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

whyit 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∈C there exist x,y∈R such that z=x+i⋅y, right? x is the real part and y is the imaginary part? Well, there also exist r,φ∈R, r≥0 such that z=r⋅(cosφ+isinφ). Here, r is called the *length* or *absolute value* of z and φ is called the *angle* or *argument*, measured counterclockwise from the positive real axis.

We can use cartesian coordinates to add complex numbers: (x1+i⋅y1)+(x2+i⋅y2)=(x1+x2)+i⋅(y1+y2)

We can use cartesian coordinates to multiply complex numbers:

(x1+i⋅y1)⋅(x2+i⋅y2)=(x1x2−y1y2)+i⋅(x1y2+y1x2)

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

(r1(cosφ1+isinφ1))⋅(r2(cosφ2+isinφ2))=(r1⋅r2)(cos(φ1+φ2)+isin(φ1+φ2))

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∘, the angle of i needs to be 90∘ or 270∘.

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