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

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Example of Complex Arithmetic

$\dfrac {\paren {1 + 2 i}^2} {1 - i} = -\dfrac 7 2 + \dfrac 1 2 i$


Proof

\(\displaystyle \dfrac {\paren {1 + 2 i}^2} {1 - i}\) \(=\) \(\displaystyle \dfrac {1 + 4 i + 4 i^2} {1 - i}\) multiplying out numerator
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {1 + 4 i - 4} {1 - i}\) Definition of Imaginary Unit
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {-3 + 4 i} {1 - i}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\paren {-3 + 4 i} \left({1 + i}\right)} {\left({1 - i}\right) \left({1 + i}\right)}\) multiplying top and bottom by $1 + i$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {-3 - 3 i + 4 i + 4 i^2} {1^2 + 1^2}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {-7 + i} 2\) simplifying

$\blacksquare$


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