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

Example of Complex Addition

$\paren {-3 + 5 i} + \paren {4 + 2 i} + \paren {5 - 3 i} + \paren {-4 - 6 i} = 2 - 2 i$

Let:

$\ds z_1$

$=$

$\ds -3 + 5 i$

$\ds z_2$

$=$

$\ds 4 + 2 i$

$\ds z_3$

$=$

$\ds 5 - 3 i$

$\ds z_4$

$=$

$\ds -4 - 6 i$

These can be depicted in the complex plane as follows:

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

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 $z_4$.

$\blacksquare$

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Graphical Representation of Complex Numbers. Vectors: $5 \ \text{(c)}$