# Take integers a,b,c,d. Show that...

• April 29th 2010, 12:59 AM
NikoBellic
Take integers a,b,c,d. Show that...
Take integers a,b,c,d. Show that (a+b*sqrt(2))/(c+d*sqrt(2)) = (ac-2bd)/(c^2-2d^2) + ((bc-ad)*sqrt(2))/(c^2-2d^2). Then choose integers s and t differing from those terms by at most + or - 1/2. Show that the norm of the difference between the expression above and s+t*sqrt(2) has norm less than 1.

Thanks!
• April 29th 2010, 09:41 AM
Bruno J.
Begin by using the fact that $(c+d\sqrt{2})^{-1}=\frac{c-d\sqrt{2}}{c^2-2d^2}$.