Question

Why is i^i a real number?I want the reasoning behind it

Answer 1

i = √-1

√-1^√-1 = 1^1 = 1

Answer 2

right?

Answer 3

I meant \[i^i\] not \[i^2\]

Answer 4

i^i = 0.207879576

Answer 5

i^i = (e^ln i)^i = e^(i*ln i) = e^(i*i (pi/2+2pi n)) = e^(-pi/2-2pi n)

Answer 6

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

Answer 7

http://boards.4chan.org/sci/res/4520555

Answer 8

You should try there.

Answer 9

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....

Answer 10

But don't you think it's strange that i^i is a real number

Answer 11

NO.....the proof is in front of you....:) it's just an amazing result of mathematics....

Answer 12

Yeah I can see the proof.But everything in math has a reasoning behind it right?

Answer 13

yeah it does...

Answer 14

Because when I first saw it I wouldn't have imagined that i^i would be real.How can we explain it?

Answer 15

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'...

Answer 16

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

Answer 17

actually when you multiply a cartesian co-ordinate with \[e ^{i \theta}\]
it rotates the curve/vector etc. by \[\theta\]

Answer 18

Yeah that's true.

Answer 19

and since \[i=e ^{i \pi/2}\] it rotates by 90 degrees...

Answer 20

Yeah.It is rotating in the imaginary plane

Answer 21

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
\[Log(z)=ln(|z|)+i\arg(z)\]