia I have no idea what ZeeTV is

innisfree

I think Zee was Sidney Coleman's student - I see similarities in their style.

Danu

5:33 PM

I don't watch television

0celo7

@innisfree I think you are correct. His adviser was Wheeler I think.

Glen The Udderboat

@Danu Then you don't know Koos Postema.

Danu

@0celo7 err... isn't he just called Anthony---Tony being reserved for friends?

@GlenTheUdderboat Indeed I don't. Who's that?

Gowtham

@Sofia i know what ZeeTV is , but i doubt many in europe do ..

0celo7

@Danu Yes, I'm saying he never writes his name in full.

Sofia

5:34 PM

@Danu ZeeTV is the most well known TV channel of India. And there is no Anthony there.

innisfree

@ocelo do you mean incorrect? That his advisor was actually Wheeler?

Glen The Udderboat

Koos Postema

Jacobus Gerrit (Koos) Postema (Rotterdam, 17 augustus 1932) is een Nederlands verslaggever en presentator. == Biografie == Koos Postema werd geboren in Rotterdam als zoon van een trambestuurder. Zijn vader overleed toen hij vier jaar was en na het bombardement op Rotterdam verhuisde het gezin in 1940 naar Vlaardingen. Postema behaalde het diploma van de Rotterdamse pedagogische academie en ging - na militaire dienst - als leraar aan de slag aan een Vormingsinstituut voor werkende jongeren in Rotterdam. Hij stond daar vijf jaar voor de klas. Postema begon zijn mediacarrière op 1 mei 1960 a...

ACuriousMind

@GlenTheUdderboat Who is Koos Postema? A Dutch actor?

Ah

nvm

Danu

@Sofia LOL, Indian television

@GlenTheUdderboat Why do I care? :P

0celo7

@innisfree I think you are correct. I was just saying that he also worked with Wheeler.

They are similarly incomprehensible at times.

Danu

5:35 PM

@0celo7 Oh, okay.

Glen The Udderboat

@Danu It is good to know random shit.

ACuriousMind

@GlenTheUdderboat verslaggever means reporter? Dutch is great :D

0celo7

@GlenTheUdderboat Is there Dutch Jeopardy?

Danu

@GlenTheUdderboat On occasion

@0celo7 No

@ACuriousMind Literally "guy who gives report"

Glen The Udderboat

@0celo7 I'm not sure. I don't watch that much either. :)

innisfree

5:37 PM

Oh right, sure.I'm interested by those connections between advisor and student. Famous physicists often have famous students. Do the best physicists supervise the best students? or do the great physicists inculcate important ideas in their students and create a new generation of great physicists? Maybe both, of course.

Danu

@innisfree I think the latter

Glen The Udderboat

@innisfree There's a website for that.

Danu

Choosing a supervisor will be so hard :\

Glen The Udderboat

@innisfree Forgot the address, but it gives you a full tree of advisor-student relations.

Danu

@GlenTheUdderboat Mathematics Genealogy

genealogy.math.ndsu.nodak.edu

Glen The Udderboat

5:39 PM

@Danu Yup, I think that's it.

innisfree

Do they reach any conclusions, though? or is it just raw data?

Danu

The student list of Hilbert is so impressive :D

0celo7

Did Einstein have any PhD students?

Danu

@innisfree Data

Glen The Udderboat

@0celo7 The site says 2.

38 years apart?!

0celo7

5:42 PM

@GlenTheUdderboat Oh it has physicists too? Never heard of either of his students.

Glen The Udderboat

@0celo7 Well, that may tell us something.

innisfree

Staff at IAS have few teaching responsibilities.

Danu

Einstein was known to hate teaching

@innisfree He wasn't always at the IAS

1933-1955

In particular, all of his famous work was already done by then :P

Glen The Udderboat

@Danu Because he didn't have good students to inspire him? :)

Danu

@GlenTheUdderboat Hah, I don't think he had (m)any students while doing his most famous work

Glen The Udderboat

5:45 PM

@Danu He walked with Gödel. Not sure if that would help. Unless you aim for the ultimate.

Danu

@GlenTheUdderboat Haha :D

Damn logicians

innisfree

Wow, David Hilbert supervised the world-champion chess player Emmanuel Lasker

4

Danu

@innisfree !!

Glen The Udderboat

@innisfree I'm taking note of that. In my random knowledge compartment.

Danu

Damn, I wish I could ping whatshisface

@ACuriousMind I appear to have forgotten the 'name' of our Luxembourgian friend :\

@innisfree I can't find Lasker in the list

Glen The Udderboat

5:48 PM

Here's another chess world champion: genealogy.math.ndsu.nodak.edu/id.php?id=61173. Not particularly interesting, I think.

Danu

Lasker was a real homo universalis! ..except for sports

@GlenTheUdderboat "veranderlijken" what a word!

Glen The Udderboat

And here's Lasker, not with Hilbert, but with Noether ('s husband?): genealogy.math.ndsu.nodak.edu/id.php?id=60257

ACuriousMind

@Danu Are you just saying he's not been here for a long while, or have you really forgotten it?

Danu

@ACuriousMind Genuinely

Glen The Udderboat

Ah, Noether's father...

ACuriousMind

5:50 PM

@Danu Phonon

Danu

I knew it was a 'P'

Glen The Udderboat

@innisfree I don't see Lasker in Hilbert's list either.

Stan Shunpike

@Danuhave u used Landau?

innisfree

i read that on wiki

Floris

"veranderlijken" sounds like a Marten Toonder word - something Professor Prlwytzkofsky would use

ACuriousMind

5:53 PM

lol, Emmy Noether's biography there says: Emmy Noether is best known for her contributions to abstract algebra, in particular, her study of chain conditions on ideals of rings.

Stan Shunpike

@Danu have u used landau?

Sry typo

ACuriousMind

I don't think any physicist would agree with that :D

Stan Shunpike

I don't even know anything and I don't agree

ACuriousMind

But yeah, Noetherian rings are awesome, if often a bit overkill as an assumption

Danu

@StanShunpike Yeah, at times

@ACuriousMind Yeah, that's funny

@innisfree @GlenTheUdderboat en.wikipedia.org/wiki/List_of_mathematicians_who_studied_chess

ACuriousMind

5:55 PM

Are mathematicians aware of Noether's theorems?

Stan Shunpike

@Danu do you like the books?

Danu

@StanShunpike Sure

@ACuriousMind Interestingly, I discussed Noether recently in the Mathematics chat

They had some vague idea, but thought it was a different Noether ;)

Glen The Udderboat

@Danu "David Hilbert encouraged Lasker to obtain a Ph.D in mathematics." That's not what we're talking about. Noether's father it is then.

Danu

@StanShunpike In particular volume 2 is really famous

However, I never used them to really learn a subject

Glen The Udderboat

So, in a (very strange) sense, Emmy and Lasker are half-siblings.

ACuriousMind

6:01 PM

@Danu Haha

That's good

0celo7

@ACuriousMind TIL "Kennedy" is a banned name in Germany?

@ACuriousMind TIL names in Germany must be gender specific and cannot be surnames?

Danu

Say what?

ACuriousMind

@0celo7 What do you mean?

I think there's a requirement that at least one of your given names be gender-specific, though, yes.

Glen The Udderboat

The internet concurs: firstnamesgermany.com/the-german-law-on-first-names

ACuriousMind

Basically, there's a giant list of names. If it's not on there, you can't name your child that.

I'm not sure what I think of that system as a whole, but it at least prevents much silliness.

user54412

6:14 PM

The more I'm in this room, the more I realize all German stereotypes are true

ACuriousMind

@ChrisWhite I'm...not debating that.

Danu

@ChrisWhite :D I'm proud of the role I've played in exposing these.... truths.

@ACuriousMind Wow, that's absolutely stupid. My name is actually Danu

0celo7

Maria as a second name for boys?

user54412

@Danu I was chatting with Freeman Dyson one day, and he suggested that it was maybe a mistake for Einstein to go to the IAS over e.g. Caltech. It allowed him to embrace his miserly hermit side too much.

Danu

@ChrisWhite Oh man, I'm so jealous of you right now!

user54412

6:18 PM

@0celo7 not uncommon with Hispanic names: en.wikipedia.org/wiki/Jos%C3%A9_Mar%C3%ADa

Danu

@0celo7 This is pretty standard (comes from biblical influences, methinks)

I admire Dyson quite a bit

And perhaps he's right on that

user54412

wow, I just realized Dyson is over 90

Danu

Yeah! He's so old, but appears very lively

user54412

he still goes to colloquia too

Danu

I watched a 3 hours-and-something video interview with him

very interesting to see

user54412

6:22 PM

he's given a lot of thought to a lot of things, not the least of which is Astrochicken

ACuriousMind

@Danu I think if it is a valid name in another language, it will also be permitted, but you have to do some bureaucracy, as always ;)

@0celo7 Mostly as tribute to some female ancestor named Maria.

Danu

@ACuriousMind Hmkay. I really like having a unique name though, I don't want it to be in a book so others can imitate me!

For the Euler $\varphi$ function, does one count 1 as a number that's relatively prime?

ACuriousMind

@Danu ::German stares blankly at you, trying to comprehend the concept of uniqueness::

alarge

@ChrisWhite Which is why I was surprised to see him still contributing to science at a high level (in game theory, though) just a couple of years ago.

user54412

@Danu It must, in order to have nice properties like being multiplicative for relatively prime values

0celo7

6:31 PM

Is it possible to be agoraphobic and claustrophobic at the same time?

user54412

$\varphi(mn) = \varphi(m) \varphi(n)$ if gcd(m,n)=1

ACuriousMind

@0celo7 I sincerely hope not

Sean

@Danu Morning comes before noon... We don't all live in Europe, so at the time I posted that message it was a little before one. And according to the time stamps on chat, your fun exchange with Sofia happened around 11 EST

Oops, that was supposed to be @ACuriousMind

^

Oh well

Haha

ACuriousMind

@Sean I know your time is different, my eurocentric mind just isn't able to remember what time your morning is in my time zone

Glen The Udderboat

@alarge I remember that vaguely. Infinitely repeated prisoners' dilemma, right?

Sean

6:45 PM

@ACuriousMind What time is it right now for you?

I'm eastern daylight time, so i'm probably 4 or 5 hours behind you

ACuriousMind

@Sean It's almost 8pm

alarge

@GlenTheUdderboat A new strategy for it, yes, that I think in some way outperformed tit-for-tat (which I think was believed to be the optimal strategy).

But there was some followup to it, showing that the Dyson strategy wasn't stable or something. Can't remember anymore.

Glen The Udderboat

@alarge No, it wasn't, but it was popularised by somebody as the winning strategy in a live competition. And even there, I don't think it won.

Sean

@ACuriousMind And it's almost 3 here. So I guess I'm 5 hours behind

Glen The Udderboat

@alarge Axelrod

Sean

6:50 PM

Maybe you forgot about the time difference because you're so young :P

hahaha

ACuriousMind

@Sean Possibly xD

Sean

@ACuriousMind what interpretation of quantum mechanics do you subscribe to?

ACuriousMind

@Sean Shut up (and calculate)!

Also:

3 hours ago, by DanielSank

@FreeMind: Stay away from anyone who utters the phrase "interpretations of quantum mechanics".

Sean

I'm allowed to think @DanielSank is nuts haha

Danu

@Sean You can just edit older messages by pressing the "up" button on your keyboard before typing anything

Sean

6:54 PM

What's the point in calculating if you're not gonna think about what it means?

Danu

What's the point in thinking about what it means when you don't know what you're talking about? :D

6

ACuriousMind

Instant stardom, @Danu :D

Sean

Also, if you don't bother talking about interpretations, what are you gonna do when questions about bells inequalities come along?

alarge

@GlenTheUdderboat Then I must misremember as I had thought that it won the computer competition, too.

Danu

@ACuriousMind You're god damn right!

ACuriousMind

6:56 PM

@Sean Bell's inequalites are statements about how certain probabilities behave in certain types of theories. I don't need interpretations for them.

Danu

@Sean Plug my ears and shout NANANANA DECOHERENCE DECOHERENCE NANANANA

Glen The Udderboat

@alarge I read that now on Wikipedia (source: Varoufakis, the current Greek finance minister). But I'm fairly sure it didn't...

Stan Shunpike

@ACuriousMind Okay so yesterday I tried to make a list of the properties that could be described using state vectors. And you simply responded "all properties can be described that way." Let me ask a separate question. What are the fundamental properties of elementary particles?

Sean

@ACuriousMind You do if you're trying to figure out whether entanglement violate locality or not

Glen The Udderboat

@StanShunpike What do you mean by 'fundamental'?

Sean

6:58 PM

It seems to be like Bell's inequalities, and wheelers delayed choice experiment, and quantum erasers require some kind of interpretation

Danu

@StanShunpike That they're not fundamental, but just excitations of fields ;D

Sean

to me**

ACuriousMind

@Sean Oh, that. I think the issue goes away when you define rigorously what you mean by locality

Stan Shunpike

@GlenTheUdderboat properties um...what we have to keep track of in order to know what is going on in the system. Its stuff we find useful to measure.

Maybe I'm asking the wrong kind of question

Glen The Udderboat

@StanShunpike Is an unmeasured property also a property?

Stan Shunpike

7:00 PM

@GlenTheUdderboat yes, so that shoots a hole right through what I just said

user54412

@Sean Ultimately, it comes down to the fact that no quantum interpretations disagree as to experimental results, by construction. Therefore distinguishing them is unavoidably unscientific in the Popperian falsifiability sense.

Glen The Udderboat

@alarge "Yet it won the tournaments by consistently scoring a strong second-place with a variety of partners." So, it did, but only according to the specific tournament setup.

But not one-on-one.

Stan Shunpike

Two uses I can think of for "fundamental" properties are (1) to characterizes differences between particles and (2) to specify what's happening to a system.

0celo7

Mario is in 180 games.

Glen The Udderboat

@StanShunpike Define the scope of that 'system'.

Danu

7:08 PM

@ChrisWhite So $\varphi(1)=1$?

user54412

@Danu yes

alarge

@GlenTheUdderboat Right, so I think it depends on the application which is really the best strategy to go by. Like in evolution, which is one of the most important applications of game theory, you are interested in stable strategies, and I think it was this condition or maybe something similar that the Dyson strategy was not able to meet. Maybe. If you look it up, do let me know.

Danu

@ChrisWhite But then we have to accept $\exists n \in \mathbb{N}$ s.t. $n$ is relatively prime to itself

Isn't that totally ridiculous?

user54412

@Danu what do you want "relatively prime" to mean other than "gcd = 1" ?

Danu

@ChrisWhite Very good point.

Stan Shunpike

7:16 PM

@GlenTheUdderboat this may make no sense. :D I'm going to say the scope of a system is defined by its conservation laws. When a system has no conserved properties, measurement becomes difficult, but if there are conserved quantities, then we can characterize a system by how it transforms into another state while keeping those quantities fixed.

@Danu wait, is spin an excitation of a field?

Like what is an excitation?

Danu

@StanShunpike Spin is a property of the field

Stan Shunpike

So its a fundamental property lol?

Danu

Sure it is. This stuff comes from group theory mostly

I think we classify fields---and consequently particles---by the representations of the Poincare group or something

I'll know more about this in about 5 months

Stan Shunpike

What happens in 5 months?

ACuriousMind

The Poincare/Lorentz representation approach is known as Wigner's classification.

Danu

7:20 PM

@StanShunpike I'm taking courses on this soon

@ACuriousMind There we go

ACuriousMind

@StanShunpike Sorry, but that doesn't really make sense. Conservation laws follow from symmetries of the system, but you can measure any observable, conserved or not

Danu

So, why is $\varphi(2)=1$ true? @ChrisWhite

I thought, since $\varphi(1)=1$, we'd have to count $n$ as well---so prime numbers would be counted as relatively prime to themselves, i.e. $\varphi(2)=2$?

Anyone?

I guess $n$ cannot contribute, since $n$ can never be invertible $\mod n$.

This does make $\varphi(1)=1$ weird.

@ChrisWhite What do you reply to the objection that, by that logic, $\varphi(2)=2$

Oh wait, I guess one should say gcd(2,2)=2

so it's all okay

[end of monologue]

Danu

7:55 PM

We've passed the 50.000 question mark!

3

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