anonymous
  • anonymous
Why is i^i a real number?I want the reasoning behind it
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
i = √-1 √-1^√-1 = 1^1 = 1
anonymous
  • anonymous
right?
anonymous
  • anonymous
I meant \[i^i\] not \[i^2\]

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anonymous
  • anonymous
i^i = 0.207879576
anonymous
  • anonymous
i^i = (e^ln i)^i = e^(i*ln i) = e^(i*i (pi/2+2pi n)) = e^(-pi/2-2pi n)
anonymous
  • anonymous
I was asking why is it so?I mean all the math gives you the answer but I want to know why i^i would result in a real number
anonymous
  • anonymous
http://boards.4chan.org/sci/res/4520555
anonymous
  • anonymous
You should try there.
Savvy
  • Savvy
write in euler form.... \[i = e ^{i\pi/2}\] so \[i ^{i}\] = \[e ^{i ^{2}\pi/2}\] = \[e ^{-\pi/2}\] =\[1/e ^{\pi/2}\] which is real for obvious reasons....
anonymous
  • anonymous
But don't you think it's strange that i^i is a real number
Savvy
  • Savvy
NO.....the proof is in front of you....:) it's just an amazing result of mathematics....
anonymous
  • anonymous
Yeah I can see the proof.But everything in math has a reasoning behind it right?
Savvy
  • Savvy
yeah it does...
anonymous
  • anonymous
Because when I first saw it I wouldn't have imagined that i^i would be real.How can we explain it?
Savvy
  • Savvy
i can understand that....but sorry man i can't explain that in any of those logical ways....because we as students still don't know much about 'i'...
anonymous
  • anonymous
Yeah I know.But here's somethng interesting.Recently I read something about what is i.Apparently i is something you multiply with to get a 90 degree rotation of a vector.I did not completely understand it yet.But it's good to see that there is an explanation.Thanx for the reply
Savvy
  • Savvy
actually when you multiply a cartesian co-ordinate with \[e ^{i \theta}\] it rotates the curve/vector etc. by \[\theta\]
anonymous
  • anonymous
Yeah that's true.
Savvy
  • Savvy
and since \[i=e ^{i \pi/2}\] it rotates by 90 degrees...
anonymous
  • anonymous
Yeah.It is rotating in the imaginary plane
anonymous
  • anonymous
can also think as follows, what does \[b^x\] mean? one definition that works for all real numbers would be \[e^{x\ln(b)}\] and so what is the meaning of \[i^i\]? one meaning is \[e^{i\ln(i)}\] now log is not a single valued function for complex numbers, but one value is \[\ln(i)=\ln(1)+i\frac{\pi}{2}\] as in general for a complex number \[Log(z)=ln(|z|)+i\arg(z)\]

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