# Definition:Rational Number/Fraction

## 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 $1$.

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### Vulgar Fraction

A **vulgar fraction** is a rational number whose numerator and denominator are both integers.

### Proper Fraction

A **proper 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 proper 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 proper 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.

## Also known as

Some sources use the more unwieldy term **fractional number**.

## Historical Note

The consideration of **fraction** was the next development of the concept of a number after the natural numbers.

They arose as a matter of course from the need to understand the process of measurement.

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 $\ds {3 \atop 4}$.

## 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

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.In particular: Vulgar or proper?If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

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*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 - 1971: Wilfred Kaplan and Donald J. Lewis:
*Calculus and Linear Algebra*... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers - 1974: Murray R. Spiegel:
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*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: The Dark Ages? - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**fraction**