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 2004-05-20, 12:53 #1 mfgoode Bronze Medalist     Jan 2004 Mumbai,India 22×33×19 Posts Imaginary or real? I quite enjoyed browsing thru a thread on 'The Reimann Hypothesis' by Thom Ruley in math forum posted last year ('03). Many joined in the fray as it digressed to the definition of the sqr. rt. of -1 denoted by 'i' To those who participated and others interested I set the foll problem. 1) What is the value of i^i ? Is it an imaginary, real or complex no.? 2) How about i^-i ? A surprise is in store for many who can work these two out. :surprised Mally.
 2004-05-20, 13:20 #2 jinydu     Dec 2003 Hopefully Near M48 2·3·293 Posts How do I post that "blackout" function, so that users have to highlight in order to see what I type?
 2004-05-20, 13:29 #3 akruppa     "Nancy" Aug 2002 Alexandria 2,467 Posts Admittedly I had to look up ln(i) = i*Pi/2. From there, it's easy: i^i = e^(i*ln(i)) = e^(i*i*Pi/2) = e^(-Pi/2) ~= 0.20788, a real number, surprisingly enough! i^-i is just the reciprocal, ~4.8105 Alex PS: blacked out text goes into spoiler tags: [ s p o i l e r ] and [ / s p o i l e r ] (without the spaces) Last fiddled with by akruppa on 2004-05-20 at 13:30
2004-05-20, 13:48   #4
R.D. Silverman

Nov 2003

22·5·373 Posts

Quote:
 Originally Posted by mfgoode I quite enjoyed browsing thru a thread on 'The Reimann Hypothesis' by Thom Ruley in math forum posted last year ('03). Many joined in the fray as it digressed to the definition of the sqr. rt. of -1 denoted by 'i' To those who participated and others interested I set the foll problem. 1) What is the value of i^i ? Is it an imaginary, real or complex no.? 2) How about i^-i ? A surprise is in store for many who can work these two out. :surprised Mally.
The question is not quite posed correctly. i^i does not have a unique value.
The question should either be: what is the value of i^i assuming the
principal branch of the logarithm function, OR "what is the smallest possible
absolute value of i^i" OR "classify all possible values of i^i" etc.

2004-05-20, 17:04   #5

"Richard B. Woods"
Aug 2002
Wisconsin USA

170148 Posts

Quote:
 Originally Posted by akruppa Admittedly I had to look up ln(i) = i*Pi/2.
... or remember the polar-coordinate "cis theta" form (i = e[sup]i * theta[/sup] = 0 * cos theta + i * sin theta), obtaining theta = pi/2 + 2 * pi * N for N = ..., -1, 0, 1, ... {thank you, Mr. Silverman}

 2004-05-20, 20:29 #6 S80780   Jan 2003 far from M40 12510 Posts I first PMed this to Mally: - can someone give a -sketchy?- (dis)proof for the last question? - 1) What is the value of i^i? i^i = e^[(2k+0.5)*pi*i*i] = e^[-(2k+0.5)*pi], k in Z, so it is real. 2) What is the value of i^-i? i^-i = (i^i)^-1 = e^[(2k+0.5)*pi], k in Z, so it is real. I expect the mainvalue is k=0. :question: A more interesting question here would be, if e^pi is in Q. Benjamin
 2004-05-20, 20:42 #7 biwema     Mar 2004 5758 Posts different way to hide Maybe it is much more subtle, to hide the messages that way: It is much more subtle. This is now the hidden message. On the other hand, some people might not find the text at all. You can still make it visible the same way.
 2004-05-21, 00:30 #8 jinydu     Dec 2003 Hopefully Near M48 2×3×293 Posts How did you do that?
 2004-05-21, 01:05 #9 jinydu     Dec 2003 Hopefully Near M48 2·3·293 Posts Also, is this correct, Bob Silverman? "Robert Silverman is a senior research scientist at RSA Laboratories in Bedford, MA. He has an A.B. from Harvard in Applied Mathematics and a Masters (an ABD) from the University of Chicago in Operations Research. he spent four years at Data Resources Inc. and ten years at the MITRE Corporation where he was a Principal Scientist. His research interests include parallel and massively distributed computing, computational number theory, algorithmic complexity theory and general design and analysis of numerical algorithms. He is a member of the American Mathematical Society."
2004-05-21, 06:21   #10
biwema

Mar 2004

3·127 Posts

Quote:
 Originally Posted by jinydu How did you do that?
Just change the textcolor in a way that it is the same as the background.

use the tag in [ ]:

COLOR=#F5F5FF

text

/COLOR

2004-05-21, 11:28   #11
R.D. Silverman

Nov 2003

11101001001002 Posts

Quote:
 Originally Posted by S80780 I first PMed this to Mally: - can someone give a -sketchy?- (dis)proof for the last question? - snip A more interesting question here would be, if e^pi is in Q.[/spoiler] Benjamin
e^pi is easily proved transcendental (unlike e*pi or e+pi) via the Gelfond
Schneider theorem. Note that i^i satisfies the conditions for the theorem.

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