If there a formal definition for the sum of two irrationals is rational if it is only the additive inverse plus some rational? If $x$ and $y$ are two irrational numbers and $x + y$ is a rational number, I know one of them has to be the additive inverse with/without some rational term I believe. I know it is not always the case that the sum of two irrational numbers is irrational.
for example:
$(\sqrt2 -1) + (-\sqrt2 +4) = 3$
But I can only seem to come up with examples that are of this nature and that seems to be the general consensus. But is there a formal definition stating it has to be of this form or is there a counterexample I am ignorantly unaware of.
then if $x+y\in\mathbb{Q}$ would it be true that $x-y$ would always just produce a multiple of that irrational plus some rational implying that if $x$ and $y$ are irrational and $x+y\in\mathbb{Q}$ then $x-y\not\in\mathbb{Q}$
You would not look for a formal definition but for a theorem, and also let's not talk about consensus. (Namely, an argument in math is either right or wrong, regardless on how many people are &quot;for&quot; or &quot;against&quot; it.)
So if $\alpha, \beta$ are irrational and $\alpha+\beta=q$ - rational, then $\beta=-\alpha+q$ - i.e. indeed $\beta$ is the inverse of $\alpha$ plus a rational constant, as you have claimed. Then $\alpha-\beta=2\alpha-q$, which must then be irrational. (Because, otherwise you would add $q$ and conclude that $2\alpha$ would be rational, and then you would halve it and conclude that $\alpha$ would be rational - contradiction!)
Yes, but it is trivial.  Say $x,y$ are two rational numbers, but $x+y = r$ is rational.  Then the sum is of the form
$$
x + y = (x+0) + (-x+r) = r
$$