# Definition:Rational Number/Fraction

## Contents

## Definition

By definition, a rational number is a number which can be expressed in the form:

- $\dfrac a b$

where $a$ and $b$ are integers.

A **fraction** is a rational number such that, when expressed in canonical form $\dfrac a b$ (that is, such that $a$ and $b$ are coprime), the denominator $b$ is not $1$.

### Vulgar Fraction

A **vulgar fraction** is a rational number whose absolute value is less than $1$, expressed in the form $r = \dfrac p q$, where $p$ and $q$ are integers.

### Improper Fraction

An **improper fraction** is a rational number whose absolute value is greater than $1$, specifically when expressed in the form $r = \dfrac p q$ where $p$ and $q$ are integers such that $p > q$.

### Mixed Number

A **mixed number** is a rational number whose absolute value is greater than $1$, expressed in the form $r = n \frac p q$ where:

- $p$ and $q$ are integers such that $p < q$
- $r = n + \dfrac p q$

## Examples

- $(1): \quad \dfrac 1 2$ is a vulgar fraction.

- $(2): \quad \dfrac 3 1$ is
*not*a**fraction**, as $b = 1$, and so $\dfrac 3 1 = 3$ which is an integer.

- $(3): \quad \dfrac 4 2$ is
*not*a**fraction**.

Although $b \ne 1$, $\dfrac 4 2$ is not in canonical form as $2$ divides $4$, meaning they have a common factor of $2$.

Furthermore, when $\dfrac 4 2$ expressed in canonical form is $\dfrac 2 1$ which, by example $(2)$, is an integer and so not a **fraction**.

- $(4): \quad \dfrac 5 2$ is an improper fraction.

It can be expressed as a mixed number as follows:

- $\dfrac 5 2 = \dfrac {4 + 1} 2 = \dfrac 4 2 + \dfrac 1 2 = 2 \frac 1 2$

- $(5): \quad \dfrac {24} {36}$ is a vulgar fraction, although not in canonical form.

It is found that when $\dfrac {24} {36}$ *is* expressed in canonical form:

- $\dfrac {24} {36} = \dfrac {12 \times 2} {12 \times 3} = \dfrac 2 3$

its denominator is not $1$.

Hence $\dfrac {24} {36}$ is indeed a vulgar fraction.

## Numerator and Denominator

### Numerator

The term $a$ is known as the **numerator** of $\dfrac a b$.

### Denominator

The term $b$ is known as the **denominator** of $\dfrac a b$.

A helpful mnemonic to remember which goes on top and which goes on the bottom is "**N**umerator **O**ver **D**enominator", which deserves a "nod" for being correct.

## Historical Note

The convention where a bar is placed between the numerator and denominator was introduced to the West by Fibonacci, following the work of the Arabic mathematicians.

Previous to this, **fractions** were written by the Hindu mathematicians without the bar.

Thus $\dfrac 3 4$ would have been written $\displaystyle {3 \atop 4}$.

## Linguistic Note

The word **fraction** derives from the Latin **fractus** meaning **broken**.

This is in antithesis to the concept of integer, which derives from the Latin for **untouched**, in the sense of **whole**, or **unbroken**.

Colloquially, informally and rhetorically, the word **fraction** is typically used to mean **a (small) part of a whole**, and *not* in the sense of improper fraction.

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.1$. Number Systems - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Integral Domains: $\S 1$. Introduction - 1974: Murray R. Spiegel:
*Theory and Problems of Advanced Calculus*(SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Real Numbers: $3$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.2$: The set of real numbers - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): Chapter $1$: Complex Numbers: The Real Number System: $3$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $3$: Notations and Numbers