Complex Arithmetic/Examples/(4+2i)^-1 (2-3i)^-1
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Example of Complex Arithmetic
- $\dfrac 1 {\paren {4 + 2 i} \paren {2 - 3 i} } = \dfrac 7 {130} + \dfrac {2 i} {65}$
Proof
\(\ds \dfrac 1 {4 + 2 i} \times \dfrac 1 {2 - 3 i}\) | \(=\) | \(\ds \dfrac {4 - 2 i} {\paren {4 + 2 i} \paren {4 - 2 i} } \times \dfrac {2 + 3 i} {\paren {2 - 3 i} \paren {2 + 3 i} }\) | multiplying top and bottom by conjugate of bottom | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {4 - 2 i} {4^2 + 2^2} \times \dfrac {2 + 3 i} {2^2 + 3^2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {4 - 2 i} \paren {2 + 3 i} } {20 \times 13}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {4 \times 2 - \paren {-2} \times 3} + \paren {4 \times 3 + \paren {-2} \times 2} i} {20 \times 13}\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {14 + 8 i} {260}\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 7 {130} + \dfrac {2 i} {65}\) | simplification |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $1 \ \text{(iv)}$