Complex Algebra/Examples/(z-1) (z-i)^-1 = 2 over 3

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 $2 \ \text{(ii)}$