Mathematics

10:12 AM

@JonasTeuwen No you told me that I had to have an xmassy avatar and I told you that I couldn't be bothered to spend time on making one. Then a way of making one in less than a minute occurred to me : )

user19161

@chris You really have no idea how your avatar changed even though you didn't change your login?

Matt N.

Hi btw.

user19161

@MattN. Nice bear there. I guess you ain't gonna email me, but that's OK. =)

Chris's sister

@JasperLoy: honestly? no, but this perhaps I didn't focus on it that much .:D

user19161

I wonder how Nameless types so quickly.

Matt N.

10:16 AM

@JasperLoy Thanks. Well, I don't have a reason to email you.

Old John

@JasperLoy I hate typing races, since I always seem to lose - I should stick to answering unpopular questions that most people aren't interested in

Chris's sister

@anon: maybe you're interested in iugaza.edu.ps/ar/periodical/articles/…

user19161

@OldJohn I think maybe he uses MathType like Pedro.

Old John

@JasperLoy Maybe, yes

user19161

@OldJohn How's the kitchen?

Old John

10:20 AM

@JasperLoy All finished, and all working beautifully, thanks!

user19161

@OldJohn Hmm, I think I should be able to reach 12k this year. After that, I'll see if I want to continue.

Old John

@JasperLoy I have no targets - not really interested in gaining rep as an aim :)

user19161

@OldJohn I just discovered that Math and TeX opted out for the Christmas hats!

user19161

If you go to Eng you will see that some avatars have strange hats on them.

BenjaLim

@OldJohn There's exciting stuff here: math.uconn.edu/~kconrad/blurbs/gradnumthy/totram.pdf

Dumb Cow

10:25 AM

I'm just sitting at 1k :(

user19161

Not constructive. math.stackexchange.com/questions/264054/…

Old John

@BenjaLim I will try to read that later - but it looks like he is finding the ring of integers of a cubic field in a simpler way than I have ever seen before :)

@BenjaLim whenever I get a link like that, I often have a look at what else can be found there by deleting the last part of the URL - like this :)

Chris's sister

I just upvoted the guy.

I don' t understand at all the downvoting attitude. They are all questions.

user19161

He should just delete the question and gain back the rep.

Old John

Must go and do stuff -later folks

Chris's sister

10:34 AM

@JasperLoy: maybe this is the way he wants to learn mathematics. It's a start. :-)

Dumb Cow

Proof by contradiction?

user19161

@DumbCow Yes. What about it?

Dumb Cow

> If $x$ and $y$ are two arbitrary real numbers with $x < y$, prove that there is always a real $z$ such that $x < z < y$.

Will this be a proof by contradiction?

user19161

@DumbCow You can prove this by coming up with a specific z.

Dumb Cow

@JasperLoy specific $z$?

user19161

10:45 AM

@DumbCow Well, $x<\frac{x+y}{2}<y$. QED.

Dumb Cow

I think I got it...

@JasperLoy no, a rigorous proof. As you could see in yesterday's question, I am a fan of rigorous proofs =)

user19161

Look, this is a rigorous proof.

user19161

If you already know all the usual properties of the real numbers that will suffice.

Dumb Cow

Okay, let me get my proof here:

user19161

Otherwise you have to say which axioms you used and what you are allowed or not allowed to use.

Dumb Cow

10:47 AM

Since the set of real numbers is unbounded above, we will always have a $z$ such that $x < z$.

user19161

No, no, no.

Dumb Cow

Okay.

user19161

I have said everything in the above already.

user19161

@DumbCow If you want to use that, have you shown that it is unbounded above?

user19161

That is why I say you really need to tell us what axioms you use and what results you can or cannot use.

Dumb Cow

10:49 AM

I can.

But I have a little doubt.

user19161

What's wrong with using the inequalities in R?

user19161

What's not rigorous about it?

Dumb Cow

Actually I got it =)

user19161

In fact, it is not only rigorous, it is the best proof as it is constructive.

Dumb Cow

Thanks @JasperLoy

user19161

10:51 AM

We even explicitly construct such a z.

user19161

Often we prove existence and uniqueness theorems, but these don't give us an explicit object!

user19161

The exhibition of an explicit z settles the matter conclusively. QED.

Dumb Cow

Okay, what if the statement was changed?

user19161

To what?

Dumb Cow

> Prove that there are infinitely many such $z$'s...

user19161

10:53 AM

Ah, same thing, just use the same technique repeatedly.

Dumb Cow

Oh.

user19161

Keep bisecting.

Dumb Cow

I see

user19161

@DumbCow Did you check out Paul's notes?

user19161

I think those are good for you. You should read those first.

Dumb Cow

10:56 AM

Yes of course.

Is there a proofs section there?

user19161

Hmm, I don't know.

Dumb Cow

At least I haven't seen it.

user19161

But the best way to learn proofs is to look at proofs!

user19161

Study more proofs first before you study about proofs!

Dumb Cow

The only thing I hate about Paul's Notes is the website he is connected to.

user19161

10:57 AM

Don't fly before you walk.

user19161

I see many people trying to fly here, I think their wings will break very soon...

Dumb Cow

But a cow doesn't have wings, bro. I'm not a unicorn...

user19161

They can talk about very deep topics, but eventually they will get stuck and not be able to produce anything...

Dumb Cow

I started Calculus at Paul's Notes.

Transcript for

$Mathematics$ Mathematics