Definition:Division/Standard Number Field
This page is about Division over Standard Number Field. For other uses, see division.
Definition
The concept of division over a field is usually seen in the context of the standard number fields:
Rational Numbers
Let $\struct {\Q, +, \times}$ be the field of rational numbers.
The operation of division is defined on $\Q$ as:
- $\forall a, b \in \Q \setminus \set 0: a / b := a \times b^{-1}$
where $b^{-1}$ is the multiplicative inverse of $b$ in $\Q$.
Real Numbers
Let $\struct {\R, +, \times}$ be the field of real numbers.
The operation of division is defined on $\R$ as:
- $\forall a, b \in \R \setminus \set 0: a / b := a \times b^{-1}$
where $b^{-1}$ is the multiplicative inverse of $b$ in $\R$.
Complex Numbers
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 $b$.
Also see
- Definition:Integer Division for how the concept can be applied to integers
- Results about division over a standard number field can be found here.
Linguistic Note
The verb form of the word division is divide.
Thus to divide is to perform an act of division.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $1$. Scalar and Vector Quantities
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): division
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): division