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

Example of Complex Division

$\dfrac {2 - 3 i} {4 - i} = \dfrac {11} {17} - \dfrac {10} {17} i$

Proof

\(\ds \dfrac {2 - 3 i} {4 - i}\)

\(=\)

\(\ds \dfrac {\paren {2 - 3 i} \paren {4 + i} } {\paren {4 - i} \paren {4 + i} }\)

multiplying top and bottom by $4 + i$

\(\ds \)

\(=\)

\(\ds \dfrac {8 - 12 i + 2 i - 3 i^2} {4^2 + 1^2}\)

simplifying

\(\ds \)

\(=\)

\(\ds \dfrac {11 - 10 i} {17}\)

simplifying

$\blacksquare$

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: $53 \ \text {(e)}$