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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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Realted math.stackexchange.com/questions/157245/… –  user17762 Jan 11 '13 at 23:57
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3 Answers

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

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

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This answer illustrates perfectly why mathematics is not a spectator sport. Anyone could get to that answer.+1 –  user4140 Jan 11 '13 at 23:14
    
This seems to be the ideal MSE answer to this question. –  Joel Reyes Noche Jan 11 '13 at 23:45
    
For example if we have $\mathcal{K}=\mathcal{Q}/{0}$ and $a,b$ are some irrationals such that theyc an only be identified as $a=(k)^{1/n}$. –  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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Are you sure that is a hard problem? If a == b then... –  Malvolio Feb 15 at 0:54
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