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
- 1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 1$. Complex Numbers: $\text {(iv)}$ The definition of division
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: $(1.13)$
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Multiplication and Division: $3.7.13$
- 1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur: Quantum Mechanics ... (previous) ... (next): Chapter $2$: Mathematical Background: $2.1$ The Complex Field $C$