Complex Arithmetic/Examples/(i^4 + i^9 + i^16) (2 - i^5 + i^10 - i^15)^1
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Example of Complex Arithmetic
- $\dfrac {i^4 + i^9 + i^{16} } {2 - i^5 + i^{10} - i^{15} } = 2 + i$
Proof
| \(\ds \dfrac {i^4 + i^9 + i^{16} } {2 - i^5 + i^{10} - i^{15} }\) | \(=\) | \(\ds \dfrac {\paren {i^4} + \paren {i^4}^2 \times i + \paren {i^4}^4} {2 - \paren {i^4} \times i + \paren {i^4}^2 \times i^2 - \paren {i^4}^3 \times i^3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {1 + i + 1} {2 - i + i^2 - i^3}\) | $i^4 = 1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {2 + i} {2 - i - 1 + i}\) | $i^2 = -1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {2 + i} 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 + i\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $53 \ \text {(i)}$