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**Why does ln(i) = (1/2pi)i????**

I was bored the other day and wondered whether or not it would be possible to find out the natural log of the imaginary number

*i*. Typed it into my TI-84 and it said the answer was 1.57079632

*i*. I wondered why the might be the case, thought about it for a while and noticed that 1.5707 is equal to 1/2pi. Decided to ask my calculator what ln(-1) equaled, which was as predicted pi*

*i*.

ln(i) = 1/2ln(-1)

ln(-1

^{1/2}) = 1/2ln(-1)

1/2ln(-1) = 1/2ln(-1)

Makes enough sense to me.

What confuses/interests me is this: What is it that ties the number

*i*, the natural logarithm, and pi together? How does 1/2pi(

*i*) get spit out? I had no idea that pi and

*i*could possibly be related to each other in any way. Why is pi = [ln(1)/

*i*]?

Further, can the perimeter/are of a circle be defined in terms of

*i*?

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