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Old 2004-05-20, 12:53   #1
mfgoode
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Lightbulb 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.
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Old 2004-05-20, 13:20   #2
jinydu
 
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How do I post that "blackout" function, so that users have to highlight in order to see what I type?
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Old 2004-05-20, 13:29   #3
akruppa
 
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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
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Old 2004-05-20, 13:48   #4
R.D. Silverman
 
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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.

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Old 2004-05-20, 17:04   #5
cheesehead
 
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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}
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Old 2004-05-20, 20:29   #6
S80780
 
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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
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Old 2004-05-20, 20:42   #7
biwema
 
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Default 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.
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Old 2004-05-21, 00:30   #8
jinydu
 
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How did you do that?
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Old 2004-05-21, 01:05   #9
jinydu
 
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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."
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Old 2004-05-21, 06:21   #10
biwema
 
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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
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Old 2004-05-21, 11:28   #11
R.D. Silverman
 
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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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