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

Example of Complex Arithmetic

$3 \paren {\dfrac {1 + i} {1 - i} }^2 - 2 \paren {\dfrac {1 - i} {1 + i} }^3 = -3 - 2 i$

Proof

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

\(=\)

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

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds 3 i^2 - 2 \paren {-i}^3\)

\(\ds \)

\(=\)

\(\ds -3 - 2 \paren {-1}^3 i^3\)

\(\ds \)

\(=\)

\(\ds -3 - 2 \paren {-1} \times \paren {-i}\)

\(\ds \)

\(=\)

\(\ds -3 - 2 i\)

$\blacksquare$

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