A rational number $d$ is called a decimal fraction if $10^{k}d$ is an integer for some non-negative integer $k$. For example, any integer, as well as rationals such as

are all decimal fractions. Rational numbers such as

are not.

There are two other ways of characterizing a decimal fraction: for a rational number $d$,

1. 1.

   $d$ is as in the above definition;

2. 2.

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

3. 3.

   $d$ has a terminating decimal expansion, meaning that it has a decimal representation

   $$a.d_{1}d_{2}\cdots d_{n}000\cdots$$

   where $a$ is a nonnegative integer and $d_{i}$ is any one of the digits $0,\ldots,9$.

A decimal fraction is sometimes called a decimal number, although a decimal number in the most general sense may have non-terminating 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.

• As inherited from $\mathbb{Q}$, $D$ has a total order structure. It is easy to see that $D$ is dense: for any $a,b\in D$ with $a<b$, there is $c\in D$ such that $a<c<b$. 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 non-negative 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 non-zero 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 $k(d)=\max(m,n)$.

• For each non-negative 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

  $$D=D(0)\cup D(1)\cup\cdots\cup D(n)\cup\cdots.$$

  Note that $D(0)=\mathbb{Z}$. Another basic property is that if $a\in D(i)$ and $b\in D(j)$ with $i<j$, then $a+b\in D(j)$.