Complex Algebra/Examples/(z-1) (z-i)^-1 = 2 over 3
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Example of Complex Algebra
Let $z \in \C$ be a complex number such that:
- $\dfrac {z - 1} {z - i} = \dfrac 2 3$
Then:
- $z = 3 - 2 i$
Proof
\(\ds \dfrac {z - 1} {z - i}\) | \(=\) | \(\ds \dfrac 2 3\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds z - 1\) | \(=\) | \(\ds \dfrac {2 \paren {z - i} } 3\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 z} 3 - \dfrac {2 i} 3\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds z - \dfrac {2 z} 3\) | \(=\) | \(\ds 1 - \dfrac 2 3 i\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3 z - 2 z\) | \(=\) | \(\ds 3 - 2 i\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds 3 - 2 i\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $2 \ \text{(ii)}$