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
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1 \ \text{(m)}$