To prove that $\sqrt{2}$ is irrational, we will proceed by contradiction. That is, we will assume that $\sqrt{2}$ is rational and show that this assumption leads to a contradiction.

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Proof by Contradiction:

1. Assumption:
   Assume $\sqrt{2}$ is rational. This means that it can be written as a fraction of two integers: $\sqrt{2} = \frac{p}{q}$ where $p$ and $q$ are integers with no common factors (i.e., the fraction $\frac{p}{q}$ is in its simplest form). Also, $q \neq 0$.

2. Square both sides:
   $2 = \frac{p^2}{q^2}$ Multiply both sides by $q^2$ to get: $2q^2 = p^2$ This shows that $p^2$ is even, since it is equal to $2q^2$.

3. Implication that $p$ is even:
   Since $p^2$ is even, $p$ must also be even. (If $p$ were odd, then $p^2$ would be odd as well, which contradicts our result.)
   So, let $p = 2k$ for some integer $k$.

4. Substitute $p = 2k$ into the equation $2q^2 = p^2$:
   $2q^2 = (2k)^2 = 4k^2$ Divide both sides by 2: $q^2 = 2k^2$ This shows that $q^2$ is also even, which means that $q$ must be even as well.

5. Contradiction:
   Since both $p$ and $q$ are even, they have a common factor of 2. This contradicts our assumption that $\frac{p}{q}$ is in its simplest form (i.e., that $p$ and $q$ have no common factors).

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Conclusion:

Since our assumption that $\sqrt{2}$ is rational leads to a contradiction, the original assumption must be false. Therefore, $\sqrt{2}$ is irrational.

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Would you like a deeper breakdown of any part of the proof?

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Here are 5 related questions for further exploration:

1. How can this proof method be generalized for $\sqrt{n}$, where $n$ is not a perfect square?
2. What is the definition of an irrational number, and how does it differ from a rational number?
3. Can similar proofs show that $\sqrt{3}$ or $\sqrt{5}$ are irrational?
4. What is the connection between irrational numbers and real numbers?
5. How does this proof relate to the fundamental theorem of arithmetic?

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Tip: Proofs by contradiction are powerful tools in mathematics, especially when dealing with irrational numbers and other properties that seem counterintuitive at first.