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I need to find few examples about 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 $ain mathbb z/0, 1 in mathbb C $ if $a^x =a^y$ then $x=y$ is false

but these examples "not cool enough feels like so trivial" can you recommend some properties like these

Thanks.

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.

The field of complex numbers is algebraically closed, but the field of real numbers is not.

Here's an interesting "global" difference, which is unfortunately a bit abstract but hopefully still interesting:

Let's look at just the additive and multiplicative behavior - that is, we're considering $mathbbR$ and $mathbbC$ as fields. An automorphism of a field is a bijection from the field to itself which preserves the structure - e.g. $alpha(x+y)=alpha(x)+alpha(y)$, and so forth.

Every field has at least one automorphism, namely the identity (the trivial automorphism). $mathbbC$ has one obvious nontrivial automorphism, namely conjugation $$a+bimapsto a-biquad (a,binmathbbR),$$ and assuming the axiom of choice it has a lot more (although they're quite wild). By contrast, $mathbbR$ has no nontrivial automorphisms whatsoever! The key observation is that the ordering on $mathbbR$ is definable (which has a precise general meaning, incidentally) just from the field structure: $xge y$ iff there is some $z$ such that $y+z^2=x$. From this, together with the fact that each rational must be fixed by each field automorphism (a good exercise) and the density of $mathbbQ$ in $mathbbR$, we rule out any nontrivial automorphisms.

The two points where this breaks down for $mathbbC$ are:

• The relation $exists z(y+z^2=x)$ does not define an ordering on $mathbbC$ (indeed, $mathbbC$ as a field cannot be definably ordered at all).




• $mathbbQ$ is not in fact dense in $mathbbC$. Even without a definable ordering, if $mathbbQ$ were dense in $mathbbC$ we could at least conclude that there were no continuous automorphisms, but this prevents us from even saying that much (and indeed conjugation is continuous).

Considering the integers (which are included in the reals and complex) an interesting fact is that $5, 13, cdots$ are prime integers in the real field but they are not prime in the complex field (Gaussian Integers).