4. A number, its fractional part, integer part, and itself form a geometric sequence, then the number is

保留源文本的换行和格式，所以翻译结果如下：

4. A number, its fractional part, integer part, and itself form a geometric sequence, then the number is
$$
x=\frac{1+\sqrt{5}}{2}
$$
4. 【Analysis and Solution】Let the number be $x$, then its integer part is $[x]$, and the fractional part is $x-[x]$. From the given, we have $x=(x-[x])=[x]^{2}$, where $[x]>0, 0<x-[x]<1$. Solving this, we get $x=\frac{1+\sqrt{5}}{2}[x]$. From $0<x-[x]<1$, we know that $0<\frac{\sqrt{5}-1}{2}[x]<1, 0<[x]<\frac{1+\sqrt{5}}{2}<2$, which means $[x]=1, x=\frac{1+\sqrt{5}}{2}$.