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

Example of Complex Addition

$\paren {3 + 4 i} + \paren {5 + 2 i} = 8 + 6 i$

Proof 1

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

\(=\)

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

Definition of Complex Addition

\(\ds \)

\(=\)

\(\ds 8 + 6 i\)

$\blacksquare$

Proof 2

Let the complex numbers $3 + 4 i$ and $5 + 2 i$ 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 $8 + 6 i$, which is the sum of $3 + 4 i$ and $5 + 2 i$.

$\blacksquare$