
Algebra
if a and b are integers and (a+b* sqrt(11) )> 1 and (a+b* sqrt(11) )* (ab* sqrt(11) )=1, how do i show that a and b are positive integers? I was able to prove b>0 , but i can't seem to get the a >0 part, which makes me doubt whether i really showed b>0 .
Thanks

Suppose $\displaystyle a+b\sqrt{11} > 1 $. Also, $\displaystyle (a+b\sqrt{11})(ab\sqrt{11}) = 1 $. Let $\displaystyle c := a+b \sqrt{11} $. Then $\displaystyle ab \sqrt{11} = \frac{1}{c} $. Then $\displaystyle cacb \sqrt{11} = 1 $. Or $\displaystyle \sqrt{11} =  \frac{\frac{1}{c}a}{b} $. Or $\displaystyle \frac{1}{\sqrt{11}} = \frac{b}{\frac{1}{c}a} $.
We know that $\displaystyle 0 < \frac{1}{c} < 1 $. So $\displaystyle a > 0 $. Also $\displaystyle b > 0 $, which forces the quotient to be positive. If $\displaystyle b < 0 $ then $\displaystyle a < 0 \implies a+b \sqrt{11} < 1 $. So $\displaystyle a >0 $ so that the quotient is negative.