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I am asking the proof of the statement if it is true. I know that if the two irrational numbers have different signs their sum or product can be rational, but how about the sum of two positive irrational numbers? How to prove that the sum is also irrational?

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    $\begingroup$ What about $2-\sqrt{2}$ and $\sqrt{2}$? $\endgroup$ – GReyes Oct 28 '19 at 6:29
  • $\begingroup$ $\pi+(100-\pi)$ $\endgroup$ – mathworker21 Oct 28 '19 at 6:29
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Consider that $5+\pi$ and $5-\pi$ are both positive and irrational but their sum is $10$, which is rational.

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Pick $a>b>0$ with $a$ rational and $b$ irrational. Then $c:=a-b>0$. Assume $c$ is rational. Then also $b=a-c$ would be rational, which is absurd. Hence $c$ is irrational. Thus $b$ and $c$ are two positive irrationals with rational sum $a$.is a sum of t

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