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Given an integer $n$ and an irrational $r$, $n>r$, $n-r$ is irrational but $r + (n - r)$, the sum of two positive irrationals, is an integer. Is that the only way that two irrationals can sum to an integer?

What if the question is rephrased using rationals instead of integers? Is the only way two irrationals can sum to a rational is by using the form $r + (a/b - r)$?

Can $r_{1} + (a/b - r_{2})$ ever be a rational? An integer?

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    If $x$ and $y$ are irrationals such that $x+y$ is an integer (say, $n$), then $y= n-x$. So yes, that's the only way in which you can write $n$ as the sum of irrationals: pick one irrational and the second is just $n$ minus the first.2011-11-18
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    We don't even know if $e+\pi$ is rational or not...2011-11-18
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    What on earth do you mean by "the only way two irrationals can sum to a rational"? Yes, $a+b=c$ if and only if $b$ can be written as $c-a$. That has nothing to do with being rational or irrational.2011-11-18

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