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- Chapter 2: The Core Python Language I
- Questions
- Complex numbers: $i^i$

Determine the value of $i^i$ as a real number, where $i = \sqrt{-1}$.

Click here for a solution

First let's get the "mathematical" answer: we can use Euler's formula, $e^{ix} = \cos x + i\sin x$ with $x=\pi/2$ to obtain $e^{i\pi/2} = i$. Therefore,
$$i^i = e^{i^2\pi/2} = e^{-\pi/2} = 0.207879576\cdots.$$
This is the *principal value*: $i^i$ is multi-valued and in general
$$i^i = e^{-\pi/2 + 2n\pi}, \;\; n=0,1,2,\cdots.$$

In the Python shell, the principal value is returned as a complex number object even though the answer is real:

>>> 1j**1j (0.20787957635076193+0j)

but we can get the real part as a `float`

by accessing its `real`

attribute:

>>> (1j**1j).real 0.20787957635076193