Complex Arithmetic/Examples/(3i^30 - i^19) (2i - 1)^-1

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Example of Complex Arithmetic

$\dfrac {3 i^{30} - i^{19} } {2 i - 1} = 1 + i$


Proof

\(\ds \dfrac {3 i^{30} - i^{19} } {2 i - 1}\) \(=\) \(\ds \dfrac {3 \paren {-1}^{15} - \paren {-1}^9 i} {2 i - 1}\) as $i^2 = -1$
\(\ds \) \(=\) \(\ds \dfrac {-3 + i} {2 i - 1}\) as $\paren {-1}^n = -1$ for odd $n$
\(\ds \) \(=\) \(\ds \dfrac {\paren {-3 + i}\paren {-1 - 2 i} } {\paren {2 i - 1}\paren {-1 - 2 i} }\) multiplying top and bottom by $-1 - 2 i$
\(\ds \) \(=\) \(\ds \dfrac {3 - i + 6 i - 2 i^2} {2^2 + 1^2}\) Definition of Complex Multiplication
\(\ds \) \(=\) \(\ds \dfrac {5 + 5 i} 5\) simplifying
\(\ds \) \(=\) \(\ds 1 + i\) simplifying

$\blacksquare$


Sources