







CTK Exchange
kfom
guest

Jul0606, 08:19 PM (EST) 

7. "RE: Real or complex number"
In response to message #0

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! 

Alert  IP 
Printerfriendly page 
Reply 
Reply With Quote  Top 



JJ
guest

Jul0706, 07:34 AM (EST) 

8. "RE: Real or complex number"
In response to message #7

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)


Alert  IP 
Printerfriendly page 
Reply 
Reply With Quote  Top 






JJ
guest

Jul0906, 05:47 AM (EST) 

10. "RE: Real or complex number"
In response to message #9

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) 

Alert  IP 
Printerfriendly page 
Reply 
Reply With Quote  Top 



You may be curious to have a look at the old CTK Exchange archive. Please do not post there.
Copyright © 19962018 Alexander Bogomolny

