Which among the following is the most appropriate ‘Method of Proof’ for proving $\sqrt{2}$ is an irrational number ?

The correct option is Contradiction.

Step-by-Step Explanation:

1. Understanding Proof by Contradiction:
Proof by contradiction is a standard mathematical technique where, to prove a statement is true, we temporarily assume that the statement is false. We then logically deduce consequences from this assumption until we reach an impossible situation—a contradiction. Because a logical sequence of valid steps led to an impossibility, the initial assumption that the statement was false must itself be false, thereby proving the original statement true.

2. Application to proving $\sqrt{2}$ is irrational:
To prove that $\sqrt{2}$ is irrational using contradiction:

• Assumption: Assume the opposite, i.e., $\sqrt{2}$ is a rational number.

• Formulation: If $\sqrt{2}$ is rational, it can be written as a fraction:

$\sqrt{2}=\frac{a}{b}$

where $a$ and $b$ are coprime integers (they share no common factors other than 1) and $b\ne 0$.

• Derivation: Squaring both sides gives:

$2=\frac{a^2}{b^2}\Rightarrow a^2=2b^2$

This implies that $a^2$ is even, which means $a$ must also be even. Thus, we can express $a=2k$ for some integer $k$.

• Re-substituting: Substituting $a=2k$ back into the equation:

${\left(2k\right)}^2=2b^2\Rightarrow 4k^2=2b^2\Rightarrow b^2=2k^2$

This means $b^2$ is also even, so $b$ must be even as well.

• Reaching the Contradiction: If both $a$ and $b$ are even, they both share a common factor of 2. However, this contradicts our initial assumption that $a$ and $b$ are coprime (sharing no common factors).

• Conclusion: Since assuming $\sqrt{2}$ is rational leads to a logical contradiction, our assumption must be false. Therefore, $\sqrt{2}$ is an irrational number.

Thus, Contradiction is the most appropriate method of proof.