# 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?

## Answers and Replies

cristo
Staff Emeritus
Science Advisor

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
Science Advisor
Homework Helper

I think the OP meant prove that the sum of an irrational and rational number must be irrational but wasn't sure whether the result should be "is irrational" or "is rational".

Of course, if m is rational and x is irrational, then x+ m= n with n rational leads immediately to x= m-n, a contradiction.