4. A positive number, if its fractional part, integer part, and the number itself form a geometric sequence, then the number is $\qquad$ .
4. $\frac{1}{2}(\sqrt{5}+1)$.

Let this positive number be $x$, the fractional part be $\alpha$, and the integer part be $N$, then $x=N+\alpha, 0 \leqslant \alphaN^{2}$, so $N=1 . \alpha^{2}+\alpha-1=0$, solving for $\alpha=\frac{\sqrt{5}-1}{2}$ or $\alpha=-\frac{\sqrt{5}+1}{2}$ (discard). $x=1+\frac{\sqrt{5}-1}{2}=\frac{1+\sqrt{5}}{2}$, hence the number is $\frac{1}{2}(\sqrt{5}+1)$.