Complex Arithmetic/Examples/(5 + 5i) (3 - 4i)^-1 + 20 (4 + 3i)^-1
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Example of Complex Arithmetic
- $\dfrac {5 + 5 i} {3 - 4 i} + \dfrac {20} {4 + 3 i} = 3 - i$
Proof
\(\ds \dfrac {5 + 5 i} {3 - 4 i}\) | \(=\) | \(\ds \dfrac {\paren {5 + 5 i} \paren {3 + 4 i} } {\paren {3 - 4 i} \paren {3 + 4 i} }\) | multiplying top and bottom by $3 + 4 i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {15 + 15 i + 20 i + 20 i^2} {3^2 + 4^2}\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {-5 + 35 i} {25}\) | simplifying |
Then:
\(\ds \dfrac {20} {4 + 3 i}\) | \(=\) | \(\ds \dfrac {20 \paren {4 - 3 i} } {\paren {4 + 3 i} \paren {4 - 3 i} }\) | multiplying top and bottom by $4 - 3 i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {80 - 60 i} {4^2 + 3^2}\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {80 - 60 i} {25}\) | simplifying |
Finally:
\(\ds \dfrac {-5 + 35 i} {25} + \dfrac {80 - 60 i} {25}\) | \(=\) | \(\ds \dfrac {\paren {-5 + 80} + \paren {35 - 60} i} {25}\) | Definition of Complex Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {75 - 25 i} {25}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 - i\) | 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: $1 \ \text{(l)}$