4. A positive number, if its fractional part, integer part, and the number itself form a geometric sequence, then the number is . $\qquad$
4. Let the number be $x$, then its integer part is $[x]$, and the fractional part is $x-[x]$. From the given information, we have
$$
x=(x-[x])=[x]^{2},
$$

where $[x]>0,0<x-[x]<1$, solving this gives
$$
x=\frac{1+\sqrt{5}}{2}[x] \text {. }
$$

From $0<x-[x]<1$, we know that
$$
0<\frac{\sqrt{5}-1}{2}[x]<1,0<[x]<\frac{1+\sqrt{5}}{2}<2 \text {, }
$$

which means
$$
[x]=1, x=\frac{1+\sqrt{5}}{2} .
$$