Polar form

Every complex number $z=x+iy$ can be written as $z=r{e}^{iheta}$, where $r=\sqrt{x^2+y^2}$ is the modulus and $\theta$ is the argument. The principal argument lies in $(-\pi ,\pi ]$. Euler's formula gives $e^{i\theta }=\cos \theta +i\sin \theta$. Multiplying by $i$ rotates a complex number by $\pi /2$ anticlockwise, so $iz=i(x+iy)=-y+ix$. To find square roots, halve the argument and square-root the modulus.

Worked example: Find the complex number $w$ such that $w^2=-4i$ with argument in $(0,\pi )$.

Step 1: Write $-4i$ in polar form.
Step 2: Its modulus is $4$ and its principal argument is $-\pi /2$.
Step 3: The square roots have modulus $\sqrt{4}=2$ and arguments $-\pi /4$ and $3\pi /4$.
Step 4: The root with argument in $(0,\pi )$ is $2e^{3i\pi /4}$.
Step 5: Using Euler's formula, $w=2\left(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4}\right)=-\sqrt{2}+i\sqrt{2}$.