Flash Flash Revolution: Community Forums

Flash Flash Revolution: Community Forums (http://www.flashflashrevolution.com/vbz/index.php)
-   Critical Thinking (http://www.flashflashrevolution.com/vbz/forumdisplay.php?f=33)
-   -   real critical thinking (http://www.flashflashrevolution.com/vbz/showthread.php?t=47171)

Deeks 07-31-2006 06:27 PM

real critical thinking
 
prove to me that the square root of two is an irrational number. First one to tell me a correct answer wins 100 creds, dont leave any reasoning unproved.

Patashu 07-31-2006 06:28 PM

Re: real critical thinking
 
A rational number can be expressed in the form p/q, where both p and q are non-zero integers.

The square root of two...ah, forget it.

JKPolk 07-31-2006 06:28 PM

Re: real critical thinking
 
easy
2^.5 * 2^.5 = 2

where's my credits

Deeks 07-31-2006 06:30 PM

Re: real critical thinking
 
aww patashu you know it. say it all, jkpolk not even close, patashu gave everyone a start

T0rajir0u 07-31-2006 06:44 PM

Re: real critical thinking
 
The polynomial x^2 - 2 is irreducible over Q[x] by Eisenstein's criterion.

Pay up, bitch.

The_Q 07-31-2006 06:47 PM

Re: real critical thinking
 
And this is why I left CT. This is NOT CT.

Q

Deeks 07-31-2006 06:48 PM

Re: real critical thinking
 
haha thats a bunch of bs. and i want the "square root of two" you guys are trying to prove "two squared" is an irrational number, and its not, its 4

Tps222 07-31-2006 06:48 PM

Re: real critical thinking
 
Q for CT mod.

I'm actually going to look into that though.

Check Wiki for facts blah blah blah debates only read stickies etc.

Deeks 07-31-2006 06:49 PM

Re: real critical thinking
 
it is too CT, you have to criticaly think to come up with this answer

T0rajir0u 07-31-2006 06:53 PM

Re: real critical thinking
 
You obviously had no idea what I was saying.

Eisenstein's Criterion is a method for determining whether a polynomial with integer coefficients can be factored into two non-constant polynomials with integer coefficients. Specifically,

For a polynomial a_n x^n + a_{n-1} x^{n-1} + ... + a_0 to be irreducible over Q[x] (edit: it's the same as being irreducible over Z[x] by Gauss's lemma), it is sufficient that ther exists a prime p such that:

p | a_i for i = n-1, n-2, ... 0
p does not divide a_n
p^2 does not divide a_0

In my particular example of the polynomial x^2 - 2, the coefficients are 1, 0, -2. p = 2 is a prime that divides every coefficient except the leading one, and 4 does not divide -2. By Eisenstein's criterion, x^2 - 2 is irreducible over Q[x].

This means that it cannot be factored into two linear factors, which means that x^2 - 2 = 0 has no rational roots.

Guess what this means.

The square root of two isn't rational.



Edit: You probably still don't understand what I was saying, so here's the traditional proof.

Suppose by contradiction that \sqrt{2} is rational. Then \sqrt{2} = \frac{p}{q} for some integers p, q. Reduce the fraction to lowest terms (in other words, gcd(p, q) = 1). Then

2q^2 = p^2

But 2 | p^2 implies that 2 | p, so let p = 2p_2. Then

q^2 = 2p_2^2

But 2 | q^2 implies that 2 | q, which contradicts our assumption that \frac{p}{q} was in lowest terms. Hence no such integers p, q exist. QED.

(There are several other basically equivalent methods, such as using infinite descent to generate a contradiction, or showing that the power of two that divides one side is even while the power of two that divides the other is odd.)

(There is a third proof basically equivalent to the first that makes use of the Rational Root Theorem.)

Tibs 07-31-2006 07:04 PM

Re: real critical thinking
 
owned by math

gg t0r <3

The_Q 07-31-2006 07:06 PM

Re: real critical thinking
 
No, you had to follow the rules of math. Critical Thinking is about stretching your mind. Anyone well trained in arithmetic can solve that. Critical Thinking involves things that can change, things that can change people's minds.

For instance, your post in the Economy thread below this one is NOT CT material because you didn't support anything. Just made a statement.

Please...read the stickies.

Q

T0rajir0u 07-31-2006 07:08 PM

Re: real critical thinking
 
I don't know who you are but I love you.

TheRapingDragon 07-31-2006 07:11 PM

Re: real critical thinking
 
Why are you guys doing his homework for him?

Also, The_Q is right on all aspects. This isn't CT stuff, this is problem solving. Critical Thinking tends to cause problems through thoughts heh.

Deeks 07-31-2006 07:14 PM

Re: real critical thinking
 
thank you t0r, ima send you your creds now. =) i just needed the whole proof.

T0rajir0u 07-31-2006 07:14 PM

Re: real critical thinking
 
I don't think it's his homework. As far as I can make it out, this is what's happening.

Deeks: omg rofl my e-penis is so huge hi should post a hrad question on ffr and watcno one get it roflroflimso cool lolololololmaonrauto
People: rofl i dunno
Deeks: 8=============D
Me: Fucen pwned.
Deeks: 8=======/=====D
Q: This shouldn't be here anyway.
Deeks: 8/D

Deeks 07-31-2006 07:17 PM

Re: real critical thinking
 
no need to be mean buddy. i was just bored, and it was something i learned a couple weeks ago and wanted to see if anyone else knew it

TheRapingDragon 07-31-2006 07:19 PM

Re: real critical thinking
 
Well "buddy", you need to realise CT is one of the areas you need to be pretty tuned into in order to post in. You can't just post here when bored, use Chit Chat or Garbage Bin for that. This is a place for good intelligent conversation and debates.

So please, next time you make a thread here, just try to make it better, listen to what The_Q said.

Deeks 07-31-2006 07:23 PM

Re: real critical thinking
 
dragon stfu i didnt say anything back to you, i heard enough it was a bad post and im sorry. calm down about it

scottish 07-31-2006 07:26 PM

Re: real critical thinking
 
Woo. Haven't seen the Q in awhile, how ya doing buddy?
Yeah, I believe CT is more for differences in persepectives that go beyond the "normal" typical debate or question. I'm not saying you were wrong for posting this, and yes it does take thinking to solve this, but try to look around at the other topics before posting.


All times are GMT -5. The time now is 11:38 PM.

Powered by vBulletin® Version 3.8.1
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.
Copyright FlashFlashRevolution