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
\(\ds \dfrac {\paren {1 + 2 i}^2} {1 - i}\) | \(=\) | \(\ds \dfrac {1 + 4 i + 4 i^2} {1 - i}\) | multiplying out numerator | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1 + 4 i - 4} {1 - i}\) | Definition of Imaginary Unit | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {-3 + 4 i} {1 - i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {-3 + 4 i} \left({1 + i}\right)} {\left({1 - i}\right) \left({1 + i}\right)}\) | multiplying top and bottom by $1 + i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {-3 - 3 i + 4 i + 4 i^2} {1^2 + 1^2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {-7 + i} 2\) | simplifying |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: Example $1$.