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?

If $x>1$ then $0< \frac 1x < 1$ so $\lfloor \frac 1x \rfloor = 0$ and the equation reduces to
$$x + \frac{1}{x} = 1 + \lfloor x \rfloor$$
Any number $x>1$ can be written on the form $x = n + r$ where $n$ is an integer and $0<r<1$. Take this form for $x$ in the equation above. Since $\lfloor n + r\rfloor = n$ the equation then reduces to
$$ r^2 + (n-1)r - (n-1) = 0$$
Solve the quadratic equation for $r$ and pick the root that is in $(0,1)$. $x = n + r$ will then be an irrational number that satisfy your equation for any value of the integer $n > 1$.
