Question

The imaginary part of $i^i$ is

• A. $0$
• B. $1$
• C. $2$
• D. $-1$

Answer 1

The correct option is A

$0$

Find the imaginary part of the given expression

Given expression, $i^i$

We know that ,

$e^{i\theta }=\cos \theta +i\sin \theta$

$\therefore$ $e^{\frac{i\mathrm{\pi }}{2}}=\cos \frac{\mathrm{\pi }}{2}+i\sin \frac{\mathrm{\pi }}{2}$

$\therefore$ $e^{\frac{i\mathrm{\pi }}{2}}=i$

Let $i^i=u+iv$

Take natural logarithm on both sides

$\ln \left(u+iv\right)=\ln \left(i^i\right)$

$=i\ln \left(i\right)$

$=i\ln \left(e^{\frac{i\mathrm{\pi }}{2}}\right)$

$=i\times \frac{i\mathrm{\pi }}{2}$

$=\frac{-\mathrm{\pi }}{2}$

$\therefore$ $u+iv=e^{\frac{-\mathrm{\pi }}{2}}$

On comparing, we get that the imaginary part of the number $i^i$ is $0$.

Hence option $\left(A\right)$ $0$, is the correct answer.