Prove that if n is a natural number, then √2 + √n is irrational.?

Question

NOTE: we know that √2 is irrational.

This is how I did it. I just want to make sure that if I did it right or not.

Proof by Contrapositive:
Suppose √2 + √n = r, a rational number
Then √n = r - √2
We square both sides
n = (r - √2)²
expand the right hand side
n = r² - 2*r*√2 + 2

therefore, √2 = (n – 2 - r²) / (-2*r)

numerator and denominator are rational, so √2 is rational, which is not true. Therefore √2 + √n is irrational.

if I am wrong, pls provide me with right proof.

one more question...

give an example of a positive irrational number, x, so that √2 + x is rational.

Answer 1

Additional question: x = 2 - sqrt2.

sqrt2 + x = 2 → rational!

Answer 2

Your proof is fine.
For the second question, 2-√2 will work.
Here is another way: Take each digit, x, of √2
and replace it by 9-x. Call the new number n.
Then √2 + n = 9.9999999... = 10, which is rational.

Answer 3

The proof is valid. Well done.

Answer 4

your method is 100% rite. thats how we solved such probs in school too

Answer 5

Yep Yep, totally valid proof.
Nice one.
:)