Algebra Problem with Rationals and Proofs

[silvermane]

Hello fellow forum buddies :)

Homework Statement

a.) Prove that if a+b\sqrt{2} = c+d\sqrt{2} with a,b,c,d all in Q, then
a = c and b = d.
b.) Prove that a^2 - 2b^2 with a, b in Q is nonzero unless a=b=0

The Attempt at a Solution

I really don't know where to start. Any tips would be nice. I just need to see something that I'm not seeing at the moment. :(
I don't need an answer, just a little jolt lol. It will be greatly appreciated! :)

 

[fzero]

For part a, is (a-c) in Q? Is (b-d)\sqrt{2}?

For part b, factor the equation and try to use the result from part a.

 

[╔(σ_σ)╝]

I guess he ^ already took care of you ;-).

Post again if you need more clearification.

 

[silvermane]

For part a, is (a-c) in Q? Is (b-d)\sqrt{2}?

For part b, factor the equation and try to use the result from part a.

AHHH... wow I'm glad I see it now. So I just need to solve a and c with respect to the fact that they are rational, then b and d with the fact that they're not rational?

 

[silvermane]

I guess he ^ already took care of you ;-).

Post again if you need more clearification.

Awe phanku :)

 

[╔(σ_σ)╝]

AHHH... wow I'm glad I see it now. So I just need to solve a and c with respect to the fact that they are rational, then b and d with the fact that they're not rational?

b and d are rational, however, the product with sqrt(2) is not. I believe this is what you meant.

I believe you can see that if b-d was non-zero a contradiction ensues. :-)

EDIT

lmao.
U r velcome :-).

 

[fzero]

AHHH... wow I'm glad I see it now. So I just need to solve a and c with respect to the fact that they are rational, then b and d with the fact that they're not rational?

Well if x = a - c is nonzero, can we find y such that x = y \sqrt{2}? Is such a y compatible with y = b-d?

 

[silvermane]

Ahhhh that should do it lol. I wrote that down and when I saw each of your replies, it was confirmed. I think I can finish it from here. Thanks so much! :)

 

[╔(σ_σ)╝]

Okay. Post if you get stuck in part b or something.