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I've been doing a lot of thinking about imaginary numbers lately. My first question was "What is sqr(i)?".
I thought it was unsolvable until I punched it into my trusty (and often right) caclulator and found out it was (sqr(.5)i + sqr(.5))^2
So obvious now. Of course.
Anyways, a while later, I thought about series of powers (the same way a power is a series of multiplications and those a series of additions). (We'll call a series of power ^^)
So my question is:
x^^2 = x^x = -4
What is x?
My calculator tells me: false. But I have this tendency not to trust it sometimes, especially when it won't give me an answer.
Also, is i ^ i defined? ((-1)^.5)^((-1)^.5) = (-1)^(i/2) = ?
I thought it was unsolvable until I punched it into my trusty (and often right) caclulator and found out it was (sqr(.5)i + sqr(.5))^2
So obvious now. Of course.
Anyways, a while later, I thought about series of powers (the same way a power is a series of multiplications and those a series of additions). (We'll call a series of power ^^)
So my question is:
x^^2 = x^x = -4
What is x?
My calculator tells me: false. But I have this tendency not to trust it sometimes, especially when it won't give me an answer.
Also, is i ^ i defined? ((-1)^.5)^((-1)^.5) = (-1)^(i/2) = ?