Complex Arithmetic/Examples/3(1+i) + 2(4-3i) - (2+5i)/Proof 2
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Example of Complex Arithmetic
- $3 \paren {1 + i} + 2 \paren {4 - 3 i} - \paren {2 + 5 i} = 9 - 8 i$
Proof
We have:
\(\ds 3 \paren {1 + i}\) | \(=\) | \(\ds 3 + 3 i\) | ||||||||||||
\(\ds 2 \paren {4 - 3 i}\) | \(=\) | \(\ds 8 - 6 i\) | ||||||||||||
\(\ds -\paren {2 + 5 i}\) | \(=\) | \(\ds -2 - 5 i\) |
These can be depicted in the complex plane as follows:
To find the required sum, proceed as in the following diagram:
Construct $8 - 6 i$ with its initial point placed at the terminal point of $3 + 3 i$.
Construct $-2 - 5 i$ with its initial point placed at the terminal point of this instance of $8 - 6 i$.
The required resultant $3 \paren {1 + i} + 2 \paren {4 - 3 i} - \paren {2 + 5 i}$ is therefore represented by the terminal point of $-2 - 5 i$.
$\blacksquare$
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: $61 \ \text {(d)}$