But if you’re not allowed to assume that the answer is true, what are you? Can you assume that sqrt(2) is irrational, or do you have to prove that? I feel like that question (prove that 7-sqrt(2) is irrational) could be a reasonable homework question on a college intro to proofs sort of class! It’s definitely not trivial.

]]>If the point of the problem was simply “cite an appropriate theorem you learned”, as the first student did, then the second student did so as well, twice: She observed correctly that 2 is not a perfect square, cited the true theorem that square roots of non-perfect-squares are irrational, and then (like the first student) cited the theorem that the difference of a rational and irrational number is irrational.

I also agree that one would actually like to see a problem that asked students to grapple with the meanings of the words “rational” and “irrational”. But honestly, if the first student is correct I don’t see why the second one shouldn’t be.

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