I was playing around with iota in my Math lecture today and I observed something very, very strange.
We can write a complex number z, in polar form as
$$ z = r e^{\iota\theta} $$
where e is the base of the natural logarithm.
So, going in the same fashion, we can also express iota itself in this form, as
$$ i = e^{\iota\pi/2} $$
And then we have the expression of iota that itself equals to the power of the base of the natural logarithm (multiplied by pi/2)
So I wanted to ask if this recurrence (or dependency, pardon me I couldn't really get the right word for it), is just a mere coincidence, or has some mathematical implications as well? Because this recurrence implies that we can express the complex number z
$$ z = re^{\iota\theta} $$
as
$$ z = re^{e^{\iota\pi/2}\theta} $$
And in the similar manner, as
$$ z = re^{e^{e^{\iota\pi/2}\pi/2}\theta} $$
And so on.
So does this recursive plugging of the value of iotas really have any implications or consequences, or any meaning perhaps?
Thankyou.