Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
Realted math.stackexchange.com/questions/157245/… –  user17762 Jan 11 '13 at 23:57

3 Answers 3

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

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

share|improve this answer
    
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

No on 1. $ (2+\sqrt{5}) + (1 + \sqrt{5}) = 3 + 2\sqrt{5}$, while $ (2+\sqrt{5}) - (1 + \sqrt{5}) = 1$.

share|improve this answer

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....

share|improve this answer
    
Are you sure that is a hard problem? If a == b then... –  Malvolio Feb 15 at 0:54

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.