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

Example of Complex Addition

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

Proof 1

\(\ds \paren {2 + 3 i} + \paren {4 - 5 i}\)

\(=\)

\(\ds \paren {2 + 4} + \paren {3 - 5} i\)

Definition of Complex Addition

\(\ds \)

\(=\)

\(\ds 6 - 2 i\)

$\blacksquare$

Proof 2

Let the complex numbers $2 + 3 i$ and $4 - 5 i$ be represented by the points $P_1$ and $P_2$ respectively in the complex plane.

Complete the parallelogram with $OP_1$ and $OP_2$ as the adjacent sides.

By the Geometrical Interpretation of Complex Addition, the point $P$ represents the complex number $6 - 2 i$, which is the sum of $2 + 3 i$ and $4 - 5 i$.

$\blacksquare$