Question

Prove that, there does not exist a rational number r such that r² = 2.​

Answer 1

your solution :::::

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Answer 2

Answer:

Step-by-step explanation:

Assume r is rational. Then r = p/q, where p, q are integers, and p & q are relatively prime.

r^2 = (p/q)^2 = 2. (We are told that r^2 = 2)

p^2 / q^2 = 2 means p^2 = 2 * q^2. So, p^2 is even (because 2 divides p^2).

If p^2 is even then p is even. (If p were odd, then p’s prime factorization would exclude 2. Taking its square does not produce “2” in the prime factorization.)

So, p being even means there exists an integer, k, such that 2*k = p

Replacing p with 2k yields: 4k^2 = 2q^2. So, 2k^2 = q^2.

By the same reasoning stated above, q^2 is even and so q is even.

But that’s impossible because then both p & q have 2 as a common factor, which contradicts the assumption of being relatively prime.

So, r^2 = 2 means r must be irrational.