1. ## complex nos problem

which is greater

i^i or i^-i

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

2. Can complex numbers be ordered?

3. 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.

4. Originally Posted by pickslides
Can complex numbers be ordered?
Yes, e.g. by the lexiographic order. However, there is no ordered complex field.

5. $\displaystyle i^i = \left(e^{\frac{\pi}{2}i}\right)^i$

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

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

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 $\displaystyle i^{-i} > i^i$.