decimal fraction
A rational number^{} $d$ is called a decimal fraction if ${10}^{k}d$ is an integer for some nonnegative integer $k$. For example, any integer, as well as rationals such as
$$0.23123,\frac{3}{4},\frac{236}{125}$$ 
are all decimal fractions. Rational numbers such as
$$\frac{1}{3},\frac{227}{12},2.\overline{312}$$ 
are not.
There are two other ways of characterizing a decimal fraction: for a rational number $d$,

1.
$d$ is as in the above definition;

2.
$d$ can be written as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers, and $q={2}^{m}{5}^{n}$ for some nonnegative integers $m$ and $n$;

3.
$d$ has a terminating decimal expansion, meaning that it has a decimal representation
$$a.{d}_{1}{d}_{2}\mathrm{\cdots}{d}_{n}000\mathrm{\cdots}$$ where $a$ is a nonnegative integer and ${d}_{i}$ is any one of the digits $0,\mathrm{\dots},9$.
A decimal fraction is sometimes called a decimal number, although a decimal number in the most general sense may have nonterminating decimal expansions.
Remarks. Let $D\subset \mathbb{Q}$ be the set of all decimal fractions.

•
If $a,b\in D$, then $a\cdot b$ and $a+b\in D$ as well. Also, $a\in D$ whenever $a\in D$. In other words, $D$ is a subring of $\mathbb{Q}$. Furthermore, as an abelian group, $D$ is $2$divisible and $5$divisible. However, unlike $\mathbb{Q}$, $D$ is not divisible (http://planetmath.org/DivisibleGroup).

•
As inherited from $\mathbb{Q}$, $D$ has a total order^{} structure^{}. It is easy to see that $D$ is dense (http://planetmath.org/DenseTotalOrder): for any $a,b\in D$ with $$, there is $c\in D$ such that $$. Simply take $c={\displaystyle \frac{a+b}{2}}$.

•
From a topological point of view, $D$, as a subset of $\mathbb{R}$, is dense in $\mathbb{R}$. This is essentially the fact that every real number has a decimal expansion, so that every real number can be “approximated” by a decimal fraction to any degree of accuracy.

•
We can associate each decimal fraction $d$ with the least nonnegative integer $k(d)$ such that ${10}^{k(d)}d$ is an integer. This integer is uniquely determined by $d$. In fact, $k(d)$ is the last decimal place where its corresponding digit is nonzero in its decimal representation. For example, $k(1.41243)=5$ and $k(7/25)=2$. It is not hard to see that if we write $d={\displaystyle \frac{p}{{2}^{m}{5}^{n}}}$, where $p$ and ${2}^{m}{5}^{n}$ are coprime^{}, then $k(d)=\mathrm{max}(m,n)$.

•
For each nonnegative integer $i$, let $D(i)$ be the set of all $d\in D$ such that $k(d)=i$. Then $D$ can be partitioned into sets
$$D=D(0)\cup D(1)\cup \mathrm{\cdots}\cup D(n)\cup \mathrm{\cdots}.$$ Note that $D(0)=\mathbb{Z}$. Another basic property is that if $a\in D(i)$ and $b\in D(j)$ with $$, then $a+b\in D(j)$.
Title  decimal fraction 

Canonical name  DecimalFraction 
Date of creation  20130322 17:27:15 
Last modified on  20130322 17:27:15 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 1101 
Related topic  RationalNumber 
Defines  decimal number 