Complex Algebra/Examples/(2+i)z + i = 3

Example of Complex Algebra

Let $z \in \C$ be a complex number such that:

$\paren {2 + i} z + i = 3$

Then:

$z = 1 - i$

Proof

\(\ds \paren {2 + i} z + i\)

\(=\)

\(\ds 3\)

\(\ds \leadsto \ \ \)

\(\ds \paren {2 + i} z\)

\(=\)

\(\ds 3 - i\)

\(\ds \leadsto \ \ \)

\(\ds z\)

\(=\)

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

\(\ds \)

\(=\)

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

multiplying top and bottom by $2 - i$

\(\ds \)

\(=\)

\(\ds \dfrac {6 + i^2 - 3 i - 2 i} {2^2 + 1^2}\)

simplification

\(\ds \)

\(=\)

\(\ds \dfrac {5 - 5 i} 5\)

simplification

\(\ds \)

\(=\)

\(\ds 1 - i\)

simplification

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $2 \ \text{(i)}$