Are these Real numbers? $e^{i}$ , $i^{i}$ , $\sqrt{i}$

Algebra Level 3

Fill in the Blank:

Some of the following 3 expressions are multi-valued, while the rest has/have only one value: $e^{i},\quad i^{i},\quad \sqrt{i}.$ Also, precisely $\text{\_\_\_\_\_\_}$ of them has/have a value that is strictly real.

Notations:

$e \approx 2.718$ denotes the Euler's number .
$i = \sqrt{-1}$ denotes the imaginary unit .

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1 solution

Chew-Seong Cheong
Sep 17, 2017

Using Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$ , we have:

$\begin{aligned} e^i & = \cos 1 + i \sin 1 & \color{#D61F06} \text{Complex} \\ i^i & = \left(e^{i\frac \pi 2}\right)^i = e^{-\frac \pi 2} & \color{#3D99F6} \text{Real} \\ \sqrt{i} & = i^\frac 12 = \left(e^{i\frac \pi 2}\right)^\frac 12 = e^{i\frac \pi 4} = \cos \frac \pi 4 + i \sin \frac \pi 4 & \color{#D61F06} \text{Complex} \end{aligned}$

Therefore, there is only $\boxed{1}$ , $i^i$ , has a real value.

Are these Real numbers? e i e^{i} e i , i i i^{i} i i , i \sqrt{i} i ​

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Are these Real numbers? $e^{i}$ , $i^{i}$ , $\sqrt{i}$