Question

Let $z$ being a complex number whose argument is $\frac{\pi }{4}$ and modulus is $1$. Then find the complex number $z^2+z$.

Answer 1

Given, $z$ being a complex number whose argument is $\frac{\pi }{4}$ and modulus is $1$.

Then the number can be written as

$z=1.e^{\frac{i\pi }{4}}=\frac{1}{\sqrt{2}}(1+i)$.

Now,

$z^2+z$

$=\frac{1}{2}\left(2i\right)+\frac{1}{\sqrt{2}}(1+i)$ [ Since $(1+i{)}^2=2i$]

$=i+\frac{1}{\sqrt{2}}(1+i)$

$=\frac{1}{\sqrt{2}}+(1+\frac{1}{\sqrt{2}})i$.