complex nos problem

• Oct 11th 2010, 07:43 PM
prasum
complex nos problem
which is greater

i^i or i^-i

i^i^i or i^(pi/2)
• Oct 11th 2010, 07:46 PM
pickslides
Can complex numbers be ordered?
• Oct 11th 2010, 11:34 PM
Opalg
Quote:

Originally Posted by prasum
which is greater

i^i or i^-i

i^i^i or i^(pi/2)

Enter these expressions as Google searches if you want them evaluated. For example, a search for i^i will get the response i^i = 0.207879576. In fact, i^i and i^-i are real numbers, so it makes sense to ask which is greater. But i^(i^i) and i^(pi/2) are complex. As pickslides points out, there is no natural sense in which one is greater than the other.
• Oct 11th 2010, 11:58 PM
raphw
Quote:

Originally Posted by pickslides
Can complex numbers be ordered?

Yes, e.g. by the lexiographic order. However, there is no ordered complex field.
• Oct 12th 2010, 12:45 AM
Prove It
$\displaystyle i^i = \left(e^{\frac{\pi}{2}i}\right)^i$

$\displaystyle = e^{\frac{\pi}{2}i^2}$

$\displaystyle = e^{-\frac{\pi}{2}}$.

Meanwhile $\displaystyle i^{-i} = (i^i)^{-1}$

$\displaystyle = \left(e^{-\frac{\pi}{2}}\right)^{-1}$

$\displaystyle = e^{\frac{\pi}{2}}$.

Clearly $\displaystyle e^{\frac{\pi}{2}} > e^{-\frac{\pi}{2}}$, so $\displaystyle i^{-i} > i^i$.