Question

Given z1 = 1 + i, z2 = 1 + i, find w = z1 ⋅ z2, and graph z1, z2, and w.

Answer 1

We draw complex numbers as points, where the x,y coordinates are the real and imaginary part:

$z = a+bi \mapsto (a,b)$

So, we have

$z_1 = z_2 = 1+i \mapsto (1,1)$

So, both $z_1$ and $z_2$ are represented by the point (1,1).

Their product is

$w=z_1\cdot z_2 = (1+i)^2 = 1+2i+i^2 = 1+2i-1=2i$

And its representation is

$w=2i = 0+2i \mapsto (0,2)$