Indeed, @Sriva @Asa

Are questions about math software really off-topic? math.stackexchange.com/questions/83174

I thought they were on topic

@AsafKaragila exactly, I cannot use Rob, Robj is not nice and Robjo sounds cool

@robjohn: only if you're not against it

@Gortaur doesn't matter to me.

I wonder if Rob doesn't return, how long it is until @Rob pings me again?

I have to go out to lunch in a bit so I may have to drop off line for a while.

@Sriva: Two more points :-P

@AsafKaragila Yeah, I will hunt for more typos soon =)

Poor J.M. He is missing all these points =)

Continously chasing typos?

@Sriva: which two points?

@Srivatsan: a new smallish project for you...

@Rob: enjoy your lunch

@tb People make innovative mistakes. =)

Thanks, tb

Thanks everybody.

@Srivatsan that last edit was somehow incomplete...

@JonasTeuwen You know, you shouldn't be here...

(there's a real life out there somewhere, or so I heard...)

@tb I have had dinner with some nice people. Now I'm home and exhausted! :-).

@Jonas Is the defense over?

[Must be; it's the 17th.]

@Srivatsan It is. I've got a 9/10 :-).

Congrats, @Jonas!

I don't know the significance of the score; I assume it's a good thing.

Well, 10 is the maximum and 9 is "excellent".

Nice, good job.

How are you celebrating it? =)

I have paid good monies for a dinner.

You paid? That sounds strange.

good = big?

Yes, I have paid.

Plus I've got some presents.

@Gortaur 597€.

@JonasTeuwen whut???

@JonasTeuwen 597 for dinner?

Is that for everyone?

@robjohn Yes.

@JonasTeuwen wait, did you have to organize it? or it was just for your friends?

My god. How many people?

@JonasTeuwen Ah, so you are taking others out for your celebration rather than them throwing you a party.

@robjohn That's what is surprising me.

@Srivatsan yes, but not unheard of.

And to expect someone to shell out 600 euros... That's a little hard on the person.

@Srivatsan either I do not understand it at all, or it is surprising indeed

@Gortaur It was for my friends.

@Srivatsan 12.

@robjohn Yes. Throwing a party is too intense.

@JonasTeuwen ok, that's nice that you didn't have to organize it )

Plus I've got some money as a present and I've got a salary.

I've organized it as well! But that was easy. I've called the restaurant and they said: Fine!

@JonasTeuwen aaah, I understand it now. In Russia we call it проставиться: that means that whenever someone has some success he takes his friends/collegues to the restaurant and pay for everything

Wow, that must have been some party. Is it that expensive? (600/12 = 50 euros per head...)

Food + wine.

@Gortaur How does that translate?

@JonasTeuwen like 'put on the table'. It's not obligatory, but people may be offended if you won't do it. Strange, but ce la vie

@Rob: would you like to talk about division by zero?

Indeed. I only graduate once in my life.

@JonasTeuwen Twice counting Ph.D.

@Gortaur ok

@Srivatsan Yes, that is only in four years.

@Srivatsan PhD is without a grade. only your postdoc uni counts )

@Rob how often do you divide by zero?

never

@Gortaur :-p

Today in commutative algebra we were taught that you can divide by zero.

@AsafKaragila Ze Horror.

You just need to complete the integers with respects to {0}...

@Gortaur Well, there'll be a committee and a defense. You'll pass the defense. It's pretty much the same thing once again... =)

Although you'd get the trivial ring...

@AsafKaragila at least that they have zero divisors, and why have them if you don't intend to use them?

@robjohn Huh?

off to lunch cul8r

@Rob then ask Asaf, he knows it now

@AsafKaragila if I have to explain, it really wasn't a good joke.

Well, in the trivial ring {0} we have that 0=1, and really all the elements are 0...

@Srivatsan there is no grade ) also, usually you know result before the defense. no adrenalin ;)

@Rob: you see what happens? that is what you wanted?

Why do you involve me into the story?

@AsafKaragila you're not involved.

in the case he really wants to know it though, you may help him

If the art of doing mathematics is forgetting about the superfluous information, then obviously division by 0 is superfluous information ... 0 =?= 1

Is that how a mathematician would think about it Gortaur?

A mathematician would think whether or not in ZF, ICF implies AC.

Okay, so curiosity got the better of me and I went to find out what the thing Rob has written about division by zero is.

WTF is ICF?

Imagine my surprise: It's actually quite orthodox.

@HenningMakholm Where has Rob written about division by zero? // And what's quite orthodox?

@Srivatsan here

@MartinSleziak mathoverflow.net/questions/81204/…

I was expecting a long rambling screed about how the mathematical establishment is suppressing his brilliant discovery of how one can divide by zero.

@Rob: To be absolutely fair, I have no idea how you managed to get to 19 points of reputation.

I guess some people will upvote anything.

@HenningMakholm LOL!!!

I liked this sentence. "Division is not always possible in the system of numbers consisting of the integers (6 is divisible by 2 and 3 but not by 5), but in those cases where it is, the result is always uniquely determined."

@Rob Well, that is what one tends to expect from somebody who pops in out of the blue and demands an intelligent discussion about division by zero.

@HenningMakholm I apologize for my overly aggressive demand for intelligent discussion Sir

I just wanted someones opinion on my orthodox approach to this question Sir

@AsafKaragila In that MO question you have $2^X$ on several places, where I would expect $2^{|X|}$.

Is it just a typo, or $2^X$ can have som e different meaning that I do not know?

@MartinSleziak It's me being lazy...

I could have written |2^X| as well, but the context is clear...

@HenningMakholm Normally students are not given the proof of the multiplicative property of 0 to begin with and this leads to " a long rambling screed about how the mathematical establishment is suppressing his brilliant discovery of how one can divide by zero."

@Rob Teach proofs to students? What kind of a crazy suggestion is that... =)

@Srivatsan I found a web site once called, I think a "theorem a day?" that never even defined what a proof is

This one?

Yes, I think that is it. I tried to find the definition of "proof" and it was not there!

Why would it need to? It's not "definition of the day", and doesn't purport to be a self-contained anything.

Also, the usual definition of "proof" is "whatever practicing mathematicians will accept as a valid argument even after trying hard to find flaws in it".

@HenningMakholm good point

Even after?

If they accept it without looking for flaws, well... something is wrong with the system.

Well, if you show a page of text to a mathematician and ask: "Is this (a) a proof, (b) informal discussion, (c) an axiom system, or (d) an excerpt from a post-modernist novel", he may well call it a proof without checking for the details.

I'd call it "(e) Who are you and why are you here again?"

Demanding that the checks had always happened would also render meaningless a sentence such as "Such-And-Such's proof of the Whatchamicallit Conjecture turned out to be flawed".

But -- as always in natural language -- there's a range of wider and narrower meanings of the word, which are usually disambiguated by context.

I hate to interrupt this wonderfully entertaining dialogue, but could you please give me your opinion(s) on the orthodox piece of logical reasoning that I presented in "Division by 0"

Okay, very quickly: It looks basically correct, but does not seem to offer any clear added value over the earlier answers. Its focus on (quasi)-formal reasoning from a particular axiom system might be technically impeccable, but still not a very satisfying answer for the original asker, who would be wondering why exactly those axioms were chosen as valid in the first place. For that, the informal arguments in the other answers are more likely to enlighten the asker.

In conclusion, I would (and did) neither upvote nor downvote it.

@HenningMakholm Thank you for your explanation.

@robjohn Sorry, do you still have the link about the tangential stuff? :-).

the metric in spherical coordinates

"... is a homoegenous quadratic form and not a sum of differentials of the three coordinates"

I have modified my profile! :-).

can someone explain what the author is trying to say in the above quoted sentence?

@JonasTeuwen "For the moment I have a master of science degree in mathematics" -- do you expect to lose it anytime soon?

Well... I'll add "only".

I hope not to lose it. It has cost me more than five years!

Would flow better if instead you took out "For the moment"

Okay. Thanks!

@JonasTeuwen Nice.

Is there a specific reason you have expanded MSc to master of science? // I feel that Master of Science should be capitalized. Let me check...

Yes, that's right: en.wikipedia.org/wiki/Master_of_Science

Can someone answer me a quesion?That is, do you have to study for master degree in math before you study a math PHD in US?

Could you tell us which country you're from?

@Srivatsan Thanks :-).

And aren't you a high school student?

@JonasTeuwen Looks good.

@Srivatsan I have expanded it because MSc degrees are (I think) uncommon in the VS.

So expanding it might explain better what it is.

@JonasTeuwen So I imagine. Considering that VS itself isn't that common. =)

@Srivatsan - No,just curius how much do you pay to obtain a phd

I just get a salary.

@Victor PhD pays you!

That sounds like a Russian reversal. =)

In the US, PhD pays you!

@Srivatsan -you really get pay when you study?

I was half-joking.

But most students get a stipend. You won't live like a king, but enough for sustenance.

But you might have to work for it though: TA, RA, what not.

@Victor Short answer: No.

@Jonas Would you say that you majored in analysis for your Masters degree?

Does Abel-Ruffini implies that the solutions of some higher degree polynomial cannot be express in a radical.When i ask this i really think i should delete my recent question

@Srivatsan Yes.

@Srivatsan It is not my résumé. It might be nice for people to have an idea what I am supposed to be able to do.

@JonasTeuwen No, I was trying to explain what I wanted to say. I decided to scratch that because that only seemed more confusing. // And I think I was right in scratching it, considering that you didn't understand that bit... :)

Okay :-).

@JonasTeuwen - could you skip the master and go to PHD?

But if it can be written up in a better way, I'm open to suggestions...

@Victor In The Netherlands that can be quite hard. Certainly if you have got your BSc degree there. If you've got it from say the US it is possible.

There's a transpondian difference here, I think. In the traditional European system, the various degrees build on top of each others, so you wouldn't start working for a research degree until after a master degree. There have been some experiments in taking students out of a partially completed master's program directly into a PhD track, which is commonly said to be due to American influence.

So the impression I get is that the American tradition is that when you enroll in a graduate program you decide relatively early whether you're aiming for a master's degree exit or a PhD one. But I don't know this firsthand. (Can anyone here confirm?)

There are other influences. They now try to make people only get BSc degrees...

@HenningMakholm - does that implies master in US is waste money because you could get a PHD instead of the master

MSc is probably much shorter.

@HenningMakholm You're right I think. Unless the PhD candidate decides to drop out with a MSc degree. // I presume that in many places you decide when you apply.

@Victor Probably depends on what you want to use it for. If your aim is an academic career, there's no substitute for a PhD.

Also, i want to ask is that BS worth more amount of money then the BA?

I'm not sure what the difference is :-). Mathematics in The Netherlands is always BSc.

So, no one here is come from US?

@JonasTeuwen In India, it's BA. I always found it strange

@HenningMakholm I would say that's true, although depending on the program you can sometimes switch to the PhD if you change your mind and do sufficiently well in the master's program. You can also get a consolation prize master's degree if you flunk out or give up on the PhD.

@Srivatsan Well, it doesn't matter much I suppose.

I'm from the U.S. What exactly do you want answered, @Victor?

:2464484 It's 6pm on the east, 3pm (I think) on the west.

(Shh. I forgot which way the Earth turned)

@Srivatsan Yep, 3:00 p.m. here.

@HenningMakholm Earth should be turned way up. I can't wait for their new album.

@MikeSpivey - How much do you pay for PHD?

@Victor I thought we went over that question :).

@Victor It depends on the program. Public universities will be cheaper than private ones, generally - often much cheaper. However, most math PhD programs will cover your tuition for you and pay you a stipend - often to teach lower-level calculus courses. You can get an idea of how much tuition and stipends are by looking at "Assistantships and Graduate Fellowships in the Mathematical Sciences," published annually by the AMS.

@JonasTeuwen springerlink.com/content/x761286035258411/fulltext.pdf

please read carefully:For Abel–Ruffini theorem, in a book it says that equations with higher than fourth degree are in general incapable of algebraic solution, but in wikipedia it says theorem says "that not all solutions of higher-degree equations can be obtained by starting with the equation's coefficients and rational constants, and repeatedly forming sums, differences, products, quotients, and radicals " Does the wikipedia miss something because incapable mean you never find out solution

Well, it's quite clear that the quintic equation x^5 - 1 has a solution expressible without radicals, additions, or multiplications...

@Victor Even simpler than Zhen's example: take the polynomial (x-1)(x-2)(x-3)(x-4)(x-5) (after expanding it out).

and for the book it only refer to coefficent , not the power of any term of the polynomial, could we use the power of the terms to write up a formula?

@MikeSpivey - thanks for your information

@Victor What do you mean by power? The number 5, which is the degree of the polynomial in both the above examples?

i mean the power, for example the power of x^5 is 5 @srivatsan

The statement of the Abel–Ruffini theorem is that there is no single expression composed of iterated radicals, sums, and products of the coefficients of a quintic equation which solves all quinitic equations. However, for certain special cases, there may be.

@zhenlin - so the theorem doesn't contradict the possibility that i could write out a formula that could solve all the solutions of a polynomial compose of radical,sums and products of the "coefficent and power"?

The powers are just constants, so they add no new information.

@ZhenLin - i mean 5x^5 + x^2 +1 is different from 5x^5+1 because the first one have power of 5 and 2

Well, yes, obviously. But then you are splitting it into cases and it's obvious that some can be solved and others can't. In fact we have a complete understanding of when it can or cannot be done (Galois theory).

Galois theory only works if you assume that it works. It does not work if you assume that it is wrong, for instance is 0=1.

@ZhenLin - So, eventually, some of polynomial couln't be solved?

If by solved you mean constructing the roots by hand, then no.

Yes. There are known examples of polynomials which have no solution by radicals.

Some polynomials cannot be solved.

In fact, I can give you a very nice example, which will always work for degree >4.

Take the Taylor polynomial of e^x, the expansion for terms of degree >4 is not solvable.

Isn't that one of the most amazing things? That you can actually prove something like that?

@Asaf: In the trivial ring any polynomial can be solved :p

@ZhenLin Oh, I meant 0=1 added to ZFC actually.

@Asaf: Any polynomial can be solved in that case too. And not solved.

Yeah.

actually does transdental number could not express by radical?

Almost by definition, a number which can be expressed by radicals of rational numbers is not transcendental.

Almost?

I thought that a number is algebraic if it can be expressed by radicals; and a transcendental number is just a number which is not algebraic...

An algebraic number is precisely a number which is a zero of some polynomial with rational (or integer) coefficients. So it doesn't necessarily have to be expressible by radicals.

I guess this means one thing.

I have to go to bed.

@zhenlin - So,from what you said, is that possible that we could find the root express by transcendental for some polynomial that could not express by radical?

No, any zero of any polynomial with rational coefficients is algebraic, i.e. not transcendental.

@zhenlin - thank you, but how do you prove your statement?

It's by definition.

@ZhenLin As any other proof in mathematics ;-)

Well, of course.

@AsafKaragila - what is a website that i could find the proof? also, can all irrational number express by the radical?

Huh?

@AsafKaragila - i mean radical of a number

Again, huh?

@Victor: It seems you are a bit confused about definitions. Any transcendental number is irrational but cannot be expressed by radicals.

@AsafKaragila @ZhenLin - so can a root of a polynomial be irrational but canot express by radical?

Only when I have slept enough hours.

If it is rational then it can by "expressed by radicals" (understood here as an abbreviation for "expression composed of iterated radicals, sums and products")

You mean algebraic, no?

No, I mean rational. It's a degenerate case of being expressible by radicals.

Are you calling rational people degenerate?

Sure, why not, they don't exist, after all. :p

Or do you mean only radicals are degenerates?

So what is different between irrational and transcendental?

does irrational mean it is not algebraic?

No, there are algebraic irrational numbers.

Irrational means not of the form p/q where p,q are integers and q is nonzero.

Perhaps a Venn diagram would be useful here...

Transcendental means "not a root of a polynomial whose coefficients are rationals".