10/12/2007, 09:15 PM
Dear Matt! I got you!
If you mean (i,i,3), the outcome is rather ... civilized (but, nevertheless, complex):
In fact, (i,i,3) = i^(i^i) = i#3 = i-tower-3 and we may proceed as follows:
i#1 = i;
i#2 = i^(i#1) = i^i = e^(Pi*i*i/2) = e^(-Pi/2) = 0.207879572..;
i#3 = i^0.207879572..= e^(i*0.326536474..) = 0.947158998.. + i . 0.320764449..;
Then: (i,i,3) = 0.947158998.. + i . 0.320764449.. . So far, so good!
On the contrary, if you actually mean:
(i,i,i) = i ^ .... (i^i)[i times] = i-tower-i = i-penta-2, then the problem is "really complicated", instead of simply being only ... complex.
Nevertheless, we are not afraid of anything, since we even found an "infinite tower" with the height equal to the imaginary unit. In fact, we have:
i = e^(Pi*i/2), which is self-explanatory, as far its value is concerned. However, we can write it as: i = (e^(Pi/2))^i = k^i , which defines an infinite tower (i = k#oo), with "base" k=4.810477381.. (real), and with a height equal to ... "i".
I go and drink another one. See you soon!
GFR
If you mean (i,i,3), the outcome is rather ... civilized (but, nevertheless, complex):
In fact, (i,i,3) = i^(i^i) = i#3 = i-tower-3 and we may proceed as follows:
i#1 = i;
i#2 = i^(i#1) = i^i = e^(Pi*i*i/2) = e^(-Pi/2) = 0.207879572..;
i#3 = i^0.207879572..= e^(i*0.326536474..) = 0.947158998.. + i . 0.320764449..;
Then: (i,i,3) = 0.947158998.. + i . 0.320764449.. . So far, so good!
On the contrary, if you actually mean:
(i,i,i) = i ^ .... (i^i)[i times] = i-tower-i = i-penta-2, then the problem is "really complicated", instead of simply being only ... complex.
Nevertheless, we are not afraid of anything, since we even found an "infinite tower" with the height equal to the imaginary unit. In fact, we have:
i = e^(Pi*i/2), which is self-explanatory, as far its value is concerned. However, we can write it as: i = (e^(Pi/2))^i = k^i , which defines an infinite tower (i = k#oo), with "base" k=4.810477381.. (real), and with a height equal to ... "i".
I go and drink another one. See you soon!
GFR