To prove that $\sqrt{n}$ is irrational for any positive integer $n$ that is not a perfect square, we can use a proof by contradiction. Here's how the proof goes:

Proof by Contradiction:

1. Assume the opposite: Suppose $\sqrt{n}$ is rational for some positive integer $n$, but $n$ is not a perfect square.

2. Express $\sqrt{n}$ as a fraction: Since $\sqrt{n}$ is rational, it can be expressed as: $\sqrt{n} = \frac{a}{b}$ where $a$ and $b$ are integers with no common factors other than 1 (i.e., the fraction is in its lowest terms), and $b \neq 0$.

3. Square both sides to remove the square root: $n = \frac{a^2}{b^2}$

4. Rearrange the equation to: $n \cdot b^2 = a^2$ This equation states that $a^2$ is equal to $n$ multiplied by $b^2$.

5. Analyze the implications: Since $a^2$ is a perfect square, $n \cdot b^2$ must also be a perfect square. For $n \cdot b^2$ to be a perfect square, $n$ itself must be a perfect square because $b^2$ is already a perfect square.

6. Contradiction: However, we assumed that $n$ is not a perfect square. Therefore, this implies that $n \cdot b^2$ cannot be a perfect square, which contradicts the requirement that $a^2$ must be a perfect square.

Conclusion:

Since our assumption that $\sqrt{n}$ is rational leads to a contradiction, the original assumption must be false. Therefore, $\sqrt{n}$ must be irrational if $n$ is not a perfect square.

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Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What is an example of a number $n$ that is a perfect square, and why is $\sqrt{n}$ rational in this case?
2. How can you prove the irrationality of $\sqrt{2}$ using a similar method?
3. What are some other common irrational numbers, and how can their irrationality be proven?
4. Can you describe the difference between rational and irrational numbers?
5. How would the proof change if $n$ were a negative integer?

Tip: When dealing with proofs by contradiction, it's essential to carefully examine the logical steps that lead to a contradiction to ensure the proof's validity.