Definition:Complex Algebra

Definition

Complex algebra is the branch of algebra which specifically involves complex arithmetic.

Thus it is the branch of mathematics which studies the techniques of manipulation of expressions in the domain of complex numbers.

Examples

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:

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

Linguistic Note

The word algebra originates from the Arabic word al-ğabr, meaning balancing, reduction or restoration.

It originates from the name of a book, circa 825 C.E., by Muhammad ibn Musa al-Khwarizmi:

Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala (The Compendious Book on Calculation by Completion and Balancing)

Also see

• Results about complex algebra can be found here.