Complex Subtraction/Examples/(7 + i) - (4 - 2i)/Proof 2
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Example of Complex Subtraction
- $\paren {7 + i} - \paren {4 - 2 i} = 3 + 3 i$
Proof
By definition of complex subtraction:
- $\paren {7 + i} - \paren {4 - 2 i} = \paren {7 + i} + \paren {-4 + 2 i}$
Let the complex numbers $7 + i$ and $-4 + 2 i$ 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 $3 + 3 i$, which is the sum of $7 + i$ and $-4 + 2 i$.
Hence, $3 + 3 i$ is the difference of $7 + i$ and $2 - 5 i$.
$\blacksquare$
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: $61 \ \text {(b)}$