Question

What is the argument of z=-4

Answer 1

Answer on Question #71420 – Math– Complex Analysis Question

What is the argument of $z = -4$.

Solution

We write the complex number in the form $z = |z| (\cos \varphi + i \sin \varphi)$.

$|z|$ – modulus of the complex number ($|z| = \sqrt{a^2 + b^2}$ for $z = a + bi$), $\varphi$ is called the argument of the complex number ($\cos \varphi = \frac{a}{\sqrt{a^2 + b^2}}$, $\sin \varphi = \frac{b}{\sqrt{a^2 + b^2}}$ for $z = a + bi$).

$z = -4 + i \cdot 0$, hence $|z| = \sqrt{(-4)^2 + 0^2} = 4$. Hence we obtain a system of equations

$$\left\{ \begin{array}{l} \cos \varphi = -1 \\ \sin \varphi = 0 \end{array} \right. \rightarrow \varphi = \pi \text{ (radian) or } \varphi = 180{}^\circ \text{ (degrees)}$$

Answer: $\varphi = \pi$ (radians) or $\varphi = 180{}^\circ$ (degrees).

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