Choosing Topics

[James Crook]

Has everyone visited at some time the khan academy  maths sections, and if so, what were your reactions?

Is it helpful, or does it just get in the way?

This isn't just about the maths.  We also want to look at how to improve maths teaching for people who are streets ahead of the people around them of the same age.

Are there useful topics in the maths sections? 

Is it all too easy?  Have you already found some text on the internet to work through that would be better?

I am more than happy to facilitate people studying a book online, or equally guide through a particular section of Khan academy.  I'd like to start mapping out a path for each person, so that we have some structure to what we do.  If there are online books or online lectures you would like to work from, that look like they are at the right level for you, please send links.  

--James.

[shr...@gmail.com]

Also interesting is OCW by MIT. I would advise you all to check it out.

[Jonathan Joo]

Here's my fact: (sorry it's so late)

There are infinities of different sizes! This was proven by Cantor, using proof by contradiction. It basically goes as follows:

If all reals were countable, then all real numbers could be placed in a one-to-one correspondence with the list of all integers. 
Now, from this list, we can then construct a number that differs from all the other real numbers in the Nth decimal place by the Nth digit. However, this is contradictory- if the original set contained all real numbers and they were one-to-one with all integers, then it shouldn't have been possible to construct this number.

Thus, it is proven that there are more real numbers than integers, and that the list of all reals are not countable. It's interesting, because who knew there were different sizes of infinity?

^If that was a bad explanation, here's another fact that's interesting, and much simpler:

.999999...=1

Proof:

Let x=.999...

10x=9.999...

10x-x=9.999...-.999...

9x=9

x=1

So, .999...=1!

That's pretty cool, I think.

-Jonathan

---

From: James Crook <james....@gmail.com>
To: l2lea...@googlegroups.com
Sent: Friday, June 17, 2011 6:13 PM
Subject: Choosing Topics

[James Crook]

Jonathan wrote:

> Here's my fact: (sorry it's so late)

> There are infinities of different sizes! This was proven by Cantor, using proof by contradiction. It basically goes as follows:

> If all reals were countable, then all real numbers could be placed in a one-to-one correspondence with the list of all integers. 
> Now, from this list, we can then construct a number that differs from all the other real numbers in the Nth decimal place by the Nth digit. 

> However, this is contradictory- if the original set contained all real numbers and they were one-to-one with all integers, then it shouldn't 

> have been possible to construct this number.

> Thus, it is proven that there are more real numbers than integers, and that the list of all reals are not countable. It's interesting, because 

> who knew there were different sizes of infinity?

Yes, that's cool.

For me it has the same kind of feel as "The set of all sets that do not contain themselves".

There are approaches to mathematics that get round the problem of multiple kinds of infinity.  Essentially they work by saying that you are not allowed to take an infinite number of steps in any mathematical process - though you can take as many steps as you like, if it is still some finite number.  That makes listing all the numbers and changing their Nth digits illegal by these rules. That the normal rules of mathematics allows you to do that relies on the axiom of choice .  So logicians explore alternative versions of mathematical rules - looking to better understand what mathematics can and cannot do. 

--James.

[James Crook]

As well as OCW, Khan and online textbooks, there are also the maths lectures at OMC. 

I'd like to hear what people would like to work on and how the group can help everyone become better mathematicians.

I'd like to start mapping out a path for each person, so that we have some structure to what we do.

--James.

[Kelsey]

Actually, I think I might want to work on real analysis if possible.
It would be very helpful for the next school year.

On Jun 18, 12:44 pm, James Crook <james.k.cr...@gmail.com> wrote:
> As well as OCW, Khan and online textbooks, there are also the maths lectures
> at OMC.
>
> I'd like to hear what people would like to work on and how the group can
> help everyone become better mathematicians.
> I'd like to start mapping out a path for each person, so that we have some
> structure to what we do.
>
> --James.
>

> On 17 June 2011 23:20, shri...@gmail.com <shri...@gmail.com> wrote:
>
>
>
>
>
>
>
> > Also interesting is OCW by MIT. I would advise you all to check it out.
>
> > ----- Reply message -----
> > From: "James Crook" <james.k.cr...@gmail.com>
> > Date: Fri, Jun 17, 2011 5:13 pm
> > Subject: Choosing Topics
> > To: <l2lea...@googlegroups.com>
>

> > Has everyone visited at some time the khan academy<http://www.khanacademy.org/>

[James Crook]

Kelsey,

That's great.  Please create a page on the wiki like the one Shri has created for differential equations.

You can leave the pre-requisites section blank for the moment, but I would like an additional section titled 'notation/terminology'.  In it we will put symbols for and explanations of:

'For all x'
'There exists'
'Such that'
'Without loss of generality'
'Without much loss of generality'
'Supremum'
'Infimum'
'Is an element of'
'The set of Natural numbers'
'The set of Real numbers'
'The set of Integers'


If you can get the symbols for the first two in already, that would be great.  There is a mnemonic for remembering those two - For ALL is an upside down A.  There EXISTS is a back to front E.  This comes from the days when books were typeset, and the printers just used these letters upside down.

Shri please give some advice/experience on how you learned how to do the notation using <math></math> in wiki.  When you want a new symbol that you don't know already, how do you find it?  Everyone will need to know this to advance in maths, so this will help everyone.

--James.

[Shri R Ganeshram]

Hi guys,

I personally love this guide: http://tobi.oetiker.ch/lshort/lshort.pdf .
Whenever I have an easy problem though, like not knowing a symbol, a google search is generally sufficient, but as a starter, having that guide opened in a new tab will help tremendously.

-Shri

________________________________________
From: James Crook [james....@gmail.com]
Sent: Tuesday, June 21, 2011 4:52 AM
To: Shri R Ganeshram
Subject: Fwd: Choosing Topics

Shri,

Shri please give some advice/experience on how you learned how to do the notation using <math></math> in wiki. When you want a new symbol that you don't know already, how do you find it? Everyone will need to know this to advance in maths, so this will help everyone.

What I mean here, is how do you learn about LaTeX. What internet resources have you found the most useful for you for learning LaTeX?

--James.

---------- Forwarded message ----------
From: James Crook <james....@gmail.com<mailto:james....@gmail.com>>
Date: 21 June 2011 01:45
Subject: Re: Choosing Topics
To: l2lea...@googlegroups.com<mailto:l2lea...@googlegroups.com>

Kelsey,

That's great. Please create a page on the wiki like the one Shri has created for differential equations.

You can leave the pre-requisites section blank for the moment, but I would like an additional section titled 'notation/terminology'. In it we will put symbols for and explanations of:

'For all x'
'There exists'
'Such that'
'Without loss of generality'
'Without much loss of generality'
'Supremum'
'Infimum'
'Is an element of'
'The set of Natural numbers'
'The set of Real numbers'
'The set of Integers'

If you can get the symbols for the first two in already, that would be great. There is a mnemonic for remembering those two - For ALL is an upside down A. There EXISTS is a back to front E. This comes from the days when books were typeset, and the printers just used these letters upside down.

Shri please give some advice/experience on how you learned how to do the notation using <math></math> in wiki. When you want a new symbol that you don't know already, how do you find it? Everyone will need to know this to advance in maths, so this will help everyone.

--James.

On 20 June 2011 22:51, Kelsey <nephyth...@gmail.com<mailto:nephyth...@gmail.com>> wrote:
Actually, I think I might want to work on real analysis if possible.
It would be very helpful for the next school year.

On Jun 18, 12:44 pm, James Crook <james.k.cr...@gmail.com<mailto:james.k.cr...@gmail.com>> wrote:
> As well as OCW, Khan and online textbooks, there are also the maths lectures
> at OMC.
>
> I'd like to hear what people would like to work on and how the group can
> help everyone become better mathematicians.
> I'd like to start mapping out a path for each person, so that we have some
> structure to what we do.
>
> --James.
>

[James Crook]

Kelsey -

I've created an outline of the page.

http://onlinemathcircle.com/wiki/index.php?title=Real_Analysis

Do you have an online book or video course you want to work from?  I've looked online and so far not found anything that I think is good. We may need to choose a book and work from that.  Possibilities include Rudin, Apostol, and Understanding analysis By Stephen Abbott.  We could also use the contents page and then use wikipedia/planet-maths for each topic, though that wouldn't be as good.

--James.

[James Crook]

Kelsey,

I've created the page, but I've not seen any edits from you or any e-mails about it.

Let me know if you want to study this still - and if so what's the next step.

--James.

[Abe Rabin]

I'm curious, would anybody know where I could find accessible open problems in geometry or ideas? I'm planning on submitting a paper for Intel for geometry but I'm having a terrible time figuring out an idea for  the project.

Thanks in advance.

[Shri Ganeshram]

Hey Abe,

There are quite a few problems in the AoPS forum, and I believe you can find some on wikipedia.

[Abe Rabin]

Oh thanks dude. Would they make good research problems?

[Shri R Ganeshram]

The open problems definitely would.
______________________________________
From: l2lea...@googlegroups.com [l2lea...@googlegroups.com] On Behalf Of Abe Rabin [honest....@gmail.com]
Sent: Friday, June 24, 2011 10:58 PM
To: l2lea...@googlegroups.com
Subject: Re: Choosing Topics

Oh thanks dude. Would they make good research problems?

On Fri, Jun 24, 2011 at 10:11 PM, Shri Ganeshram <shr...@gmail.com<mailto:shr...@gmail.com>> wrote:

Hey Abe,

There are quite a few problems in the AoPS forum, and I believe you can find some on wikipedia.

On Jun 24, 2011 8:10 PM, "Abe Rabin" <honest....@gmail.com<mailto:honest....@gmail.com>> wrote:
> I'm curious, would anybody know where I could find accessible open problems
> in geometry or ideas? I'm planning on submitting a paper for Intel for
> geometry but I'm having a terrible time figuring out an idea for the
> project.
>
> Thanks in advance.