## Precalculus (6th Edition) Blitzer

The absolute value of a complex number $a+ib$ is the shortest length from the origin to the point $a+ib$. To determine the absolute value of $a+ib$, apply the Pythagorean Theorem or distance formula from the origin $\left( 0,0 \right)$ to the point $a+ib$ $\left( a,b \right)$.
Let $z=a+ib$, then the absolute value of point $z$ is represented by $\left| z \right|$ $\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}$ Example: The above explanation can be justified with the help of an example. Let $z=4+i3$ be a complex number, then the absolute value of $z$ is, \begin{align} & \left| z \right|=\sqrt{{{4}^{2}}+{{3}^{2}}} \\ & =\sqrt{25} \\ & =5 \end{align} In this example $5$ is the absolute value of the complex number $z=4+i3$