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Raising things to i power, I came across 4.8104773809655 
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#1
Jun1005, 10:27 AM

P: 57

I was working on the general meaning of taking things to the i power  I was pondering the meaning of
e^pi*i = 1 The proof for this thing is well established, I was musing about meaning. An obvious question was: What to the i power goes to zero? As I was hunting for that number (on my TI83 while driving to work  and it is very hard to press [2nd] i on that thing with one hand), I came across this number: 4.8104773809655 Namely: 4.8104773809655^i => i ...at least on my TI83. 4.81047738096535^i on the google calculator. What is this number, and who has done work on it? Regards, Steve Rives 


#2
Jun1005, 10:51 AM

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P: 1,123

For:
[tex]x^i = i[/tex] The solutions for x are: [tex]e^{\frac{\pi}{2} + n\pi} \quad \forall n \in \mathbb{Z}[/tex] So if you hadn't already guessed: [tex]e^{\frac{\pi}{2}} \approx 4.8104773809653516554730356667 \ldots[/tex] As for the equation: [tex]i^x = 0[/tex] Well there exists no complex solution for x. In fact the only time: [tex]a^b = 0[/tex] For a and b in complex numbers is when [itex]a=0[/itex] and [itex]b \neq 0[/itex]. 


#3
Jun1005, 10:54 AM

P: 57

Of coruse! Only, I found it the hard way just now:
ln(4.81047738...) is 1.5707963 And that's pi/2 


#4
Jun1005, 07:12 PM

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P: 9,488

Raising things to i power, I came across 4.8104773809655
he did not ask when i^x = 0, he asked (what)^i >0?
so we might look at a^i = e^(i ln(a)) = cos(ln(a)) + i sin(ln(a)). but again it is clear that this number has absolute value 1, so cannot approach zero, at least not for real a. now for complex a, we just need to solve for when e^z goes to 0. but e^(n)>0 for example, so therefore e^i(in) >0 too. so x = in satisfies e^(ix) goes to zero as n goes to infinity, i.e. as x goes to i.infinity. 


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