Power of i
Power of i#
A day ago I was nerd sniped by a YouTube video asking what the value of the following expression is:
\[\sqrt[i]{i}\]
I had never thought of raising any number to the \(i\)th power, let alone the \(\frac{1}{i}\)th power.
This led me down the rabbit hole of figuring how to raise arbitrary numbers to the \(i\)th power. There were all sorts of interesting identities surrounding the complex number \(i\). For example,
\[i = e^{\frac{i \pi}{2}},\]
which implies
\[\ln (i) = \frac{i \pi}{2}.\]
There is also this gem,
\[i^i = e^{-\frac{\pi}{2}}\]
For those interested in the general case, there’s a whole Wolfram page devoted to the topic.
I haven’t been this distracted since diving into dual numbers a few years ago.