Proof using sqrt(2) is irrational

Prove that for any rationals x, y, with y not equal to 0, x+y.sqrt(2) is an irrational number.

For this, I wrote:

Suppose x+y.sqrt(2) is rational. Then, since x and y are rational, y.sqrt(2) is also rational, since the difference of two rationals is numbers is also rational.

Then from there I proved by contradiction that sqrt(2) is irrational, and said that this implied that x+y.sqrt(2) is also irrational.

Is this the correct method? Any help would be appreciated :)

Re: Proof using sqrt(2) is irrational

Quote:

Originally Posted by

**sakuraxkisu** Prove that\for any rationals x, y, with y not equal to 0, x+y.sqrt(2) is an irrational number.

Suppose x+y.sqrt(2) is rational. Then, since x and y are rational, y.sqrt(2) is also rational, since the difference of two rationals is numbers is also rational.

If each of $\displaystyle r~\&~s$ is a rational number and $\displaystyle (r+s)\sqrt2=t$ where $\displaystyle t\in\mathbb{Q}$ then $\displaystyle \sqrt2=\frac{t}{r+s}$.

What is wrong with wrong with that?

Re: Proof using sqrt(2) is irrational

i believe he means x + y√2, not (x + y)√2.

since we know that y ≠ 0, we have, if x + y√2 = t, a rational number:

√2 = (t - x)/y, which is rational.