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CTK Exchange
kfom
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Jul-06-06, 08:19 PM (EST) |
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7. "RE: Real or complex number"
In response to message #0
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I know this sounds really bizarre... I read some book a long time ago, something like "The Story of sqrt(-1)" anyway, all I can remember is the first digit:
i^i = 0.2...
It's real!
It's mindblowing! |
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JJ
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Jul-07-06, 07:34 AM (EST) |
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8. "RE: Real or complex number"
In response to message #7
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This isn't surprising at all. Operations and/or functions involving complex numbers often leads to real number.
For example, the most simplest cases are :
i*i = -1
i^i = exp(-pi/2)
ln(i)=pi/2
cos(i)=cosh(1)
cosh(i)=cos(1)
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JJ
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Jul-09-06, 05:47 AM (EST) |
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10. "RE: Real or complex number"
In response to message #9
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First : computation of real and imaginary parts of the logarithm of a complex number
ln(a+i.b)=x+i.y hence
a+i.b = exp(x+i.y) = exp(x).exp(i.y) = exp(x).(cos(y)+i.sin(y))
a = exp(x).cos(y) and b = exp(x).sin(y)
a^2 + b^2 = exp(2x) and b/a = tan(y)
x = (1/2)ln(a^2 +b^2)
y = atan(b/a)
Case of a=0, b=1 : x=0 and y=pi/2
hence : ln(i) = i.pi/2
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Second : computation of i^i
c^p = exp(p.ln(c)) hence i^i = exp(i.ln(i))
ln(i)=i.pi/2 hence i^i = exp(i.(i.pi/2))
i^i = exp((i^2)pi/2) = exp((-1)pi/2)
i^i = exp(-pi/2) |
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Copyright © 1996-2018 Alexander Bogomolny
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