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Today in class we saw that the sum of an irrational with a rational gives an irrational, but not necessarily that the sum of irrationals gives an irrational.

And I was wondering what sufficient or necessary conditions you would have to satisfy two irrational numbers $a$ and $b$ for their sum to be irrational. Could someone help me with an answer.

user1098937
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    This is not a simple thing. In general it is still an open problem. For example, it is not known whether $\pi +e$ is irrational. A search for "sum of irrational numbers" will get you many details. – MJD Oct 29 '22 at 03:22
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    $\pi$ is irrational. $-\pi$ is irrational aswell. Their sum is $0$, which is rational. Even for multiplication (take $\sqrt2$ for example). – Hamdiken Oct 29 '22 at 03:25
  • Irrationality proofs are usually extremely hard , there are however some sufficient conditions that guarantee that a number is even transcendental , for example the condition used in the Gelfond Schneider theorem. – Peter Oct 29 '22 at 08:31

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