A rational number $d$ is called a decimal fraction if $10^kd$ is an integer for some non-negative integer $k$ . For example, any integer, as well as rationals such as $$0.23123,\qquad \frac{3}{4},\qquad \frac{236}{125}$$ are all decimal fractions. Rational numbers such as $$\frac{1}{3},\qquad -\frac{227}{12}, \qquad 2.\overline{312}$$ 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 $\displaystyle{\frac{p}{q}}$ , where $p$ and $q$ are integers, and $q=2^m5^n$ for some non-negative integers $m$ and $n$ ;
3. $d$ has a terminating decimal expansion, meaning that it has a decimal representation $$a.d_1d_2\cdots d_n000\cdots$$ 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 $d=\displaystyle{\frac{p}{2^m5^n}}$ , where $p$ and $2^m5^n$ 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 $a+b\in D(j)$ .