Question

Prove that there is no n ∈ ℕ such that n² = 2

Answer 1

To prove that there is no n ∈ ℕ such that n² = 2, we can use a proof by contradiction. The assumption that an n exists leads to a contradiction when considering the polynomial equation x^4 - 4 = 0. By checking all possible rational roots, we can conclude that there is no solution.

Step-by-step explanation:

To prove that there is no n ∈ ℕ such that n² = 2, we can use a proof by contradiction.

1. Assume that there exists an n ∈ ℕ such that n² = 2.
2. Square both sides to get n^4 = 4.
3. By rearranging the equation, n^4 - 4 = 0.
4. Now, consider the polynomial equation x^4 - 4 = 0.
5. Using the rational root theorem, we can see that the only possible rational roots are ±1 and ±2.
6. By substituting these values into the equation, we find that none of them satisfy the equation.
7. Therefore, there is no n ∈ ℕ such that n² = 2.