TITLE: Can any two irrational numbers NOT of the form (m+A) and (n-A) be added to produce a rational number?
QUESTION [5 upvotes]: $m$ and $n$ being rational numbers, A being an irrational number.
I was wondering if two irrational numbers when added always yield an irrational number. All the counter-examples I could find were of the form $(m+A) + (n-A) = m+n$.
Are there any counter-examples NOT of this form?

REPLY [2 votes]: Let $i_1$, $i_2$ be irrational, and $i_1 + i_2 = a$; with $a$ rational (the assumption).
Then
$$
i_1 = \frac{a}{2} - \frac{b}{2}\\
i_2 = \frac{a}{2} + \frac{b}{2}
$$
where $$b \equiv i_2-i_1$$
Case 1: $b$ is irrational.
If $b$ is irrational, then $i_1$ and $i_2$ are of the form that we are trying to avoid.  This is because if $a$ is rational, so is $a/2$; if $b$ is irrational, so is $b/2$.
Case 2: $b$ is rational.
Substituting $a-i_2$ for $i_1$ into the definition of $b$, we get
$$i_2 = \frac{a+b}{2}$$
But, since $a$ and $b$ are rational in Case 2, then their sum must be rational.  This would mean $i_2$ is rational, which violates our original assumption.  (A similar line leads to $i_1$ having to be rational in Case 2, as well.)  This is a contradiction, so we learned $b$ must be irrational.
So, putting it together, if two irrationals sum to a rational, then they are of the form we are trying to avoid.  (So, the answer to the question as asked is "no.")