Do there exist two irrational numbers $a$ and $b$ ($a > b$) such that their sum and product are equal to the same integer, and both $a$ and $b$ are greater than $\frac{1}{2}$ and less than 4? If they exist, find these two numbers; if not, explain the reason.
Solution: Let $a+b=ab=t, t$ be an integer, then $a, b$ are the two roots of the equation $x^{2}-t x+t=0$.
Let $f(x)=x^{2}-t x+t$.
Since $a, b$ are both greater than $\frac{1}{2}$ and less than 4, then
$$
\left\{\begin{array}{l}
\Delta = t^{2} - 4 t > 0, \\
\frac{1}{2} < \frac{t}{2} < 4, \\
f\left(\frac{1}{2}\right) = \frac{1}{4} - \frac{1}{2} t + t > 0, \\
f(4) = 16 - 4 t + t > 0,
\end{array} \text { we get } \left\{\begin{array}{l}
t > 4, \\
1 < t < 8, \\
t < \frac{16}{3}.
\end{array}\right.\right.
$$

Therefore, $t=5$.
Thus, $x^{2}-5 x+5=0$.
Solving, we get $a=\frac{5+\sqrt{5}}{2}, b=\frac{5-\sqrt{5}}{2}$.