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Is the sqrt of -1 equal to this: ?

Since i (the sqrt of -1) can be written as e^(iθ), and θ is pi/2 here, 'i' can be written as e^(pi/2 i) which equals (e^(pi/2)^i). e^pi/2 = 4.81 to 3 sig fig.
It follows that 4.81^i = i.

Fascinating? Are there any other such examples of this type: a^n = n ?

(edited 8 years ago)

Reply 1

B_9710

Original post by calculador

This looks right. There are many other results like this that can be shown in this way. A good one is i^i =(e^(iπ/2))^i = e^(-π/2). Which of you work out is about 0.208. This is not the only answer. Equally i^i=(e^(i5π/2))^i and so i^i=e^(-5π/2) which is about 3.9 x 10^-4 So this is a seemingly weird and fascinating property about complex numbers.

(edited 8 years ago)

That is correct.

Oh, maths makes my head spin sometimes. Like the fact that there is a number r that is irrational, but r to the power of the square root of 2 is rational. It's mental how numbers can have such properties, but such is maths...