Irrational Square Roots Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Our textbook presents a proof that √2 is irrational. This is a classic proof, but not particular easy to understand on first reading. To help you make sense of the proof, and to gain additional experience with proof by contradiction, this discussion invites you to explore similar proofs.

Discuss how you might prove that √6 is irrational. The proof is similar to the textbook’s proof about √2. As usual, I’m not asking any one person to do the whole proof, but rather for multiple people to offer thoughts, questions, etc. about how the textbook’s proof might be adapted to √6, until someone can pull all the ideas together into a complete proof (or we do it in class).

Does similar reasoning work to show that √8 is irrational? How about √10? How about square roots of odd numbers, such as √3 or √5? Can you characterize what integers can be shown to have irrational square roots using a similar argument to the book’s about √2?