Mathematics - 2015-05-16 (page 5 of 6)

9:03 PM

Is this all to avoid using exact answers, Owatch? :) I.e., to just plug $\arctan (1/\sqrt{3})$ into a calculator, and hope to fix it later?

teadawg1337

If I became a math teacher, I would never accept decimal answers.

pjs36

You don't know how many times I saw people write $\sin(\pi/6) = 0.009$ in a single semester of pre-calc, @teadawg1337 ... Just to go one step beyond decimal answers, but nonsense decimal answers :)

Owatch

@teadawg1337 I don't even like fractions anyway.

Karim Mansour

@pjs36 you know what my prof suggested to me

Owatch

I'll even put 1/2 in decimal form.

teadawg1337

9:09 PM

@pjs36 It took me a second to realize that answer is in degrees...

Karim Mansour

he suggested that I solve the problems even before I read the chapter that way it will even increase my problem solving skills

@pjs36

pjs36

Interesting, @Karim, what class was this?

Paul Plummer

-

teadawg1337

@Owatch That will come back to bite you, if it hasn't already.

Karim Mansour

Algebra @pjs36 he told me that I should try that way too because he knows that I am doing DF this summer

teadawg1337

9:12 PM

Hello @TedS

Ted Shifrin

hi mr @teadawg :)

Karim Mansour

hi @TedShifrin

long time no seen

Ted Shifrin

hi @Karim .... yeah, real long time

Karim Mansour

what do you think of this method @TedShifrin did you try it before?

solving problems before reading the content? based on just the definitions you have?

Ted Shifrin

When you first start to learn mathematics, it's about knowing definitions, and writing basic proofs using only definitions. But that soon becomes inadequate, and one really needs to understand foundational theorems and how to apply them.

Owatch

9:14 PM

Im making so many mistakes.

Small ones, I'm scared

Ted Shifrin

@Owatch, you too need to take a break.

teadawg1337

@Owatch Those kinds of mistakes add up pretty quickly

Owatch

I did.

I was gone for like an hour.

teadawg1337

Wow, that pun was actually unintentional...

Owatch

Maybe more.

Balarka Sen

9:16 PM

I make silly mistakes all the time, @Owatch.

Owatch

Not like me.

Ted Shifrin

@Balarka: It's really past your bedtime.

teadawg1337

I made a silly mistake when I was helping you earlier today

Balarka Sen

yes, I know.

Owatch

Multiplying X by -Y, well, instead of it being x(-y), I read it as x - y

Ted Shifrin

9:17 PM

Parentheses are so important, or the lack thereof, @Owatch. I put big red marks on my students' papers for such things.

Karim Mansour

@Owatch one time during my first year of uni I took first year mechanics I had 100 up until the final then during the final I made one stupid mistake of trignometry analyzing the angle as $gcos(\theta)$ instead of $gsin(\theta)$ and another mistake that made me get 94 in the class

I always do that though during semester I am super well then get to final and do stupid mistakes

haha

Owatch

Oh no. It brought you down to 94

Karim Mansour

yeah

Ted Shifrin

Shocking, isn't it, @Owatch? :D

teadawg1337

@Owatch That sounded like sarcasm. Imagine what would happen at around a 75

Owatch

9:20 PM

One step from Seppuku Karim?

Ted Shifrin

@teadawg: I'm sorry for your "altercations" with you-know-who. It seems her self-congratulatory attitude has removed all of us except robjohn.

Karim Mansour

yeah @Owatch

well I do that all the times xD I always get tired by end of the semester and do bad mistakes

Balarka Sen

If it was only her self-congratulatory attitude, it'd be ok. She's pretty good at the stuff she does, so I don't have that much of a problem with it.

Mike Miller

morning

Karim Mansour

evening @MikeMiller

Mary Star

9:23 PM

Could some one take a look at my question:

0

Q: How could we continue to show the inequality?

Let $\Omega$ a bounded space. Let $u_1$ the solution of the problem $$-\Delta u_1(x)=f(x), x \in \Omega \\ u_1(x)=g_1(x), x \in \partial{\Omega}$$ and $u_2$ is the solution of the problem $$-\Delta u_2(x)=f(x), x \in \Omega \\ u_2(x)=g_2(x), x \in \partial{\Omega}$$ Using the maximum principle I...

Balarka Sen

morning, @Mike

teadawg1337

@TedShifrin Eh, I've been through worse arguments before

Mary Star

?

Ted Shifrin

I already said goodnight, @MikeM, but goodnight^2.

Karim Mansour

good thing chat is alive today

I might ask algebra question later today

questions *

Mike Miller

9:24 PM

Alumni day is too loud. They're singing or something, and I can somehow hear it from my office.

Ted Shifrin

Speaking only for myself, @Karim, I'm not alive.

So how did your talk go, @MikeM?

:S @TedShifrin

It's in an hour.

oh

@Karim Find a nonabelian group with every subgroup normal.

Mike Miller

9:25 PM

But I can tell you it'll be a tragedy ahead of time, @Ted :D

Ted Shifrin

too easy, @Balarka

Terrible attitude, @MikeM

Karim Mansour

$Q_8$

Mike Miller

I don't actually think that. But I think I will try to avoid non-mathematician talks from now on.

Karim Mansour

I actually just read about $Q_8$ in DF it is very interesting

Balarka Sen

ok, that was fast.

Ted Shifrin

9:27 PM

It's a good skill for developing good colloquium skills.

Karim Mansour

I just read about them in DF @BalarkaSen

Mike Miller

That's what the general graduate student seminar is for, @Ted

Balarka Sen

@KarimMansour oh, I see.

Owatch

I divide by pi and see what fractions I come close to. Works meh.

Karim Mansour

you know what I find super cool is constructing homomorphisms by seeing if the elements of group H satisfy the relations that the generators satisfy for a particular presentation.

very nice idea @BalarkaSen and @TedShifrin

Ted Shifrin

9:28 PM

That wasn't exactly well-stated, @Karim, but I think I agree it's a good method

teadawg1337

@Owatch Horrible method. How would you go about recognizing $\frac{8\pi}{13}$?

Owatch

I wouldn't, stays in decimal form.

teadawg1337

But..... sighs

Ted Shifrin

Reliance on calculators is the demise of math students.

pjs36

Nah, teadawg, it's so much easier to spend a few minutes fiddling with a calculator constantly, rather than learn the values of 5 whole angles. That's like 15 things!

And it's not like there are any patterns or anything, to help you out...

anon

9:31 PM

@TedShifrin I was 1-on-1 with a student and watched them type 2-3 into the calculator once

Owatch

I'd do it.

Take no chances during tests.

anon

that's exactly the mindset :P

Karim Mansour

I think calculators should be bannd

banned

I mean when you do calculations in your mind your mind becomes faster

teadawg1337

But there's no risk involved when subtracting 3 from 2, @Owatch

Owatch

My HS exams only permitted calculators for certain sections.

Mike Miller

9:32 PM

on the complex analysis midterm I proctored someone took out a calclator. i told them to put it away because there's nothng they need a calculator for, and they told me they needed it to evaluate the answer ($\sin(16\pi^2)$)

lol

^2?

yes

haha

@teadawg1337 You wouldn't think so. But you don't know me.

Mike Miller

9:32 PM

otoh I can't calculate tips on my own because I'm bad at arithmetic

anon

I just round up to the nearest dollar or 0.50

Ted Shifrin

@anon: What's worse is that they have no intuition for what $1/2 + 1/3$ should be ... so if the calculator says 4.3, that's what they write.

anon

I might bet that's pretty standard amongst mathematicians

Mike Miller

so do I, but I'm awful at getting it close

Ted Shifrin

@MikeM: Does that mean I have to leave the tip when you (finally) take me out to dinner?

teadawg1337

9:33 PM

Does $\sin(16\pi^2)$ even have an alternate form?

anon

@MikeMiller the less close you are the more thankful your server will be :)

not really teadawg

Balarka Sen

our highschools don't let us use calculators, fortunately.

Ted Shifrin

@anon, only if he's on the + side, rather than the - .

anon

mike and I both agreed we round up ^

Owatch

That's a bit of a shame, calculators can be very useful.

teadawg1337

9:34 PM

@anon I know, that's the point I was trying to make :P

Ted Shifrin

But if you aren't sure of the calculation, how're you sure if you're rounding up from the correct point? :D

teadawg1337

@Owatch For CHECKING answers, not FINDING answers

Owatch

What's wrong with using them to find answers?

Mike Miller

@anon but if I'm consistently off I eat out less :p

anon

heh

teadawg1337

9:35 PM

Because you're not learning any problem-solving skills, you're being entirely reliant on a pocket computer

Paul Plummer

I was not invited to the big ignore party :(

Balarka Sen

hahaha

Mike Miller

I'll ignore you @PaulPlummer

Owatch

Makes solving things like definite integrals much easier for me.

teadawg1337

@PaulPlummer To be fair, it was announced last-minute

Owatch

9:36 PM

Because I make lots of small mistakes, especially when it comes to that sort of stuff.

Paul Plummer

Oh how sweet @MikeMiller

@teadawg1337 I guess I can forgive you guys, I probably would not have been able to make it anyways

Balarka Sen

making rows can be fun

Ted Shifrin

I can ignore you, if you'd like, @Paul.

Balarka Sen

and you missed all of it, @Paul

Chris's sis

I need to stay away from chat even when I'm here, especially when I'm incredibly productive like these days. My research is more important than any other discussion, no matter who.

pjs36

9:38 PM

Here's a fun table to help you remember special trig values.

\begin{array}{c|ccccc}
\theta & 0 & \pi/6 & \pi/4 &\pi/3 &\pi/2 \\ \hline
\sin \theta & \sqrt{0}/2 &\sqrt{1}/2&\sqrt{2}/2&\sqrt{3}/2&\sqrt{4}/2 \\
\cos \theta & \sqrt{4}/2 &\sqrt{3}/2&\sqrt{2}/2&\sqrt{1}/2&\sqrt{0}/2 \\
\end{array}

Mike Miller

I already said that, @TedShifrin

Balarka Sen

yes, it is.

Ted Shifrin

Well, two are better than one, @MikeM.

Mike Miller

Good point. We should ignore each other while we're at it

Owatch

@pjs36 Nice, thanks.

Balarka Sen

9:39 PM

Hey, I'm being left out.

Who'll ignore me?

Mike Miller

Everyone already does.

Ted Shifrin

@Balarka: You weren't very happy when either Mike or I ignored you.

Karim Mansour

paul hammer

Balarka Sen

@TedShifrin No, I wasn't, because it was very serious.

Paul Plummer

Alright @TedShifrin , although it is less sweet than @MikeMiller's offer, because you obviously stole his idea, but I guess it is the thought that counts

Ted Shifrin

9:40 PM

@PaulP: I actually did not see that he had already offered.

Paul Plummer

Sure ;)

Ted Shifrin

But you're a better boy for it, @Balarka :D

Balarka Sen

It did heaps of good to me though.

Paul Plummer

@MikeMiller Not sure if you found out, but I finished that post finally

Balarka Sen

I am really starting to learn math, for one.

Mike Miller

9:42 PM

so I heard, but I haven't had a chance to look yet

it looked interesting

Balarka Sen

oh, ps : we got hit by another, smaller, earthquake.

it was about 5.7.

Ted Shifrin

well, @Balarka, I don't know about "starting."

Damn at the earthquakes.

Paul Plummer

Are these all aftershocks?

Balarka Sen

No, this is a new one.

As I said, there is something seriously wrong with the plates.

Paul Plummer

Its pretty crazy, it also looks like Papua New Guinea was hit by a couple big ones too (last I checked, not sure if they are still happening)

Balarka Sen

9:43 PM

yeah, the whole plate is tilting

a new island woke up at Hokkaido, for example. and Andaman-Nikobar islands got hit by a few quakes too.

Barren, a long thought-to-be-dead volcano erupted after a long time the end of the last month.

all kinds of crazy stuff is happening here

Paul Plummer

@BalarkaSen Curious do people around there say these earthquakes are being caused by sinners and gays(which are the same thing as sinners)?

teadawg1337

The world is a crazy place. For one, the midwest is being hammered by the rain that California so desperately needs

Paul Plummer

Or is that just an American thing

Ted Shifrin

no, @PaulP, we're only a small fraction of the sinners.

right @teadawg

Balarka Sen

@PaulPlummer lol, why in the world would we say that?

Mike Miller

9:46 PM

@teadawg1337: It has been raining pretty hard out here in the last couple days.

It'd be nice if it were consistently so, though.

Paul Plummer

I don't know, you have some crazy religious nuts that blame everything on the gays

I guess it is just an American thing

Balarka Sen

i have never heard of that :P

Ted Shifrin

yeah, there's plenty of international bigotry and religious hatred, but Americans have the bulk of the market on that one.

Clarinetist

^

Paul Plummer

@TedShifrin I don't know, at least they are the sinners that cause everthing, all the other sins don't really matter

Ted Shifrin

9:48 PM

hi @Clarinetist ... you sayin' something?

anon

@PaulPlummer don't worry, we've worked on exporting it in recent years

Mike Miller

Have the bulk of the market, @Ted? At least we don't have Golden Dawn.

Clarinetist

Oh I use that to approve of your statement @TedShifrin

Karim Mansour

lol the world would be a better place if we collect like non-religous rational thinkers in one place and then nuke the rest or something. @TedShifrin

teadawg1337

@MikeMiller Some is better than none, I suppose

anon

9:48 PM

also muslim-majority countries also seem to in the market too

Ted Shifrin

well, @PaulP, just remember that I'm part of that "they"

Balarka Sen

@anon sure thing

Karim Mansour

yeah

Ted Shifrin

there's a ton of hatred all over the world, mostly based on religion

Paul Plummer

@TedShifrin Why are you causing these earthquakes (I know Ted)

Balarka Sen

9:49 PM

they're, like, killing everyone else around the world who writes anything against anything

Ted Shifrin

Because I'm pissed off that California has no water, of course.

Paul Plummer

Oh, okay that makes sense

Karim Mansour

I wonder if like religion wasn't in earth how advanced would human be

anon

@KarimMansour nah. today's china's not really good on human rights for instance, and definitely not mao's. or stalin's. religious fervor doesn't require actual religion.

Balarka Sen

stalin and human rights don't fit together, @anon

:P

Mike Miller

9:50 PM

depends on the humans

pjs36

I think that's his point, Balarka :)

Balarka Sen

@MikeMiller of course, only the communists are the humans

Karim Mansour

yeah I guess there always greed that will bring also humans down etc.

Paul Plummer

Stalin had a lot of human rights...

Balarka Sen

@pjs36 yes, and I'm trying to stess upon it :P

anon

9:52 PM

@PaulPlummer heh

pjs36

oh-ho, now I see!

Balarka Sen

thinking about posting a few snippets from tintin

Paul Plummer

I have never read tintin, heard its good though

Balarka Sen

the first book is pretty good

all of them are good. :)

Clarinetist

All I know about Tintin is that John Williams created the movie score

Mike Miller

9:56 PM

@BalarkaSen lol

Balarka Sen

i dunno what's funny.

Karim Mansour

The axioms for a vector space V over a field F include the two axioms that the
multiplicative group p x act on the set V . Thus vector spaces are familiar examples
of actions of multiplicative groups of fields where there is even more structure (in
particular, V must be an abelian group) which can be exploited. why must V be abelian ? in this case?

I know its in the definition of vector space

but I want to see why given this action why must it be abelian ?

anon

V is abelian by definition of the + operation in it

Ted Shifrin

If you have non-commutative vector addition, that'll destroy the parallelogram law for adding vectors.

anon

lol

Karim Mansour

10:01 PM

@anon I know that see above lol

anon

@KarimMansour huh?

well, if you know the answer to your question, why are you asking it?

Karim Mansour

I mean according to them from the definition of multiplication as group action they can derive that V must be abelian @anon @TedShifrin

that is what I am asking

but I don't see how

Balarka Sen

@KarimMansour because i'd want to make my life easier

anon

@KarimMansour who is "them"?

Karim Mansour

DF @anon

anon

10:02 PM

where

Clarinetist

Dang I wish I didn't have DF packed right now in a box

Karim Mansour

(in particular, V must be an abelian group) maybe I am not interpretating this correctly.

Ted Shifrin

Yeah, mine's in my office in a box, ready to be shipped.

Karim Mansour

maybe because V must be abelian so we can use that in the action somehow

anon

@KarimMansour are you going to tell me where or not?

Paul Plummer

10:03 PM

Is that one of the books you are keeping? @TedShifrin

Karim Mansour

its in section 1.7 of DF

Ted Shifrin

Yes, @PaulP, until I quit thinking about math entirely.

Karim Mansour

example 2 @anon

no that is not allowed @TedShifrin

Paul Plummer

How many are you keeping? Curios to know what are the "must have" books of the great @TedShifrin

Clarinetist

Curious as well

Ted Shifrin

10:04 PM

you can skip the sarcasm or whatever that is, @PaulP.

Karim Mansour

me 2

Ted Shifrin

I'm keeping about 4 boxes' worth. Partly sentimentality ... the collected works of both my advisers, figuring I might try to read some of it.

anon

@KarimMansour no, DF never once says that the definition of multiplication as a group action can be used to prove V is abelian

all it says is that V is abelian

Karim Mansour

yeah I see I interpreted that wrong my mind always try to associate things together

Paul Plummer

Well I am curious @TedShifrin Sorry it did not come out right, or funny

Karim Mansour

10:06 PM

yeah I see now @anon

anon

in fact, the phrase right before the parentheses which you omitted (grrr...) in your quote says there is "even more structure" meaning it's outside of the group action stuff

Karim Mansour

I see

Clarinetist

I might have to get my hands on Artin when I move to Madison

Karim Mansour

thank you @anon

Balarka Sen

there are well-defined field actions on nonabelian groups, just to add

Ted Shifrin

10:06 PM

@Artin is a lovely book, @Clarinetist.

Balarka Sen

you're pinging artin, lol.

Clarinetist

It will be nice to study math for a change.

Balarka Sen

I agree that it's a great book

Ted Shifrin

oops ... time to go cook dinner ... and time for Balarka to go to sleep!

Paul Plummer

It would be funny if he came in...

Clarinetist

10:07 PM

Have fun @TedShifrin

Ted Shifrin

He was one of my favorite teachers, @PaulP.

Paul Plummer

Oh cool

Karim Mansour

whats difference between artin and DF

is artin better?

Ted Shifrin

Crazily ... he must be almost 90, but he's still teaching at MIT.

Clarinetist

New Years' resolution: relearn two semesters of Abstract Algebra and/or learn C++

Balarka Sen

10:08 PM

artin takes a rather geometric approach

Karim Mansour

I mean what to move on after I finished DF ?

Paul Plummer

Was there something in particular you liked about how he taught?

Balarka Sen

finishing D-F can be tough, @Karim

Ted Shifrin

Artin tries to integrate algebra and linear algebra, and shows lots of deep mathematics intertwined with it all.

Balarka Sen

and number theory, @Ted :P

Karim Mansour

10:09 PM

yeah not this summer ofcourse maybe this year @BalarkaSen

but I am planning to do alot this summer

Paul Plummer

The unity, making connections, that is what I love about math. It is great when a teacher pulls it off

Clarinetist

@PaulPlummer Agree completely

Ted Shifrin

He communicated a more satisfying conceptual framework, whereas I find that most algebra books (particularly Herstein) can be much more formal and symbol-manipulative.

Karim Mansour

good I should also read artin seems like a good book

Ted Shifrin

I've always striven to show connections and make unifications, but most of the math curriculum is not designed that way.

Balarka Sen

10:10 PM

It sure is.

Clarinetist

@TedShifrin I find that very, very sad :/ I felt like with my undergrad lectures, the classes seemed more disjoint than they should have been

pjs36

Curriculum... what a dirty word.

Ted Shifrin

no, @pjs36, you're thinking of speculum.

Balarka Sen

Ok, now I have to go sleep.

Paul Plummer

Hahahaha

Ted Shifrin

10:11 PM

Night, @Balarka :)

pjs36

@TedShifrin I think you read my mind, I was thinking of speculi!

Karim Mansour

night @BalarkaSen

Ted Shifrin

@Clarinetist: Their political leanings notwithstandings, mathematicians tend to be very conservative and lazy in the courses they structure/teach.

Clarinetist

sigh

I was probably one of the most rebellious people in the department when I was over there

Ted Shifrin

BTW, @pjs36, you'd best work on your Latin declensions. Wrong plural.

Paul Plummer

10:13 PM

Well have a great dinner @TedShifrin, I am going to go read a bit

Ted Shifrin

LOL ... yes, I need to shaddup.

Clarinetist

Did two presentations on musical group theory (had no assistance whatsoever in making those) and another one on measure-theoretic probability and its connections to complex analysis, differential equations, and of course, undergrad probability

pjs36

OK, I'll spend a few days working on that. The things I forget :P

Clarinetist

I took advanced music theory before I took Abstract Algebra, and then I realized that musical serialism is just working with the group $(\mathbb{Z}_{12}, +)$.

That was easily the most enlightening moment of my undergraduate.

Karim Mansour

wtf

super nice @Clarinetist

I never played any instrument in my life but that would be very nice moment I would imagine !

Clarinetist

10:18 PM

Yeah, I didn't realize it, but music theorists unconciously use $(\mathbb{Z}_{12}, +)$ to construct tone rows @KarimMansour

pjs36

@TedShifrin Ah, of course; the -um goes to -a, duh!

Karim Mansour

I see very interesting !

Owatch

I'm done. I'm going to destroy something.

Last error of the evening was to take a value for T from the last question and draw my graph wrong.

I spent a ton of time writing this over neatly.

And it's in pen.

Clarinetist

@KarimMansour What music theorists do is they map each of the 12 notes of the chromatic scale to an element of $\mathbb{Z}_{12}$. Let $S = \{C, C\#, D, D\#, E, F, F\#, G, G\#, A, A\#, B\}$ be the set of notes in the 12-tone chromatic scale. The mapping $i: S \to \mathbb{Z}_{12}$ defined by $i(C) = 0$, $i(C\#) = 1$, $\dots$, $i(B) = 11$ is known in music theory as fixed-zero notation.

Karim Mansour

oh

Clarinetist

10:24 PM

@KarimMansour A tone row is a permutation of $i(S)$ (i.e., the image of $S$ under by $i$).

Karim Mansour

I guess also the more symmetric the tones are the more it will rhyme too

@Clarinetist

Clarinetist

@KarimMansour And a set (musicians have their own "set theory") is merely a subset of $i(S)$ with two or more elements.

Karim Mansour

I see

Clarinetist

Now here's what musicians like doing

Karim Mansour

I was wondering is it possible to generate music that ryhmes by computer based on permutations of i(S) that is kinda of symmetric ?

@Clarinetist ?

Clarinetist

10:29 PM

Consider a $n$-tuple of elements of $i(S)$, say $(t_0, \dots, t_n)$, $2 \leq n \leq 12$ an integer. Then the transposition operator by $k$-half steps $\mathbb{T}_{k}: \mathbb{Z}_{12}^{n} \to \mathbb{Z}_{12}^{n}$ is merely $\mathbb{T}_{k}(t_0, \dots, t_n) = (t_0 + k, \dots, t_n + k)$, all addition done in $\mod 12$.

@KarimMansour Not sure what you mean by "rhymes by computer"

Karim Mansour

I mean is it possible to generate music by computer using that method ?

Clarinetist

@KarimMansour Of course!

Whether it's pleasing to the ear though, that's a different question :P

Karim Mansour

yeah :D

Clarinetist

Here's a classic example of tone rows

Schoenberg - Three Piano Pieces, No. 1 (with sheet music)

►

pjs36

Ah, of course - with a name like Clarinetist, it just makes sense that you know this stuff! Nice :)

Clarinetist

10:32 PM

and their transposition (and whatever other bijections you may want to create)

Karim Mansour

I see

that is very interesting

I like when kinda of real math has applications like that in real world

Clarinetist

Now just a FYI, for your average listener, you do not use tone rows to create pleasing music, lol

Karim Mansour

I would like to take classes like that on music theory

but I choosed physics as my science elective

which is nice too

you know I am doing mechanics aswell now it has I see many stuff that uses abstract algebra too @Clarinetist

Clarinetist

I wish I knew at least some Physics. All I can tell you about is $F= ma$ lol :P.. and I think you can apply Calculus on this when $a$ is variable.

anon

when a is a function of time

Chris's sis

10:39 PM

No r9m, no robjohn around ...

Clarinetist

If I ever find time, I should at least learn some calculus-based physics :P

Idk how I bypassed Physics all of these years. Introductory chemistry has always been my forte

Karim Mansour

yeah that would be nice @Clarinetist

Paradox 101

Hi, can anyone explain how the closure of rationals as a subset of real numbers is the set of reals?

anon

@Paradox101 every real number is a limit point of Q, there will always be a sequence of rationals converging to it. for instance, 3, 3.1, 3.14, 3.141, 3.1415, ... will converge to pi.

Chris's sis

A book I read in one, two days (I forced myself saying two days for not being flagged)

books.google.ro/…

After I publish my book on integrals, series and limits, I make time to come here and talk some more about advanced analysis. I have many questions to ask.

I'm out.

Karim Mansour

10:51 PM

that is dedekind cut right @anon ?

or similiar to it

anon

no, cauchy sequence

Karim Mansour

I see

I never learned cauchy sequnce as a method of completing Q we learned dedekind cut in my set theory class

anon

cauchy sequences are far better

they generalize to completing any metric space

Karim Mansour

I see

Paradox 101

@anon ok thank you. also how do we know that interior of rationals in reals is empty? I know how the interior is defined but even then I have a little bit of difficulty picturing it.

anon

10:53 PM

what does it take for a point to be in the interior of Q?

Paradox 101

any point x is an interior point in D if it has an open neighborhood in D?

anon

in a metric space you might as well say an open ball instead of open nbhd

(every open ball is an open nbhd, and every open nbhd of a point contains an open ball around the point)

what are the metric balls in R? they're intervals right?

does Q contain any intervals?

Paradox 101

well yes it would but there would be only rationals in it?