Complex Algebra/Examples

Examples of Complex Algebra

Example: $\left({2 + i}\right) z + i = 3$

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

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

Then:

$z = 1 - i$

Example: $\dfrac {z - 1} {z - i} = \dfrac 2 3$

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

$\dfrac {z - 1} {z - i} = \dfrac 2 3$

Then:

$z = 3 - 2 i$

Example: $z^5 + 1$

$z^5 + 1 = \paren {z + 1} \paren {z^2 - 2 z \cos \dfrac \pi 5 + 1} \paren {z^2 - 2 z \cos \dfrac {3 \pi} 5 + 1}$

Example: $z^6 + z^3 + 1$

$z^6 + z^3 + 1 = \paren {z^2 - 2 z \cos \dfrac {2 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {4 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {8 \pi} 9 + 1}$

Example: $z^8 + 1$

$z^8 + 1 = \paren {z^2 - 2 z \cos \dfrac \pi 8 + 1} \paren {z^2 - 2 z \cos \dfrac {3 \pi} 8 + 1} \paren {z^2 - 2 z \cos \dfrac {5 \pi} 8 + 1} \paren {z^2 - 2 z \cos \dfrac {7 \pi} 8 + 1}$

Example: $\paren {1 + z}^5 = \paren {1 - z}^5$

The roots of the equation:

$\paren {1 + z}^5 = \paren {1 - z}^5$

are:

$z = \set {\dfrac {\omega^k - 1} {\omega^k + 1}: k = 0, 1, 2, 3, 4}$

That is:

$z = \set {0, \dfrac {\omega - 1} {\omega + 1}, \dfrac {\omega^2 - 1} {\omega^2 + 1}, \dfrac {\omega^3 - 1} {\omega^3 + 1} , \dfrac {\omega^4 - 1} {\omega^4 + 1} }$

where:

$\omega = \cis \dfrac {2 \pi i} 5$

Example: $\paren {z - 1}^6 + \paren {z + 1}^6 = 0$

The roots of the equation:

$\paren {z - 1}^6 + \paren {z + 1}^6 = 0$

are:

$\pm i \cot \dfrac \pi {12}, \pm i \cot \dfrac {5 \pi} {12}, \pm i$

Example: $z^4 - 3 z^2 + 1 = 0$

The roots of the equation:

$z^4 - 3z^2 + 1 = 0$

are:

$2 \cos 36 \degrees, 2 \cos 72 \degrees, 2 \cos 216 \degrees, 2 \cos 252 \degrees$

Example: $z^2 \paren {1 - z^2} = 16$

The roots of the equation:

$z^2 \paren {1 - z^2} = 16$

are:

$\pm \dfrac 3 2 \pm \dfrac {\sqrt 7} 2 i$

Example: $3 x + 2 i y - i x + 5 y = 7 + 5 i$

Let $3 x + 2 i y - i x + 5 y = 7 + 5 i$.

Then:

\(\ds x\)

\(=\)

\(\ds -1\)

\(\ds y\)

\(=\)

\(\ds 2\)

Example: $2 x - 3 i y + 4 i x - 2 y - 5 - 10 i = \paren {x + y + 2} - \paren {y - x + 3} i$

Let:

$2 x - 3 i y + 4 i x - 2 y - 5 - 10 i = \paren {x + y + 2} - \paren {y - x + 3} i$

Then:

\(\ds x\)

\(=\)

\(\ds 1\)

\(\ds y\)

\(=\)

\(\ds -2\)

Let $z_1$ and $z_2$ be arbitrary complex numbers represented in the complex plane as follows:

Example: $3 z_1 - 2 z_2$

The quantity $3 z_1 - 2 z_2$ can be depicted graphically as follows:

Example: $\dfrac 1 2 z_2 + \dfrac 5 3 z_1$

The quantity $\dfrac 1 2 z_2 + \dfrac 5 3 z_1$ can be depicted graphically as follows:

Let $z_1$, $z_2$ and $z_3$ be arbitrary complex numbers represented in the complex plane as follows:

Example: $2 z_1 + z_3$

The quantity $2 z_1 + z_3$ can be depicted graphically as follows:

Example: $\paren {z_1 + z_2} + z_3$

The quantity $\paren {z_1 + z_2} + z_3$ can be depicted graphically as follows:

Example: $z_1 + \paren {z_2 + z_3}$

The quantity $\paren {z_1 + z_2} + z_3$ can be depicted graphically as follows:

Example: $3 z_1 - 2 z_2 + 5 z_3$

The quantity $3 z_1 - 2 z_2 + 5 z_3$ can be depicted graphically as follows:

Example: $\dfrac 1 3 z_2 - \dfrac 3 4 z_1 + \dfrac 2 3 z_3$

The quantity $\dfrac 1 3 z_2 - \dfrac 3 4 z_1 + \dfrac 2 3 z_3$ can be depicted graphically as follows: