Number Theory – Irrational Numbers with a Rational Difference

Question

Are there any irrational numbers that have a difference of a rational number?

For example, if you take $\pi – e$, it looks like it will be irrational ($0.423310\ldots$) – however, are there any irrational numbers where this won't be the case?

Edit to keep up with the answers:

Cases where it won't be the case:

• $yX – y(X + n)$, where $X$ is irrational, or equivalent have been covered

• $e^{\pi i} = -1$ has been covered

• the golden ratio ($\phi$) has been covered

Are there any other cases?

$e^\pi – \pi$ comes close, but not quite – are there any cases such as this where the result is a (proper) rational number?

Answer 1

If $q$ is a rational number, and a is an irrational number, then $q+a$ is irrational, $(q+a)-a=q$.