9) If the complex number $z$ satisfies $|z|=1$, and $z^{2}=a+b \mathrm{i}$, where $a, b$ are real numbers, then the maximum value of $a+b$ is $\qquad$ .
(9) $\sqrt{2}$ Hint: From $|z|=1$ we know $\left|z^{2}\right|=1$, i.e., $a^{2}+b^{2}=1$. Therefore,
$$
(a+b)^{2} \leqslant 2\left(a^{2}+b^{2}\right)=2 .
$$

Thus $a+b \leqslant \sqrt{2}$, and the equality holds when $a=b=\frac{\sqrt{2}}{2}$. Therefore, the maximum value of $a+b$ is $\sqrt{2}$.