The discussion revolves around proving several propositions related to rational and irrational numbers, specifically involving the square root of 2. Participants are tasked with exploring the validity of these propositions and providing proofs or counterexamples.
The discussion is ongoing, with participants sharing their attempts and questioning the validity of the propositions. Some guidance is provided in the form of suggestions for counterexamples, but no consensus has been reached regarding the truth of the propositions.
Participants are working under the constraints of proving mathematical propositions and are encouraged to explore various approaches without assuming what they need to prove.
guroten said:Homework Statement
Prove these propositions
Homework Equations
If x is rational, then x + sqrt(2) is irrational.
If x + sqrt(2) is irrational, then x is rational.
If xsqrt(2) is irrational, then x is rational.
The Attempt at a Solution
I've tried a couple of ways, but I always end up at a dead end. usually, I end up assuming what I need to prove.
guroten said:After working with it some more, I've gotten all but the last. Is a counter example something like e(sqrt(2))? But I don't know if that is rational or not.