To calculate [tex]\( i^i \)[/tex], where [tex]\( i \)[/tex] is the imaginary unit and defined as [tex]\( \sqrt{-1} \)[/tex], we can use the properties of complex numbers and Euler's formula. Here is a step-by-step solution:
1. Expressing [tex]\( i \)[/tex] in exponential form:
The imaginary unit [tex]\( i \)[/tex] can be represented in exponential form using Euler's formula [tex]\( e^{i\theta} \)[/tex], which states that [tex]\( e^{i\theta} = \cos(\theta) + i\sin(\theta) \)[/tex]. For [tex]\( i \)[/tex], we have:
[tex]\[
i = e^{i\frac{\pi}{2}}
\][/tex]
This works because [tex]\( \cos\left(\frac{\pi}{2}\right) = 0 \)[/tex] and [tex]\( \sin\left(\frac{\pi}{2}\right) = 1 \)[/tex], hence:
[tex]\[
e^{i\frac{\pi}{2}} = \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right) = 0 + i = i
\][/tex]
2. Raising [tex]\( i \)[/tex] to the power of [tex]\( i \)[/tex]:
Now we need to find [tex]\( i^i \)[/tex]. Using our exponential form of [tex]\( i \)[/tex], this becomes:
[tex]\[
i^i = \left(e^{i\frac{\pi}{2}}\right)^i
\][/tex]
Using the property of exponents [tex]\( (a^b)^c = a^{bc} \)[/tex], we get:
[tex]\[
i^i = e^{i \cdot i\frac{\pi}{2}} = e^{-\frac{\pi}{2}}
\][/tex]
Here, the exponent simplifies because [tex]\( i \cdot i = -1 \)[/tex].
3. Simplifying and interpreting the result:
So, [tex]\( i^i \)[/tex] simplifies to:
[tex]\[
i^i = e^{-\frac{\pi}{2}}
\][/tex]
Evaluating [tex]\( e^{-\frac{\pi}{2}} \)[/tex] numerically:
[tex]\[
e^{-\frac{\pi}{2}} \approx 0.20787957635076193
\][/tex]
The result is a real number since the imaginary part of [tex]\( e^{-\frac{\pi}{2}} \)[/tex] is 0.
Thus, the value of [tex]\( i^i \)[/tex] is approximately [tex]\( 0.20787957635076193 \)[/tex], and there is no imaginary part. Therefore, the solution can be expressed as:
[tex]\[
i^i \approx 0.20787957635076193 + 0i
\][/tex]
