real critical thinking

[Deeks]

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]

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]

Re: real critical thinking

easy
2^.5 * 2^.5 = 2

where's my credits

[Deeks]

Re: real critical thinking

aww patashu you know it. say it all, jkpolk not even close, patashu gave everyone a start

[T0rajir0u]

Re: real critical thinking

The polynomial x^2 - 2 is irreducible over Q[x] by Eisenstein's criterion.

Pay up, bitch.

[The_Q]

Re: real critical thinking

And this is why I left CT. This is NOT CT.

Q

[Deeks]

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]

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]

Re: real critical thinking

it is too CT, you have to criticaly think to come up with this answer

[T0rajir0u]

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]

Re: real critical thinking

owned by math

gg t0r <3

[The_Q]

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]

Re: real critical thinking

I don't know who you are but I love you.

[TheRapingDragon]

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]

Re: real critical thinking

thank you t0r, ima send you your creds now. =) i just needed the whole proof.