Absolute Value and Argument
of a Complex Number Calculator

What are absolute value and argument of a complex number?

The absolute value \( \left|z\right| \) and argument \( \arg{\left(z\right)} \) of a complex number ( \(z=a+bi\) ) can be calculated as follows

\begin{align} \left|z\right|&=a^{2}+b^{2}\notag\\ \arg{\left(z\right)}&=\arctan{\left(\frac{b}{a}\right)}\notag \end{align}

Example

Example 1

Calculate the absolute value and argument of \(1+\sqrt{3}i\).

Solution

\begin{align} \left|1+\sqrt{3}i\right|&=\sqrt{1+3}\notag\\ &=\sqrt{4}\notag\\ &=2\notag\\ \notag\\ \arg{\left(1+\sqrt{3}i\right)}&=\arctan{\left(\frac{\sqrt{3}}{1}\right)}\notag\\ &=\arctan{\sqrt{3}}\notag\\ &=\frac{\pi}{3}\notag \end{align}

Example 2

Calculate the absolute value and argument of \(\displaystyle \frac{\sqrt{3}}{2}+\frac{1}{2}i \).

Solution

\begin{align} \left|\frac{\sqrt{3}}{2}+\frac{1}{2}i\right|&=\sqrt{\frac{3}{4}+\frac{1}{4}}\notag\\ &=1\notag\\ \notag\\ \arg{\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)}&=\arctan{\left(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right)}\notag\\ &=\arctan{\left(\frac{1}{\sqrt{3}}\right)}\notag\\ &=\frac{\pi}{6}\notag \end{align}