Vector Fields

Complex Field

A complex number is of the form \(a + bi \in \comp\) where \(a, b \in \real\) and \(i^{2} = -1\). Additon and multiplication is performed as

\[ \begin{aligned} (a + bi) + (c + di) &= (a + c) + (b + d)i\newline (a + bi) \times (c + di) &= a \times (c + di) + bi \times (c + di) = ac + adi + bci + bd \times (-1)\newline &= (ac - bd) + (ad + bc)i \end{aligned} \]

Complex arithmetic satisfies the following properties

• Commutativity

  \(\alpha + \beta = \beta + \alpha\) and \(\alpha \beta = \beta \alpha \quad \forall \; \alpha, \beta \in \comp\)

• Associavity

  \((\alpha + \beta) + \lambda = \alpha + (\beta + \lambda)\) and \((\alpha \beta)\lambda = \alpha(\beta \lambda) \quad \forall \alpha, \beta, \lambda \in \comp\)

• Identity Elements

  There exist identity elements \(0\) and \(1\) such that \(\alpha + 0 = \alpha\) and \(\alpha \times 1 = \alpha \quad \forall \alpha \in \comp\)

• Additive inverse

  For every \(\alpha \in \comp, \; \exists \beta \in \comp\) which is unique with \(\alpha + \beta = 0\)

• Multiplicative Inverse

  For every \(\alpha \in \comp, \; \exists \beta \in \comp\) which is unique with \(\alpha \beta = 1\)

• Distributive Property

  \(\lambda(\alpha + \beta) = \lambda \alpha + \lambda \beta \quad \forall \lambda, \alpha, \beta \in \comp\)

Subtraction and Division are also defined for complex numbers

\[ \begin{aligned} \alpha - \beta &= \alpha + (-\beta)\newline \alpha \div \beta &= \alpha * (\frac{1}{\beta}) \quad \text{where $1/\beta$ is the multiplicative inverse of $\beta$} \end{aligned} \]