When is $ a+b \phi >0$?

Question

What is an algebraic condition on rationals $a,b$ that characterises when $$ a+b \phi >0, $$ where $\phi=\frac{1+\sqrt{5}}{2}$.

Answer 1

I’m assuming that you want your condition to involve only rational numbers. We might start by rewriting your inequality as $$\frac ab>-\phi.$$ We can now multiply both sides by $2$ and add $1$, to get the equivalent statement $$\frac{2a+b}b>-\sqrt{5}.$$ Now, for $y$ positive, the statement “$x>-y$” is equivalent to the statement “either $x^2<y^2$ or $x>0$”. So, after simplifying a bit, we get that a possible set of conditions is $$\boxed{a^2+ab-b^2<0\text{ or }\frac{2a+b}b>0.}$$