Definition:Division/Field/Complex Numbers

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Definition

Let $\struct {\C, +, \times}$ be the field of complex numbers.


The operation of division is defined on $\C$ as:

$\forall a, b \in \C \setminus \set 0: \dfrac a b := a \times b^{-1}$

where $b^{-1}$ is the multiplicative inverse of $b$ in $\C$.


Notation

The operation of division can be denoted as:

$a / b$, which is probably the most common in the general informal context
$\dfrac a b$, which is the preferred style on $\mathsf{Pr} \infty \mathsf{fWiki}$
$a : b$, which is usually used when discussing ratios
$a \div b$, which is rarely seen outside grade school, but can be useful in contexts where it is important to be specific.


Specific Terminology

Divisor

Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field or a Euclidean domain.

The element $b$ is the divisor of $a$.


Dividend

Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field or a Euclidean domain.

The element $a$ is the dividend of $b$.


Quotient

Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field.

The element $c$ is the quotient of $a$ (divided) by $b$.


Examples

Example: $\dfrac {3 + 4 i} {1 + 2 i}$

$\dfrac {3 + 4 i} {1 + 2 i} = \dfrac {11} 5 - \dfrac 2 5 i$


Example: $\dfrac {1 - i} {1 + i}$

$\dfrac {1 - i} {1 + i} = -i$


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

$\dfrac {3 - 2 i} {-1 + i} = \dfrac {-5 - i} 2$


Example: $\dfrac {1 + \sin \theta + i \cos \theta} {1 + \sin \theta - i \cos \theta}$

$\dfrac {1 + \sin \theta + i \cos \theta} {1 + \sin \theta - i \cos \theta} = \sin \theta + i \cos \theta$


Also see

  • Results about complex division can be found here.


Sources

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