To prove that $\sqrt{2}$ is irrational, we use a classical proof by contradiction.

Steps of the proof:

1. Assume the opposite: Suppose $\sqrt{2}$ is a rational number. By definition, a rational number can be expressed as a fraction of two integers $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0$, and the fraction is in its lowest terms (i.e., $p$ and $q$ have no common factors other than 1, they are coprime).

   So, let's assume: $\sqrt{2} = \frac{p}{q}$ Squaring both sides, we get: $2 = \frac{p^2}{q^2}$ This implies: $p^2 = 2q^2$

2. Examine the implications: From $p^2 = 2q^2$, we see that $p^2$ is an even number because it equals $2q^2$ (which is obviously even). If $p^2$ is even, then $p$ itself must also be even (because the square of an odd number is odd). So, let $p = 2k$ for some integer $k$.

3. Substitute and simplify: Now, substitute $p = 2k$ into the equation $p^2 = 2q^2$: $(2k)^2 = 2q^2$ This simplifies to: $4k^2 = 2q^2$ Dividing both sides by 2, we get: $2k^2 = q^2$ This implies that $q^2$ is also even, which means $q$ must be even as well.

4. Contradiction: If both $p$ and $q$ are even, then they have a common factor of 2. But this contradicts our original assumption that $p$ and $q$ are coprime (they should not have any common factors other than 1).

Conclusion:

Since our assumption that $\sqrt{2}$ is rational leads to a contradiction, it follows that $\sqrt{2}$ must be irrational.

---

Would you like more details on any step or have any other questions?

Here are 5 related questions to expand your understanding:

1. What is the definition of an irrational number?
2. Can the same method be used to prove that $\sqrt{3}$ is irrational?
3. Why is it necessary for the numbers $p$ and $q$ to be coprime?
4. How does this proof relate to the Fundamental Theorem of Arithmetic?
5. Can the method of contradiction be used for other types of mathematical proofs?

Tip: In proofs by contradiction, it's important to assume the exact opposite of what you're trying to prove and carefully follow the logical implications of that assumption until a contradiction arises.