First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.2

(iv) $119+120i$

Let $z=119+120i$ .

|z|=\sqrt{119^2+120^2}=\sqrt{28561}=169

Here $x=119$ , $y=120>0$ so $\operatorname{sgn}(y)=+1$ .

\begin{aligned} \sqrt{119+120i} &=\pm\left(\sqrt{\frac{169+119}{2}}+i\sqrt{\frac{169-119}{2}}\right)\\ &=\pm\left(\sqrt{144}+i\sqrt{25}\right)\\ &=\pm(12+5i) \end{aligned}

\boxed{\sqrt{119+120i}=\pm(12+5i)}