Using an indirect proof, show that radical 2 is not a rational number?

Question

plz do step by step ty so much ^^

Answer 1

The proof was figured out by the ancient Greeks. Suppose that it IS a rational number, say P/Q, where P and Q are integers, and the fraction is in lowest terms. Then at least one of P and Q is odd. By definition, P^2 = 2Q^2. From this, P^2 is even, hence P is even. Thus Q must be odd. If P is even, then we can define R = P/2. Then we have P/Q = 2R/Q. Then 2 = 4R^2/Q^2. Which contradicts the thesis that the original fraction was in lowest terms. I don't think that this is exactly right, but it gives you the general idea.

Answer 2

Suppose sqrt2 is rational implies sqrt2=a/b for some nonzero integers a and b. So 2=(a/b) squared. Consider the prime factorization of a and b, squared. 2bsq=asq implies one of the prime factors of a is 2. The prime factors must cancel, so 2 to its power squared minus 1 equals a prime factor of b squared. This is a contradiction. Write this out in detail and you will see the contradiction. Thus, your original assumption must be false.

Answer 3

check out the link below...it gives the indirect proof the most common way i've seen it done.
http://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php