2.1.4 So­lu­tion math­c­plx-d

Ques­tion:

Find the mag­ni­tude or ab­solute value of $2+3{\rm i}$.

An­swer:

The mag­ni­tude $\vert 2+3{\rm i}\vert$ of $2+3{\rm i}$ is the square root of $2+3{\rm i}$ times its com­plex con­ju­gate $2-3{\rm i}$:

$$\vert 2+3{\rm i}\vert = \sqrt{(2+3{\rm i})(2-3{\rm i})} = \sqrt{2^2-(3{\rm i})^2}.$$

Since ${\rm i}^2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$1, $\vert 2+3{\rm i}\vert$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{13}$.