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 $1$.
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 "Numerator Over Denominator", 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
- 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: $3$
- 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