Inverse for Complex Multiplication/Examples/3+2i

Example of Inverse for Complex Multiplication

$\dfrac 1 {3 + 2 i} = \dfrac 3 {13} + \dfrac {2 i} {13}$

Proof

\(\ds \dfrac 1 {3 + 2 i}\)

\(=\)

\(\ds \dfrac {3 - 2 i} {\paren {3 + 2 i} \paren {3 - 2 i} }\)

multiplying top and bottom by $3 - 2 i$

\(\ds \)

\(=\)

\(\ds \dfrac {3 - 2 i} {3^2 + 2^2}\)

simplifying

\(\ds \)

\(=\)

\(\ds \dfrac {3 - 2 i} {13}\)

simplifying

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $1 \ \text{(ii)}$