20040520, 12:53  #1 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×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. 
20040520, 13:20  #2 
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?

20040520, 13:29  #3 
"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 20040520 at 13:30 
20040520, 13:48  #4  
Nov 2003
2^{2}·5·373 Posts 
Quote:
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. 

20040520, 17:04  #5  
"Richard B. Woods"
Aug 2002
Wisconsin USA
17014_{8} Posts 
Quote:


20040520, 20:29  #6 
Jan 2003
far from M40
125_{10} 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 
20040520, 20:42  #7 
Mar 2004
575_{8} 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. 
20040521, 00:30  #8 
Dec 2003
Hopefully Near M48
2×3×293 Posts 
How did you do that?

20040521, 01:05  #9 
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." 
20040521, 06:21  #10  
Mar 2004
3·127 Posts 
Quote:
use the tag in [ ]: COLOR=#F5F5FF text /COLOR 

20040521, 11:28  #11  
Nov 2003
1110100100100_{2} Posts 
Quote:
Schneider theorem. Note that i^i satisfies the conditions for the theorem. 

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