The discussion centers on proving three mathematical propositions regarding rational and irrational numbers involving the square root of 2. The first proposition, stating that if x is rational, then x + sqrt(2) is irrational, is confirmed as true. However, the second and third propositions are challenged, with participants seeking counterexamples to demonstrate their falsehood. The conversation highlights the difficulty in proving these statements and the need for clarity in mathematical assumptions.
PREREQUISITESMathematics students, educators, and anyone interested in number theory and mathematical proofs will benefit from this discussion.
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.