# Why is the complex plane shaped like it is?

It’s always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn’t $i$ just at some non-90 degree angle to the real number line? Could someone please explain the logic or rationale behind this? It seems self-apparent to me, but I cannot actually see why it is.

Furthermore, why is the real number line even straight? Why does it not bend or curve? I suppose arbitrarily it might be strange to bend it, but why couldn’t it bend at 0? Is there a proof showing why?

Of course, these things seem natural to me and make sense, but why does the complex plane have its shape? Is there a detailed proof showing precisely why, or is it just an arbitrary choice some person made many years ago that we choose to accept because it makes sense to us?

## Answer

There is a really important aspect of complex numbers that depends on the complex plane having exactly this shape: complex multiplication.

Complex numbers can not only be characterized in cartesian coordinates by a real part and an imaginary part, but also in polar coordinates by a length and an angle.

You know that for any $z \in \mathbb{C}$ there exist $x, y \in \mathbb{R}$ such that $z = x+i\cdot y$, right? $x$ is the real part and $y$ is the imaginary part? Well, there also exist $r, \varphi \in \mathbb{R}$, $r \geq 0$ such that $z = r\cdot(\cos\varphi + i\sin\varphi)$. Here, $r$ is called the length or absolute value of $z$ and $\varphi$ is called the angle or argument, measured counterclockwise from the positive real axis.

We can use cartesian coordinates to add complex numbers:

We can use cartesian coordinates to multiply complex numbers:

However, we can also use polar coordinates to multiply complex numbers:

So to multiply two complex numbers in polar coordinates, you multiply their lengths and add their angles. I personally think this is incredibly helpful for visualization, and this also shows why the imaginary axis needs to be at a right angle to the real axis: since the angle of $-1$ is $180^\circ$, the angle of $i$ needs to be $90^\circ$ or $270^\circ$.

Attribution
Source : Link , Question Author : user64742 , Answer Author : Anon