Complex Arithmetic/Examples/(4+2i)^-1 (2-3i)^-1

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 $1 \ \text{(iv)}$