# Definition:Fraction

## Definition

A **fraction** is an expression representing a quotient of one number (or expression) by another number (or expression).

It is usually expressed in the form:

- $\dfrac a b$

or:

- $a / b$

where $a$ and $b$ are either numbers or expressions.

The term **fraction** is usually encountered when $a$ and $b$ are integers.

In this case, the **fraction** $\dfrac a b$ represents a rational number.

### Vulgar Fraction

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

### Proper Fraction

A **proper fraction** is a **fraction** representing 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 **fraction** representing 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 (the absolute value of) the numerator is greater than (the absolute value of) the denominator: $\size p > \size q$.

### Mixed Fraction

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

- $n$ is an
**integer** - $\dfrac p q$ is a
**proper fraction**, that is, $p$ and $q$ are integers such that $p < q$.

### Complex Fraction

A **complex fraction** is a **fraction** such that the numerator or denominator or both are themselves fractions.

## Also defined as

Many sources define a **fraction** solely in the context of numbers.

Specifically, a **fraction** in such a context will be used to denote a rational number $\dfrac a b$ such that both $a$ and $b$ are integers.

Such sources will typically also demand that the **fraction** specifically represent a non-integer.

Thus $\dfrac 3 1$ and $\dfrac 4 2$ will not be considered as actual **fractions**, as they represent the integers $3$ and $2$ respectively.

## Terms of Fraction

The **terms** of a **fraction** are referred to as the **numerator** and the **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.

## Examples

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

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

It can be expressed as a mixed fraction as follows:

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

- $(3): \quad \dfrac {24} {36}$ 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.

## Also known as

Some sources introduce the more unwieldy term **fractional number** for **fraction**.

This will be the case only when such a **fraction** denotes a **rational number**.

## Also see

- Results about
**fractions**can be found**here**.

## 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

- 1938: A. Geary, H.V. Lowry and H.A. Hayden:
*Mathematics for Technical Students, Part One*... (previous) ... (next): Arithmetic: Chapter $\text I$: Decimals - 1939: E.G. Phillips:
*A Course of Analysis*(2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Number: $1.1$ Introduction - 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 - 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:
*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: $3$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**fraction** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**fraction** - 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):**fraction**