Complex Arithmetic/Examples/3(1+2i) - 2(2-3i)/Proof 2

Example of Complex Arithmetic

$3 \paren {-1 + 4 i} - 2 \paren {7 - i} = -17 + 14 i$

Proof

By definition of complex subtraction:

$3 \paren {1 + 2 i} - 2 \paren {2 - 3 i} = \paren {7 + i} + 2 \paren {-2 + 3 i}$

Let the complex numbers $3 \paren {1 + 2 i}$ and $2 \paren {-2 + 3 i}$ 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 $-1 + 12 i$, which is the sum of $3 \paren {1 + 2 i}$ and $2 \paren {-2 + 3 i}$.

Hence, $-1 + 12 i$ is the difference of $3 \paren {1 + 2 i}$ and $2 \paren {2 - 3 i}$.

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $61 \ \text {(c)}$