# What’s the difference between R2\mathbb{R}^2 and the complex plane?

I haven’t taken any complex analysis course yet, but now I have this question that relates to it.

Let’s have a look at a very simple example. Suppose $x,y$ and $z$ are the Cartesian coordinates and we have a function $z=f(x,y)=\cos(x)+\sin(y)$. However, now I change the $\mathbb{R}^2$ plane $x,y$ to complex plane and make a new function, $z=\cos(t)+i\sin(t)$.

So, can anyone tell me some famous and fundamental differences between complex plane and $\mathbb{R}^2$ by this example, like some features $\mathbb{R}^2$ has but complex plane doesn’t or the other way around. (Actually I am trying to understand why electrical engineers always want to put signal into the complex numbers rather than $\mathbb{R}^2$, if a signal is affected by 2 components)

Thanks for help me out!

$\mathbb{R}^2$ and $\mathbb{C}$ have the same cardinality, so there are (lots of) bijective maps from one to the other. In fact, there is one (or perhaps a few) that you might call “obvious” or “natural” bijections, e.g. $(a,b) \mapsto a+bi$. This is more than just a bijection:

• $\mathbb{R}^2$ and $\mathbb{C}$ are also metric spaces (under the ‘obvious’ metrics), and this bijection is an isometry, so these spaces “look the same”.
• $\mathbb{R}^2$ and $\mathbb{C}$ are also groups under addition, and this bijection is a group homomorphism, so these spaces “have the same addition”.
• $\mathbb{R}$ is a subfield of $\mathbb{C}$ in a natural way, so we can consider $\mathbb{C}$ as an $\mathbb{R}$-vector space, where it becomes isomorphic to $\mathbb{R}^2$ (this is more or less the same statement as above).

Here are some differences:

• Viewing $\mathbb{R}$ as a ring, $\mathbb{R}^2$ is actually a direct (Cartesian) product of $\mathbb{R}$ with itself. Direct products of rings in general come with a natural “product” multiplication, $(u,v)\cdot (x,y) = (ux, vy)$, and it is not usually the case that $(u,v)\cdot (x,y) = (ux-vy, uy+vx)$ makes sense or is interesting in general direct products of rings. The fact that it makes $\mathbb{R}^2$ look like $\mathbb{C}$ (in a way that preserves addition and the metric) is in some sense an accident. (Compare $\mathbb{Z}[\sqrt{3}]$ and $\mathbb{Z}^2$ in the same way.)
• Differentiable functions $\mathbb{C}\to \mathbb{C}$ are not the same as differentiable functions $\mathbb{R}^2\to\mathbb{R}^2$. (The meaning of “differentiable” changes in a meaningful way with the base field. See complex analysis.) The same is true of linear functions. (The map $(a,b)\mapsto (a,-b)$, or $z\mapsto \overline{z}$, is $\mathbb{R}$-linear but not $\mathbb{C}$-linear.)