smc

Each of the numbers $a$ and $b$ is a solution to $\left|x-\frac{1}{x}\right|=1$. Hence it is either a solution to $x-\frac{1}{x}=1$, or to $\frac{1}{x}-x=1$. Then it must be a solution either to $x^2-x-1=0$, or to $x^2+x-1=0$. There are in total four such values of $x$, namely $\frac{\pm 1\pm \sqrt{5}}{2}$. Out of these, two are positive: $\frac{-1+\sqrt{5}}{2}$ and $\frac{1+\sqrt{5}}{2}$. We can easily check that both of them indeed have the required property, and their sum is $\frac{-1+\sqrt{5}}{2}+\frac{1+\sqrt{5}}{2}=\boxed{\sqrt{5}}$.