10. The fractional part, the integer part, and the number itself of a positive number form a geometric sequence, then this positive number is
10. $\frac{1+\sqrt{5}}{2}$.

Let the integer part of this positive number be $n$, and the decimal part be $t$. Then $\frac{n}{t}=\frac{n+t}{n}=1+\frac{t}{n}$, so from $1<1+\frac{t}{n}<2$ we get $1<\frac{n}{t}<2$, thus $0<t<n<2 t<2$ which gives $n=1$. Therefore, $1=t(t+1)$, solving the equation yields $n+1:=\frac{1+\sqrt{5}}{2}$.