Complex Addition/Examples/(-3 + 5i) + (4 + 2i) + (5 - 3i) + (-4 - 6i)/Proof 2

Proof
Let:

These can be depicted in the complex plane as follows:


 * Complex-Addition-(-3+5i)+(4+2i)+(5-3i)+(-4-6i)--1.png

To find the required sum, proceed as in the following diagram:


 * Complex-Addition-(-3+5i)+(4+2i)+(5-3i)+(-4-6i)--2.png

Construct $z_2$ with its initial point placed at the terminal point of $z_1$.

Construct $z_3$ with its initial point placed at the terminal point of this instance of $z_2$.

Construct $z_4$ with its initial point placed at the terminal point of this instance of $z_3$.

The required resultant $z_1 + z_2 + z_3 + z_4$ of $z_1$ to $z_4$ is therefore represented by the terminal point of $z_4$.