Is $i$ a positive number? This question came to mind when I saw a way of finding the value of $i^i$ which included transforming it to $e^{i\ln(i)}$ and taking $\ln(i)$ as $i\frac{\pi}{2}$
I understand how we can get $\ln(i)$=$i\frac{\pi}{2}$ geometrically but this got me thinking if $i$ is a positive number as the first thing I learned about $\ln$ since high school days is that it can't take negative numbers. 

As other answers pointed out, there is no total ordering of $\mathbb C$, which is compatible with the one of the real numbers. If there was such an ordering, we could compare $i$ and $0$, so that either $i > 0$ or $i < 0$. In the first case, $i > 0$ implies that $i \cdot i = -1 > 0$, a contradiction. In the second case, $i < 0$ implies that $i \cdot i = -1 > 0$, too. 
