**Exercise 4.** Show that it is impossible to define a total ordering on $\mathbb{C}$. In other words, one cannot find a relation $≻$ between complex numbers so that:

(1) For any two complex numbers $z, w$, one and only one of the following is true: $z≻w, w≻z$ or $z=w$.

(2) For all $z_1, z_2, z_3 \in \mathbb{C}$ the relation $z_1≻z_2$ implies $z_1+z_3≻z_2+z_3$.

(3) Moreover, for all $z_1, z_2, z_3 \in \mathbb{C}$ with $z_3≻0$, then $z_1≻z_2$ implies $z_1 z_3≻z_2 z_3$.

**Solution 4.** Suppose, for a contradiction, that $i≻0$, then $-1=i \cdot i≻0 \cdot i=0$. Now we get $-i≻-1 \cdot i≻0$. Therefore $i-i≻i+0=i$. But this contradicts our assumption. We obtain a similar situation in the case $0≻i$. So we must have $i=0$. But then for all $z \in \mathbb{C}$ we have $z \cdot i=z \cdot 0=0$ Repeating we have $z=0$ for all $z \in \mathbb{C}$. So this relation would give a trivial total ordering.