Complex Arithmetic/Examples/1 2^-1 (4-3i) + 3 2^-1 (5+2i)

Example of Complex Arithmetic

$\dfrac 1 2 \paren {4 - 3 i} + \dfrac 3 2 \paren {5 + 2 i} = \dfrac {19} 2 + \dfrac 3 2 i$

Proof 1

\(\ds \)

\(\)

\(\ds \dfrac 1 2 \paren {4 - 3 i} + \dfrac 3 2 \paren {5 + 2 i}\)

\(\ds \)

\(=\)

\(\ds \paren {2 - \dfrac 3 2 i} + \paren {\dfrac {15} 2 + 3 i}\)

\(\ds \)

\(=\)

\(\ds \paren {\paren 2 + \dfrac {15} 2} + \paren {-\dfrac 3 2 + 3} i\)

\(\ds \)

\(=\)

\(\ds \dfrac {19} 2 + \dfrac 3 2 i\)

$\blacksquare$

Proof 2

Let the complex numbers $\dfrac 1 2 \paren {4 - 3 i}$ and $\dfrac 3 2 \paren {5 + 2 i}$ 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.

Using Geometrical Interpretation of Complex Addition, the point $P$ represents the complex number $\dfrac {19} 2 + \dfrac 3 2 i$, which is the sum of $\dfrac 1 2 \paren {4 - 3 i}$ and $\dfrac 3 2 \paren {5 + 2 i}$.

Hence, $\dfrac {19} 2 + \dfrac 3 2 i$ is the sum of $\dfrac 1 2 \paren {4 - 3 i}$ and $\dfrac 3 2 \paren {5 + 2 i}$.

$\blacksquare$