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#1 |
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FFR Player
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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.
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#2 |
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FFR Simfile Author
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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. |
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#3 |
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tool
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easy
2^.5 * 2^.5 = 2 where's my credits
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#4 |
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FFR Player
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aww patashu you know it. say it all, jkpolk not even close, patashu gave everyone a start
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#5 |
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FFR Player
Join Date: Aug 2005
Location: awsome
Posts: 2,946
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The polynomial x^2 - 2 is irreducible over Q[x] by Eisenstein's criterion.
Pay up, bitch.
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hehe |
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#6 |
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FFR Player
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And this is why I left CT. This is NOT CT.
Q |
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#7 |
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FFR Player
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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
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#8 |
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FFR Player
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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. |
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#9 |
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FFR Player
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it is too CT, you have to criticaly think to come up with this answer
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#10 |
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FFR Player
Join Date: Aug 2005
Location: awsome
Posts: 2,946
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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.)
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hehe Last edited by T0rajir0u; 07-31-2006 at 06:57 PM.. |
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#11 |
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FFR Player
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owned by math
gg t0r <3 |
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#12 |
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FFR Player
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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 |
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#13 |
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FFR Player
Join Date: Aug 2005
Location: awsome
Posts: 2,946
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I don't know who you are but I love you.
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hehe |
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#14 |
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A car crash mind
Join Date: Aug 2005
Posts: 9,787
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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. |
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#15 |
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FFR Player
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thank you t0r, ima send you your creds now. =) i just needed the whole proof.
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#16 |
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FFR Player
Join Date: Aug 2005
Location: awsome
Posts: 2,946
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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
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hehe |
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#17 |
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FFR Player
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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
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#18 |
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A car crash mind
Join Date: Aug 2005
Posts: 9,787
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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. |
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#19 |
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FFR Player
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dragon stfu i didnt say anything back to you, i heard enough it was a bad post and im sorry. calm down about it
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#20 |
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FFR Veteran
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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. |
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