[Math] properties that real numbers hold but complex numbers does not

Question

I need to find a few examples about the differences between real numbers and complex numbers like:

1) if $x \in \mathbb R $ then $x^2 \geq0$ is true

if $z \in \mathbb C $ then $z^2 \geq0$ is false

2) let $a \in \mathbb R/\{0, 1\} $ if $a^x =a^y$ then $x=y$ is true

let $a\in \mathbb z/\{0, 1\} \in \mathbb C $ if $a^x =a^y$ then $x=y$ is false

But these examples are not cool enough and feel very trivial. Can you suggest some other properties like these?

Thanks.

Answer 1

The relation $<$ on the real numbers is a total order that preserves order under addition and multiplication in the way we're used to, but there is no such order-preserving total order on the complex numbers.