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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 . 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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