Answer to Question #86584 in Complex Analysis for nur

Question

Question: please convert this complex number into the polar form.
1/2j

note: it is not (1/2) * j
it is 1/2*j --> j is at denominator

Answer 1

Multiply the numerator and denominator of this complex number by j

$$z=\frac{1}{2j}=\frac{1}{2j}\cdot \frac{j}{j}=\frac{j}{2{{j}^{2}}}=\frac{j}{2\cdot \left( -1 \right)}=-\frac{1}{2}j$$

So in a rectangular form we have

$$z=x+jy=-\frac{1}{2}j$$

where

$$x=0,\,\,y=-\frac{1}{2}$$

In polar form the complex number z is written as

$$z=r\left( \cos \theta +j\sin \theta \right)$$

where

$$r=\left| z \right|=\sqrt{{{x}^{2}}+{{y}^{2}}}=\sqrt{{{0}^{2}}+{{\left( -\frac{1}{2} \right)}^{2}}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$$ $$\theta =\arctan \left( \frac{y}{x} \right)=\arctan \left( \frac{-1/2}{0} \right)=\arctan \left( -\infty \right)=-\frac{\pi }{2}$$

or otherwise

$$\theta =-\frac{\pi }{2}+2\pi =\frac{3\pi }{2}$$

Substituting the obtained values, we get the polar form of this z

$$z=\frac{1}{2}\left( \cos \left( \frac{3\pi }{2} \right)+j\sin \left( \frac{3\pi }{2} \right) \right)$$