decimal fraction

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

$$0.23123,\frac{3}{4},\frac{236}{125}$$

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 ${\displaystyle \frac{p}{q}}$, where $p$ and $q$ are integers, and $q=2^m{5}^n$ for some non-negative integers $m$ and $n$;

3. $d$ 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 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 (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 $ℝ$, is dense in $ℝ$. 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

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

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