Complex Arithmetic/Examples/(z 3 - conj z 3)^5/Mistake
Jump to navigation
Jump to search
Source Work
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.)
- Chapter $1$: Complex Numbers
- Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(c)}$
Mistake
- If ... $z_3 = \sqrt 3 - 2 i$, evaluate ...:
- $\paren {z_3 - \overline {z_3} }^5$
- Ans. $1024 i$
Correction
This result actually works out to $-1024 i$:
\(\ds \paren {z_3 - \overline {z_3} }^5\) | \(=\) | \(\ds \paren {\paren {\sqrt 3 - 2 i} - \overline {\paren {\sqrt 3 - 2 i} } }^5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {\sqrt 3 - 2 i} - \paren {\sqrt 3 + 2 i} }^5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-4 i}^5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^5 \times 4^5 \times i^5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1} \times 1024 \times i\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -1024 i\) |
$\blacksquare$
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: $54 \ \text {(c)}$