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Question 1: If $a+b$ is an irrational number. Is $a-b$ an irrational number, too?

Question 2: If $\cos(a)-\sin(a)$ is irrational, Is $\sin(a)-\cos(a)$ irrational, too?

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HINT: Try for an example with $a=b$.

For the second question, note that $x-y=-(y-x)$.

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  • $\begingroup$ For example if we have $\mathcal{K}=\mathcal{Q}/{0}$ and $a,b$ are some irrationals such that theyc an only be identified as $a=(k)^{1/n}$. $\endgroup$ – Seyhmus Güngören Feb 17 '13 at 14:18
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No on 1. $ (2+\sqrt{5}) + (1 + \sqrt{5}) = 3 + 2\sqrt{5}$, while $ (2+\sqrt{5}) - (1 + \sqrt{5}) = 1$.

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Hard problem:

Is it possible to pick $a$ and $b$ such that $a+b$ is irrational but $a-b$ is rational?

Easy problem:

Is it possible to pick irrational $c$ and rational $d$ such that you can find $a,b$ such that $a+b = c$ and $a-b = d$?

They're really the same problem, of course....

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  • $\begingroup$ Are you sure that is a hard problem? If a == b then... $\endgroup$ – Malvolio Feb 15 '14 at 0:54

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