Examples for $ \text{If } \; x-\lfloor x \rfloor + \frac{1}{x} - \left\lfloor \frac{1}{x} \right\rfloor = 1 \text{, then } x \text{ is irrational.}$ I saw the answers here about how to prove:
$$ \text{If } \; x-\lfloor x \rfloor + \frac{1}{x} - \left\lfloor \frac{1}{x} \right\rfloor = 1 \text{, then } x \text{ is irrational.}$$
I understood the proof. But, can someone give me an example for an irrational $x$ that makes this equality true?

Hint:  If $k\gt2$ is an integer, then
$$\left\lfloor k+\sqrt{k^2-4}\over2\right\rfloor=k-1$$
