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

## 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

$\blacksquare$