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I am working out of Calculus by Spivak (4th ed.) and have come across a question which I would like some further insight on.

If $a$ and $b$ are both irrational, is $a+b$ necessarily irrational?

I have already proved that if $a$ is rational and $b$ is irrational then $a+b$ is irrational.

Thanks in advance!

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    $\begingroup$ $(1+\sqrt{2})+(1-\sqrt{2})=2$. $\endgroup$ – diracula Mar 5 '17 at 17:00
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    $\begingroup$ i think no take $$a=\sqrt{2},b=1-\sqrt{2}$$ and $$a+b=1$$ rational $\endgroup$ – Dr. Sonnhard Graubner Mar 5 '17 at 17:00
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    $\begingroup$ $\sqrt{2} + (-\sqrt{2}) = 0$ $\endgroup$ – Augustin Mar 5 '17 at 17:00
  • $\begingroup$ Lots of counterexamples here, so maybe you could write your own answer! We love that! $\endgroup$ – The Count Mar 5 '17 at 17:01
  • $\begingroup$ Also: math.stackexchange.com/questions/499784/…. $\endgroup$ – Martin R Mar 5 '17 at 17:05
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It might be easier to answer the contrapositive, if you don't just see the answer. The contrapositive is: "if $a+b$ is rational, is it the case that at least one of $a$ and $b$ is rational?". Since you've already proved that irrational plus rational is irrational, this is just "if $a+b$ is rational, is it the case that both $a$ and $b$ are rational?". This slightly different perspective might make it easier for you to find an example.

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No. For example $\sqrt{2}$ and $-\sqrt{2}$

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$a=1+\sqrt{2}$ and $b=1-\sqrt{2}$ are both irrational and $a+b=2$

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