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### Imaginary Numbers Raised to Imaginary Numbers

```
Date: 12/29/2001 at 23:10:01
From: Dan
Subject: Imaginary numbers raised to imaginary numbers

Hello. I was fooling around with my TI-89 graphing calculator, and I
input i^i, because of my infinite curiosity. The calculator returned
e^(-pi/2). I went to your FAQ and read all I could about imaginary
numbers and e, but it didn't really explain how or why i^i produced
that. If you could explain, it would be a great help.

Thanks a lot!
Dan
```

```
Date: 12/30/2001 at 04:51:59
From: Doctor Anthony
Subject: Re: Imaginary numbers raised to imaginary numbers

We can write

i = [cos(pi/2) + i.sin(pi/2)]

= e^(i.pi/2)

and raising both sides to power i we get

i^i = [e^(i.pi/2)]^i

= e^(i^2.pi/2)

= e^(-pi/2)

= 1/e^(pi/2)

= 0.2078795

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Imaginary/Complex Numbers
High School Imaginary/Complex Numbers

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