December 23rd, 2017, 04:05 AM  #1 
Senior Member Joined: Dec 2015 From: Earth Posts: 204 Thanks: 26  Irrational numbers
Find the two irrational numbers $\displaystyle (\alpha,\beta)$ such that $\displaystyle \alpha^2  \beta \in N $

December 23rd, 2017, 04:24 AM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 312 Thanks: 161 Math Focus: Dynamical systems, analytic function theory, numerics 
$\alpha = \sqrt{\sqrt{2}+2}, \beta = 1 + \sqrt{2}$

December 23rd, 2017, 04:24 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,232 Thanks: 2411 Math Focus: Mainly analysis and algebra 
It's not "the two irrational numbers", there are an infinity of them. One suggestion: what if $\beta$ is a solution of, for example, $x^2x1=0$ (there are infinitely many other quadratics that would work). 

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