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Geekland - Not for the faint of heart

Post by Sam Gamgee » Sun Apr 11, 2004 11:41 pm

To spare others, and make geeky conversations more concise and ordered, I have now made a Sacred Land dedicated to The Geek, to which I proudly admit my membership.

*No ridicule allowed. If YOU read this, it's YOUR own fault, because you've been explicitly forewarned*

All hail the Geeks!
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Post by Sam Gamgee » Sun Apr 11, 2004 11:42 pm

Zero^(infinity) = 2? Theremin, what are you talking about? That's not legal. And what's phi doing here? Doesn't that stand for e/6 or something having to do with a 6 and an e?

What did your teacher do, now? Are you learning Cantor theory? I thought you were taking calculus.
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Post by Theremin » Mon Apr 12, 2004 4:38 pm

I'm sorry, zero^(infinity) is a determinate form, what I meant to say is (infinity)^zero. For instance, if you take the limit as n approaches infinity of [2*(n)^(e^(-n))], you would get (infinity)^zero if you used direc t substitution of infinity. However, if you use log differentiation, you end up with an answer of 2. If the 2 were a 7, the limit would still be infinity^zero with direct substitution, but would actually be 7.
Yeah, phi is the golden ratio, 1.61803399. I was just using it as an example for a far out number. According to google, cantor theory confuses me at first glance and i have not learned it yet... What calculus are you in; I'm in AP-BC. Maybe were doing that after the AP test...
I gotta admit that I'm a little bit confused
Sometimes it seems to me as if I'm just being used
Gotta stay awake, gotta try and shake off this creeping malaise
If I don't stand my own ground, how can I find my way out of this maze?

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Post by Sam Gamgee » Mon Apr 12, 2004 6:05 pm

I finished calc last semester - we take three semesters at our school, and there are no AP classes or any kind of honors stuff, so you could be getting in to calc details i never learned. You can't directly substitute infinity! You can use it symbolically if you mean it in a very lose way, but that's not allowed. (hold that thought for a minute until the end.)

no wonder cantor theory is confusing - our math teacher says that most math majors don't touch the stuff until at least gradtuate school. I don't think it's because it's hard, necessarily, but that it's obscure. It's talking about set theory, and how you deal with infinite sets. A lot of it revolves around one-to-one correspondence, which is a way of telling "how much" without knowing it. You pair off one element from one set to another element of the second, and if you end up with nothing left, there is the same amount - you never need to know how much there is. Nevertheless, the very fact that you're asking "how much" implies that you are assigning a particular "cardinality" to the infinite set - you are giving the inifinite elements a certain number, so to speak. That's what a transfinite number is - a particular infinity. There are infinately many infinities.

EX: {natural numbers (1,2,3,...)} (N) is in one-to-one correspondence with {points in harmonic series (1, 1/2, 1/3, 1/4,...)}
You can manipulate it so that you can't pair them up if you start out the wrong way, but you only need one way in which it works for them to be in one-to-one correspondence.
Proof: Pair them off. 1, 1. 2, 1/2. 3, 1/3 (etc.)
There. I found one way, so they are in one-to-one correspondence. That means that there is the "same amount" of natural numbers and member of the points in the harmonic series.

Back to infinity to the zero or whatever: You can't do that BECAUSE there are a lot of different forms of infinity! There are infinitely many! I've given you one example, which cantor called "aleph null", but then there was another one called "aleph one" in which the interval (0,1) is a memeber. aleph one is greater than aleph null. So saying "infinity" is not specific enough.

Ok, sorry, you probably don't understand half of that - we spent a lot of time to get there, and i'm bad at explaning. Anyway, if you are interested, I'll gladly clarify, but if you're not that's fine. It's awesome stuff though. ^_^
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Post by Theremin » Mon Apr 12, 2004 9:34 pm

You can't directly substitute infinity! You can use it symbolically if you mean it in a very lose way, but that's not allowed. (hold that thought for a minute until the end.)
Yeah, that's why you have to take the limit... I know about all the stuff you are talking about besides Cantor theory, which we havn't talked about. I'm just bad at explaining it too...
The problem with AP classes is that they are constantly in danger of just becoming prep classes for the test. However, you do get a big 3-week or so chunk of cool stuff after the test, i.e. watching movies and burning stuff in parabolic mirror shells. :twisted:
I gotta admit that I'm a little bit confused
Sometimes it seems to me as if I'm just being used
Gotta stay awake, gotta try and shake off this creeping malaise
If I don't stand my own ground, how can I find my way out of this maze?

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Post by Bnielsen » Mon Apr 12, 2004 10:21 pm

Theremin wrote:watching movies and burning stuff in parabolic mirror shells. :twisted:
WOW! thats the first thing in this thread that I understood!
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Post by Theremin » Mon Apr 12, 2004 10:33 pm

Who doesn't enjoy a good burning? :lol: And if there's math involved, all the better! :D
I gotta admit that I'm a little bit confused
Sometimes it seems to me as if I'm just being used
Gotta stay awake, gotta try and shake off this creeping malaise
If I don't stand my own ground, how can I find my way out of this maze?

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Post by Sam Gamgee » Mon Apr 12, 2004 11:24 pm

Brian, leave now. Didn't you read the disclaimer? ;)

To get UBERgeeky, I just have to press this point to make sure we're at an understanding: taking a limit is NOT plugging in infinity. You can't plug in infinity. You can make a number grow without bounds, or approach a number in such a way that it could pass whatever number you wanted in accuracy, but that's different from saying that you're actually plugging infinity in. Right?

BTW, I had only been told that phi existed and had something to do with e - but what exactly is it?
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Post by Theremin » Tue Apr 13, 2004 3:59 pm

To get UBERgeeky, I just have to press this point to make sure we're at an understanding: taking a limit is NOT plugging in infinity. You can't plug in infinity. You can make a number grow without bounds, or approach a number in such a way that it could pass whatever number you wanted in accuracy, but that's different from saying that you're actually plugging infinity in. Right?
Of course! If you plug in infinity, its like...something crazy and impossible that I can't even think of!

Phi is the "Golden Ratio," approximately equal to 1.61803399, or (1+sqrt5)/2. It is found a lot in nature, such as the ratio of the distance from to top of your head to your chin to the distance from the left to right side of your head at eye level. Also, taking the limit as n approaches infinity, A-sub-n divided by A-sub-(n-1) approaches phi (if the function is the fibbonacci series). Its pretty cool stuff :D
I gotta admit that I'm a little bit confused
Sometimes it seems to me as if I'm just being used
Gotta stay awake, gotta try and shake off this creeping malaise
If I don't stand my own ground, how can I find my way out of this maze?

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Post by Sam Gamgee » Tue Apr 13, 2004 6:37 pm

Plugging in infinity is nonsense. There's more than one value of infinity - that's why you can't talk about it.

What you said about phi was cool! i never knew that. :) So, did you learn any other random fun geeky things recently?
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By his blindness to see that the warmth of his being
Is promised for his seeing, his reaching so clearly[/i][/color]

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Post by Theremin » Tue Apr 13, 2004 10:05 pm

Well, this is pretty scarey... :twisted:

If you take the area under 1/x from 1 to infinity, you get an infinite area. however, here is the scarey part (BOO!!!), if you rotate that area around the x-axis, it has a finite volume, pi units cubed... :shock: :shock: :shock: :cool:
I gotta admit that I'm a little bit confused
Sometimes it seems to me as if I'm just being used
Gotta stay awake, gotta try and shake off this creeping malaise
If I don't stand my own ground, how can I find my way out of this maze?

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Post by Sam Gamgee » Tue Apr 13, 2004 11:55 pm

Hey! We did that very thing! Our teacher taught us a very valuable lesson on a similar situation, too. Never say you won't use this stuff in real life, because

REAL LIFE SCENARIO A (and I quote from my teacher...):
"If somone tells you to build a funneloid whose base is raduis 1 and whose sides are equal to 1/x squared and area of <pi>/3, you can say 'Aha! That’s a trick question. That’s an infinite function.' It’s also a great pickup line."

REAL LIFE SCENARIO B:
"If some guy is hitting on you, just say, 'Oh, I’m sorry, I have to go build a funneloid whose base is raduis 1 and whose sides are equal to 1/x squared and area of <pi>/3.' Then he’ll go away. Unless his name is Melvin. Then he’ll say (geeky voice), 'No, you can’t do that because it’s infinite. See?'"
(you had to be there to get the full effect of the Melvin impression, but still)

This is all, of course, my cool Latin/Calculus teacher who proofread Nex Tex and whom I converted to Glass Hammerism. ^_^ Only I don't have him this year, which is sad. :cry: He would say the strangest things in class so we would write them down in a huge list and give it to everyone in the class at the end of the year. Nobody laughed harder reading it over than he did. :lol: Oh yes, this was also the teacher who told me to get a life when i was talking about learning Quenya. "Stephanie, I'm going to tell you something that starts with 'vitam' and ends in an imperative..." He had to say it to me in Latin, of couse. :lol:

But about what you were saying, the real question is, if the area is infinite, which infinity is it? :muahahahahah: :twisted:
Ok, it's aleph one. But it just sounded evil to ask.
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Post by Theremin » Wed Apr 14, 2004 3:43 pm

If some guy was hitting on me, I wouldn't be worrying about the function F(x)=1/x :lol: .


Sorry I couldn't resist...
I gotta admit that I'm a little bit confused
Sometimes it seems to me as if I'm just being used
Gotta stay awake, gotta try and shake off this creeping malaise
If I don't stand my own ground, how can I find my way out of this maze?

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Post by Sam Gamgee » Wed Jul 14, 2004 11:47 am

BRIEF SUMMARY OF CANTOR THEORY BASICS
from an unproffesional student who mostly knows what she's talking about, for Druid

1. If you want to figure out whether you have the same amount of something, but you can't count very high, there's a way to do it. Just stack your items in two piles, then take one item from one pile, and one from the other and then set them aside. Then keep doing that a pile runs out. If one runs out faster, it has less, but if they run out at the same time, that means that they are equal - they are in one-to-one correspondence.

2. example A
So let's apply this theory to infinite sets, just for kicks. I wonder what would happen if you took an infinite set like {1, 2, 3, 4,...} and paired it with an infinite set like {1, 2, 3, 5, 7, 11...}. Would we be able to find at least one way to pair them so that they'd be in 1-1 correspondence? Let's try it:
1, 2, 3, 5, 7,... (leaving out 4, 6, 7, 8, etc)
1, 2, 3, 5, 7...
Hm... This one's no good because we have a whole bunch of numbers left over in the first set - saying it's bigger. That doesn't show one-to-one correspondence. How about this:
1, 2, 3, 4, 5, 6,...
1, 2, 3, 5, 7, 11...
Look at that! They're perfecly paired! Sure, one's moving up the line of numbers slower than the other... But does it matter? We're never going to run out of numbers, so the pairing works. I mean, you tell me any number in either set, and I could tell you what it pairs up to in the other one. DING!

example b:
{whole number sqaures} with {all whole numbers (pos. and neg.)}
We need to find a way to list them...
0, 1, 4, 9, 16, 25...
1, 2, 3, 4, 5, 6... doesn't work because we left out 0 and all the negatives.

0, 1, 4, 8, 16, 25, 36...
0, 1, -1, 2, -2, 3, -3..

We've done it! Don't believe me? Well, can you find any number that I've missed? Nope. You give me any number from either set and I'll show you its proper match. They're all in there.

example c
Let's try pairing the whole numbers with the interval between 1 and 0, (0, 1)
1, 2, 3, 4, 5...
uh... well, i could start with fractions: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, etc... But then what about all those numbers in between? Like <pi>-3, .1415926535898... I would be totally missing all of those.

The fact is, you can't even begin to list the points in (0, 1). So no matter what pairing you try, you'll be missing infinitely many more. There's a proof of this, but it's kind of hard to explain online, as Druid knows well. If you really want it, I could do it again, but not guarantees on how understandable it would be.

CONCLUSION: There is a type of 1-1 correspondence among infinite sets that occurs when you can LIST the members of the sets. When you can't list the members of an infinite set, it seems to be a new type of infinity. Cantor called the listable sets "Aleph Null", and the unlistable sets "C1".

3. More sets in C1, which are really tricky to prove without visual aid:
{any interval, including, or not including the endpoints}
{the number line}
{the points on a 2 dimmensional plane}
{the points on a fifty-seventh dimmensional plane, whatever that is}
{points in any-sized circle}

Quick plausibility argument that (1,0) and (1, 2,000,000,000) are in one-to-one correspondence:
Draw one line to represent (1,0)
Now draw a bigger line below (so that their centers are in the same spot) it to represent (1, 2,000,000,000)
Now draw a line from center to center - those center points correspond
Draw a line from one third of the top line and extend it through that mid line. It will hit one and only one point on the bottom line.
Now you can keep drawing lines - you pick any point on the bottom, extend the line through, and it will hit one and exactly one point on the top. Vice versa. No matter what point you want to correspond, there is one and only one point that the line will touch, and that is the corresponded point.

See, told you I was bad at explaining these things. But that's a basic plausibility argument, even if it's not exactly a proof.

4. As I cannot prove to you because it would take way too much work (and i probably wouldn't explain it well anyway), there are also infinitely many more infinities higher than C1, but they are so beyond our comprehension we can't say much more than that.

So, you like this stuff? Cantor Theory. It rocks. And I've only mentioned the basics of it - it gets deeper and more eye-opening as you go on. I hope I haven't murdered the explanation. If you want more, take a class on it, or read a book or something. It's awesome!!

There's the piece of Geekdom for the day.
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By his blindness to see that the warmth of his being
Is promised for his seeing, his reaching so clearly[/i][/color]

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Post by Bnielsen » Wed Jul 14, 2004 11:50 am

Sam Gamgee wrote:There's the piece of Geekdom for the day.
Day?! Thats enough to last at least a month, eh? :wink:
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