Complex Addition/Examples/(-3 + 5i) + (4 + 2i) + (5 - 3i) + (-4 - 6i)
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Example of Complex Addition
- $\paren {-3 + 5 i} + \paren {4 + 2 i} + \paren {5 - 3 i} + \paren {-4 - 6 i} = 2 - 2 i$
Proof 1
\(\ds \) | \(\) | \(\ds \paren {-3 + 5 i} + \paren {4 + 2 i} + \paren {5 - 3 i} + \paren {-4 - 6 i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {-3} + 4 + 5 + \paren {-4} } + \paren {5 + 2 + \paren {-3} + \paren {-6} } i\) | Definition of Complex Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 - 2 i\) |
$\blacksquare$
Proof 2
Let:
\(\ds z_1\) | \(=\) | \(\ds -3 + 5 i\) | ||||||||||||
\(\ds z_2\) | \(=\) | \(\ds 4 + 2 i\) | ||||||||||||
\(\ds z_3\) | \(=\) | \(\ds 5 - 3 i\) | ||||||||||||
\(\ds z_4\) | \(=\) | \(\ds -4 - 6 i\) |
These can be depicted in the complex plane as follows:
To find the required sum, proceed as in the following diagram:
Construct $z_2$ with its initial point placed at the terminal point of $z_1$.
Construct $z_3$ with its initial point placed at the terminal point of this instance of $z_2$.
Construct $z_4$ with its initial point placed at the terminal point of this instance of $z_3$.
The required resultant $z_1 + z_2 + z_3 + z_4$ of $z_1$ to $z_4$ is therefore represented by the terminal point of $z_4$.
$\blacksquare$