Complex Arithmetic/Examples/z 1^2 + 2 z 1 - 3

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Example of Complex Arithmetic

Let $z_1 = 1 - i$.

Then:

${z_1}^2 + 2 z_1 - 3 = -1 - 4 i$


Proof

\(\ds {z_1}^2 + 2 z_1 - 3\) \(=\) \(\ds \paren {1 - i}^2 + 2 \paren {1 - i} - 3\)
\(\ds \) \(=\) \(\ds \paren {1 - 2 i + i^2} + \paren {2 - 2 i} - 3\)
\(\ds \) \(=\) \(\ds -2 i + 2 - 2 i - 3\)
\(\ds \) \(=\) \(\ds -1 - 4 i\)

$\blacksquare$


Sources