Is there an easy way to prove

Is there an easy way to prove....

that a rational number + an irrational number is either rational or irrational?

Just pick 2 elements and show that it the sum is rational and irrational?

cristo
Staff Emeritus

that a rational number + an irrational number is either rational or irrational?

Just pick 2 elements and show that it the sum is rational and irrational?

That's not a proof, as I explained to you before. A proof must hold true for ALL values, not just one or two.

You need to show some work in the HW forum. What have you tried?

Let $$x = \frac{p}{q}$$ and $$y$$ be an irrational number. Then show that this sum leads to a form $$p/q$$ or that it cannot be expressed in that form?

Or do a contradiction (e.g. assume that it is neither irrational nor rational)?

that a rational number + an irrational number is either rational or irrational?
Any real number is either rational or irrational, so what's the point of the statement you are trying to prove?

HallsofIvy